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Applications of zero-free regions on averages and shifted convolution sums of Hecke eigenvalues

Applications of zero-free regions on averages and shifted convolution sums of Hecke eigenvalues APPLICATIONS OF ZERO-FREE REGIONS ON AVERAGES AND SHIFTED CONVOLUTION SUMS OF HECKE EIGENVALUES JISEONG KIM Abstract. By assuming the Vinogradov-Korobov type zero-free regions and the gener- alized Ramanujan-Petersson conjecture, we give some nontrivial upper bounds of almost all short sums of Fourier coefficients of Hecke-Maass cusp forms for SL(n,Z). As an application, we obtain the nontrivial upper bounds for the averages of shifted sums for SL(n,Z) × SL(n,Z). We also give some results on sign changes by assuming some zero- free regions and the generalized Ramanujan-Petersson conjecture. Keywords. Automorphic form; Hecke eigenvalue; Shifted sum; Sign changes; Zero-free region 1. Introduction There are many interesting results in the behavior of multiplicative functions which are concerned about mean values in short intervals and their applications. For a recent account of the results, we refer the reader to [15], [4]. Hecke relations reveal that the normalized Fourier coefficients {A (m, 1,..., 1)} of Hecke-Maass forms for SL(n,Z) are multiplicative, and it is conjectured that these are divisor-bounded (generalized Ramanujan-Petersson conjecture. see subsection 3.2). In addition, there have been many investigations on an- alytic properties of automorphic L-functions which were useful to study the behavior of {A (m, 1,..., 1)}. Therefore, it is natural to try to apply the methods in multiplicative num- ber theory to automorphic L-functions. In this paper, we show that the simple methods in [13], [15] work on {A (m, 1,..., 1)} under the generalized Ramanujan-Petersson conjecture and SL(n,Z) Vinogradov-Korobov zero free regions. As applications, we get some non- trivial upper bounds of the shifted convolution sums and sign-changes of {A (m, 1,..., 1)}. Before we give some results on Hecke-Maass cusp forms for SL(n,Z), we look over the case of holomorphic Hecke cusp forms. 1.1. Holomorphic Hecke cusp forms. Let H = {z = x + iy ∶ x ∈ R,y ∈ (0,∞)},G = a b −1 SL(2,Z). Define j (z) = (cz + d) where γ = Œ ‘. When a holomorphic function c d f ∶ H → C satisfies f(γz) = j (z) f(z) for all γ ∈ SL(2,Z), it is called a modular form of weight k. It is well-known that f(z) has a Fourier expansion at the cusp ∞ (1.1) f(z) = b e(nz) n=0 2πiz where e(z) = e , and the normalized Fourier coefficients a(n) of f(z) are defined by k−1 (1.2) a(n) ∶= b n . The set of all modular forms of fixed weight k is a vector space, and it is denoted M , and arXiv:2110.06855v3 [math.NT] 18 May 2022 2 Jiseong Kim the set of all modular forms which have zero constant term is denoted S . Let T be the k n n-th Hecke operator on S such that 1 az + b (T f)(z) ∶= √ f( ). n Q Q n d ad=n b(mod d) It is known that there is an orthonormal basis of S which consists of eigenfunctions for all Hecke operators T , and these are called Hecke cusp forms. When f is a Hecke cusp form, the eigenvalues λ (n) of the n-th Hecke operator satisfy a(n) = a(1)λ (n), and λ (n) are multiplicative. For Hecke cusp forms f, it is known that +ǫ (1.3) λ (n) ≪ X f f,ǫ n=1 for sufficiently large X (see [17]). Therefore, when x ∈ [X, 2X], x+h x+h x λ (n) ≪ S λ (n)S + S λ (n)S Q Q Q f f f (1.4) n=x n=1 n=1 +ǫ ≪ x . f,ǫ for 1 ≤ h ≤ X. Also for any small fixed ǫ > 0 and h ≫ X , x+h x+h λ (n) ≪ Sλ (n)S Q Q f f n=x n=x (1.5) Sλ (p)S − 1 ≪ h (1 + ) p=1 p∈P by Shiu’s theorem (see Lemma 2.1), where P is the set of all primes. By the Sato-Tate law, Sλ (p)S − 1 8 −1 3π (1 + ) ≪ (log X) p=1 for sufficiently large X (see [11], [20]. − 1 ∼ −0.1511...). 3π In [9], Jutila proved that for any fixed ǫ > 0, +ǫ (1.6) λ (n)λ (n + l) ≪ X , f f f,ǫ n=1 where 1 ≤ l ≤ X . We now apply the above bounds to get the following theorem. Theorem 1.1. Let ǫ > 0 be small fixed, and let X be sufficiently large. Let 1 ≤ h ≪ X . Then x+h 1 1 − − +ǫ 2 6 U λ (n)U ≪ max Šh ,X � f f,ǫ n=x for almost all x ∈ [X, 2X]. Proof. For convenience we assume that X ∈ N. Consider 2X x+h 1 1 (1.7) U λ (n)U . Q Q X h n=x x=X By squaring out the inner sums, (1.7) is Zero-free regions and averages over short intervals 3 2X h x+h−l (2 − δ (l)) λ (n)λ (n + l) Q Q Q 0 f f Xh n=x x=X l=0 where δ (l) = 1 for l = 0, and 0 otherwise. The diagonal terms (l = 0) of (1.7) are 2X x+h (1.8) λ (n) . Q Q Xh n=x x=X x 2 1 By using the fact that ∑ λ (n) ∼ c X for some constant c , (1.8) is bounded by O ( ). f f f f n=1 The off-diagonal terms (1 ≤ l ≤ h) of (1.7) are h 2X x+h−l (1.9) λ (n)λ (n + l), Q Q Q f f Xh n=x l=1 x=X and by Deligne’s bound λ (n) ≪ n . Therefore, (1.9) is bounded by h 2X−h+l h 2 ǫ Š (h − l) λ (n)λ (n + l) + (h − l) X �. Q Q Q f f Xh l=1 x=X+h−l l=1 By (1.6), h 2X−h+l 2 +ǫ (1.10) (h − l) λ (n)λ (n + l) ≪ h X . Q Q f f f,ǫ l=1 x=X+h−l Therefore, (1.9) is bounded by − +ǫ ǫ−1 (1.11) O ŠX + hX �. f,ǫ By the assumption on the range of h, (1.8) is bounded by −1 − +ǫ O max Šh ,X �. f,ǫ By the standard application of the Chebyshev inequality, the proof is complete. Note that the upper bound in Theorem 1.1 is better than the trivial bound from (1.5) when h is bigger than some power of log X, and it is better than the trivial bound from +ǫ (1.4) when h ≪ X . Note that the upper bound of shifted sums was crucial for the proof of Theorem 1.1. 1.2. Zero-free regions and shifted convolution sum. To generalize Theorem 1.1 for Fourier coefficients A (m, 1,..., 1) of Hecke-Maass cusp forms F for SL(n,Z), it suffices to have some upper bounds of shifted convolution sums A (m, 1,..., 1)A (m + l, 1,..., 1) F F m=1 for some range of l, where A (m + l, 1,..., 1) is the complex conjugate of A (m+l, 1,..., 1). F F Getting nontrivial upper bounds or asymptotics of the above shifted sum has many appli- cations. But as far as the author is aware, it does not yet exist in the literature for the cases of n ≥ 3. As an alternative, let us consider the successful approaches in multiplicative number theory. Let s = σ + it where σ,t ∈ R. Define A (m, 1,..., 1) L (s) ∶= . m=1 The generalized Riemann hypothesis claims that there are no nontrivial zeros of L (s) for σ > . The present state of knowledge of the zero-free region is, there are no nontrivial zeros of L (s) where σ > 1 − log St + 3S 4 Jiseong Kim for some constant c > 0 (this c being dependent on the conductor of F ). But for the F F Riemann zeta function ζ(s), it is known that there are no nontrivial zeros where σ > 1 − (log St + 3S) log log St + 3S and StS ≫ 1 for some constant c > 0. This zero-free region (the power of the logarithm in the denominator is ) is called the Vinogradov-Korobov zero-free region, and this gives that for any A > 0, −A A+2 ≪ (log X) when t > (log X) 1+it p=P +ǫ (log X) 3 for e ≪ P ≤ Q ≪ X, any small fixed ǫ > 0. By applying the above inequality, the authors of [13] show that the nontrivial results of the Liouville function can be obtained by the method which is relatively simpler than the methods in [15] (note that the methods in [15] are sophisticate, general, and give better results when one does not have some wide zero-free regions). This motivates us to prove the analog of the above upper bound by assuming the Ramanujan-Petersson conjecture (see (3.8)) and the existence of c such that L (s) has the SL(n,Z) Vinogradov-Korobov zero-free region (1.12) σ > 1 − . (log St + 3S) log log St + 3S Lemma 1.2. Let ǫ > 0 be small fixed, and let A be sufficiently large. Assume the zero-free 2ǫ (log X) region (1.12) and the generalized Ramanujan-Petersson conjecture. Then for e ≪ t ≪ X, A (p, 1,..., 1) −A (1.13) Q ≪ (log X) 1+it p=P p∈P +ǫ 1−ǫ (log X) (log X) for all e < P < Q < e . 1 1 Proof. The proof of this basically follows from the proof of [13, Lemma 2]. Let κ = ,T = log X StS+1 . By Perron’s Formula, Q s s κ+iT A (p, 1,..., 1) 1 Q − P 1 1 = logL (s + 1 + it) ds + O(R ) S F F 1+it p 2πi κ−iT s p=P (1.14) p∈P −1 +O(P ) where 4Q 1 4P SA (m, 1,..., 1)S Q SA (m, 1,..., 1)S P F 1 F 1 κ κ R = max Š ( ) , ( ) � Q Q Q P 1 1 m m m max(1,T S log S) m max(1,T S log S) m=1 m=1 −1 (see [19, Corollary 5.3]). Note that O(P ) comes from SA (p , 1,..., 1)S k(1+it) p , k≥2,p∈[P ,Q ]∩P 1 1 by applying Lemma 3.1 (A (p , 1,..., 1) ≪ O(1)). By applying the upper bound (3.14) F F ǫ c 1 F (A (m, 1,..., 1) ≪ m ), it is easy to show that R ≪ . Let σ = . F ǫ F F.ǫ 0 2 1 (log X) 3 (log log X) 3 Zero-free regions and averages over short intervals 5 By shifting the line of integral in (1.14) to [−σ − iT,−σ + iT ] and changing variables, 0 0 Q −σ +iu −σ +iu 1 0 0 +iT Q − P A (p, 1,..., 1) 1 1 1 = log L (1 − σ + it + iu) du F 0 1+it p 2πi −iT −σ + iu p=P , 0 (1.15) p∈P 1 −1 + O ( ) + O (P ). F,ǫ F Note that a standard application of the Borel-Caratheodory theorem gives (see [2, Section 4]), (1.16) log L (σ + it + iu) ≪ S log(t + u + 2)S F 1 F for σ ∈ [1 − σ , 1 + κ]. Therefore, by applying (1.16) for bounding the integral in (1.15), 1 0 −A (1.13) is bounded by (log X) . In [7], Jesse showed that under the assumption of the generalized Lindelöf hypothesis and a weak version of the Ramanujan-Petersson conjecture, when F is a nontrivial Hecke-Maass cusp form for SL(n,Z) and A (m, 1,..., 1) are normalized Hecke eigenvalues of F, x+h 2X 1− U A (m, 1,..., 1)U dx ∼ B X F F X X m=x 1− +ǫ 1−ǫ for some B , where h ∈ [X ,X ] for some small fixed ǫ > 0. In the following subsec- 1−ǫ (log X) 1−ǫ tion, we consider the cases h ∈ [e ,X ]. 1.3. Main results. The mean-square version of Theorem 3.10. Let ǫ > 0 be small fixed, and let X 1−ǫ (log X) 1−ǫ be sufficiently large. Let e ≪ h ≪ X . Assume the generalized Ramanujan- ǫ ǫ Petersson conjecture on F and the zero-free region (1.12). Then x+h 2X 2 2 − +ǫ U A (m, 1,..., 1)U dx ≪ h (log X) . S F F,ǫ X X m=x This theorem implies the following corollaries. 1−ǫ (log X) Corollary 1.3. Let ǫ > 0 be small fixed, and let X be sufficiently large. Let e ≪ 1−ǫ h ≪ X . Assume the generalized Ramanujan-Petersson conjecture on F and the zero-free region (1.12). Then h 2X 1 2 − +ǫ Sh − lSŠ A (m, 1,..., 1)A (m + l, 1,..., 1)� ≪ (log X) . Q Q F F F,ǫ Xh l=−h m=X l≠0 By using the Rankin-Selberg bound, one can remove the weight Sh−lS at the price of losing some cancellations. The SL(n,Z) Sato-Tate conjecture with the generalized Ramanujan- Petersson conjecture implies that for sufficiently large X, SA (p, 1,..., 1)S − 1 κ −1 (1.17) (1 + ) ≍ (log X) p≤X where iθ iθ iθ iθ iθ 2 1 2 n j κ ∶= Se + e + ... + e S Se − e S dθ dθ ...dθ n M 1 2 n−1 S n−1 n−1 [0,2π) n!(2π) θ +θ +...+θ =0 1≤j<k≤n 1 2 (see [8, (1.16)]). For example, κ ≍ 0.8911,κ ≍ 0.8853, and κ ≍ 0.8863. Therefore, by a 3 4 5 variant of the Shiu’s theorem (2.3), we have h 2X 2κ −2 (1.18) A (m, 1,...., 1)A (m + h, 1,..., 1) ≪ X(log X) Q Q F F l=1 m=X 6 Jiseong Kim under the generalized Ramanujan-Petersson conjecture and (1.17). Assuming the Vinogradov- Korobov zero-free region instead of the Sato-Tate conjecture, we prove the following upper bound which is better than (1.18). 1−ǫ (log X) Corollary 1.4. Let ǫ > 0 be small fixed and let X be sufficiently large. Let e ≪ 1−ǫ h ≪ X . Assume the generalized Ramanujan-Petersson conjecture on F and the zero-free region (1.12). Then h 2X − +ǫ (1.19) Š A (m, 1,..., 1)A (m + l, 1,..., 1)� ≪ (log X) . Q Q F F F,ǫ X(2h − 1) l=−h m=X l≠0 Proof. The left-hand side of (1.19) can be represented as x+h 2X 1 1 ‰ Ž (1.20) Q A (m, 1,...1) A ([x], 1,..., 1)dx. F F X X 2h − 1 m=[x−h] m≠x By the Cauchy-Schwarz inequality and Theorem 3.10, x+h 2X ‰ Ž Q A (m, 1,...1) A ([x], 1,..., 1)dx F F X 2h − 1 m=[x−h] m≠x x+h 2 2X (1.21) 1 2 SA ([x], 1,..., 1)S F 2 ≪ Š T A (m, 1,...1)T + dx� X F S F X 2h − 1 (2h − 1) m=[x−h] − +ǫ ≪ X(log X) . F,ǫ Remark 1.5. If the zero-free region (1.22) σ > 1 − (log StS + 3) holds for some constants c > 0, θ < 1, then it is easy to show that we can replace − +ǫ θ−1+ǫ log P (log X) in Theorem 3.10 with (log X) (the cancellation comes from the log Q +ǫ (log X) truncation in Lemma 3.4. Therefore, we only need to replace P = e with P = θ+ǫ (log X) e in the beginning of Section 2). Since the Vinogradov-Korobov zero-free region is proved for the Riemann zeta function, one could get some similar cancellations on the symmetric square L-functions. As an application of (1.22), let’s talk about the sign-changes of {A (m, 1,..., 1)} . By m=1 assuming the Ramanujan-Petersson conjecture, it is proved that −1 (1.23) SA (m, 1,...1)S ≫ x(log x) . Q F m≤x Essentially, this lower bound comes from Q SA (p, 1,..., 1)S log p ≥ Q SA (p, 1,..., 1)S log p F F p≤x p≤x (1.24) = ‰1 + o(1)Žx (see [8, (1.15)]). When F is self-dual, A (m, 1,..., 1) ∈ R for all m ∈ N. Hence, for the self-dual Hecke-Maass cusp forms F, if x+h x+h A (m, 1,..., 1) = o‰ SA (m, 1,..., 1)SŽ, Q Q F F m=x m=x Zero-free regions and averages over short intervals 7 x+h then there’s at least one sign changes among {A (m, 1,..., 1)} . Therefore, we will prove m=x the following corollary in Section 3. Corollary 1.6. Let ǫ > 0 be sufficiently small and let X be sufficiently large. Let F be a self-dual Hecke-Maass cusp form for SL(n,Z). Assume the generalized Ramanujan- Petersson conjecture on F, the zero-free region (1.22) for some θ < . Then the number of X X sign changes in {A (m, 1,..., 1)} is ≫ . F F,ǫ 1−ǫ m=1 (log X) Remark 1.7. We use the generalized Ramanujan-Petersson conjecture to give the upper bounds of second moments, shifted convolution sums of Hecke eigenvalue squares over some subsets in long intervals. Remark 1.8. One may apply some methods in [4], [12] to get some results similar to the results in this paper, without assuming the SL(n,Z) Vinogradov-Korobov zero-free regions. The results from the above methods may hold for a wider range of h, but cancellations are smaller than the results in this paper. Specifically, under the generalized Ramanujan- Petersson conjecture, the best possible upper bound for Theorem 3.10 from [12, Theorem 1.7] is (0.1211...) 32+ (log X) . Note that the above bound is not sufficiently small to get a bound better than (1.18). Remark 1.9. When χ is a primitive, non-principal Dirichlet character modulo q, the arguments in the proof of Theorem 1.1 gives x+h 2 1 X Xd(q) h U Q χ(n)U ≪ max ‰ , , Ž h h X n=x q for almost all x ∈ [X, 2X], where 1 ≤ h ≤ X and d(q) is the number of divisors of q. In order to treat the off-diagonal terms, one may use Weil’s bound for Kloosterman sums. 2. Lemmas From now on, ǫ be an arbitrary but small fixed positive number, and the value of ǫ may +ǫ 1−ǫ (log X) (log X) differ from one occurrence to another. Let P = e and Q = e . Let S X 2X be the set of integers in [ , ] having at least one prime factor in [P,Q]. We denote d d ′ X 2X ′ ′ S ∶= {n ∈ [ , ] ∶ n ∉ S }. For convenience we denote S instead of S , S instead of S . d 1 d 1 d d Let P denote the set of all prime numbers. For simplicity of notation, we write Ma , Qa instead of M a , Q a p p p p p p p∈P p∈P for any a , respectively. For abbreviation, let GVK stand for the SL(n,Z) zero-free region (1.12) and let GRP stand for the generalized Ramanujan-Petersson conjecture. We apply the following lemma for the average of multiplicative functions over intervals. Lemma 2.1. (Variants of Shiu’s theorem) Let α,β ≥ 1, ǫ > 0. Let g be a non-negative multiplicative function (arithmetic function) such that v v g(p ) ≤ α for all p ∈ P, v ∈ N, (2.1) 3 g(n) ≤ βn for all n ∈ N. Then for 2 ≤ X ≤ Y ≤ X, 1 ≤ H ≤ X, X+Y X 1 g(p) − 1 (2.2) g(n) ≪ ‰1 + Ž, Q M α,β,ǫ Y p n=X p=1 8 Jiseong Kim H X+Y X g(p) − 1 2 (2.3) g(n)g(n + l) ≪ HY ‰1 + Ž Q Q M α,β,ǫ p=1 l=1 n=X Proof. See [16, Lemma 2.3]. X 2X By Lemma 2.1, it is easy to check that typical numbers in [ , ] are contained in S . By d d (2.2), X logP U{n ∈ [X, 2X] ∶ n ∉ S }U ≪ d log Q (2.4) X 1 − +ǫ ≪ (log X) . Lemma 2.2. Let h ≥ 1. For sufficiently large X, h 2X log P (2.5) 1 ′(n)1 ′(n + l) ≪ hX( ) Q Q S S ǫ log Q n=X l=1 Proof. This comes from a direct application of (2.3) by substituting 1 (n) into g(n). The following lemma will be needed after Lemma 3.6. Lemma 2.3. (Dirichlet mean value theorem) For any {a } ⊂ C, n n∈N 2X 2X T 2 (2.6) U Q U dt ≪ (T + X) Q Sa S . it 0 n n=X n=X Proof. See [18, Theorem 9.2]. 3. Proof of Theorem 3.10, and corollaries n−1 3.1. Hecke-Maass cusp forms for SL(n,Z). Let n ≥ 2, let v = (v ,v ,...,v ) ∈ C 1 2 n−1 n−1 and m = (m ,m ,...,m ,m ) ∈ Z . Let U (Z) be the set of all upper triangular 1 2 n−2 n−1 n−1 matrices of the form ⎡ ⎤ ⎢ 1 ⎥ ⎢ ⎥ ⎢ I ∗ ⎥ (3.1) U (Z) = ⎢ ⎥ n−1 ⎢ ⋱ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ b⎦ with ∑ r = n − 1, I denotes the r × r identity matrix, ∗ denotes arbitrary integer i r i i i=1 i entries (Similarly, U (R) with ∗ ∈ R). Let F(z) be a Hecke-Maass form for SL(n,Z) such that ∞ ∞ A (m ,m ,...,m ) F 1 2 n−1 F(z) = ⋯ Q Q Q Q k(n−k) n−1 m =1 m =1 m ≠0 γ∈U (Z)ÓSL(n−1,Z) 1 n−2 n−1 2 n−1 Sm S (3.2) ∏ k=1 × W (Mγ z;v,ψ n−1 ) Jacquet 1,1,...,1, Sm S n−1 where ⎡ ⎤ m m ⋯m Sm S 1 2 n−2 n−1 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ (3.3) M = ⎢ m m ⎥, 1 2 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ (3.4) γ =  , 1 Zero-free regions and averages over short intervals 9 and W is the Jacquet-Whittaker function of type v associated to a character ψ Jacquet m such that (3.5) W (z;v,ψ ) = I (w ⋅ u ⋅ z)ψ (u)d u. Jacquet m S v n m U (R) where n−1 n−1 b v i,j j I (z) = y , M M i=1 j=1 2πi(m u +m u +...+m u ) 1 1,2 2 2,3 n−1 n−1,n ψ denotes the character of U (R) such that ψ (u) ∶= e m n m where ⎡ ⎤ 1 u u ⋯ u 1,2 1,3 1,n ⎢ ⎥ ⎢ ⎥ 1 u ⋯ u ⎢ 2.3 2,n ⎥ ⎢ ⎥ (3.6) u = ⎢ ⋱ ⋮ ⎥, ⎢ ⎥ ⎢ ⎥ 1 u n−1,n ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ and [ ] ⎡ ⎤ (−1) ⎢ ⎥ ⎢ ⎥ ⎢ 1 ⎥ w = ⎢ ⎥. ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ For more details, we refer the reader to [3, Chapter 5, Chapter 9]. 3.2. Generalized Ramanujan-Petersson conjecture (GRP). In order to apply Lemma 2.1, it is necessary to assume the GRP. L (s) has an Euler product −s −2s L (s) = Š1 − A (p,..., 1)p + A (1,p,..., 1)p F M F F p∈P −1 n−1 −(n−1)s n −ns − ⋯ + (−1) A (1,..., 1,p)p + (−1) p � (3.7) −1 −s = Š1 − a (p)p � M M F,j p∈P j=1 for some a (p) ∈ C. GRP claims F,j (3.8) Sa (p)S = 1. F,j Therefore, we get the following lemma. Lemma 3.1. Assume GRP. Then for any k ∈ N and p ∈ P, A (p , 1, 1..., 1) = O (1). F n Proof. It is known that (see [21], [1]) for any k ∈ N and p ∈ P, (3.9) A (p , 1,..., 1) = S (a (p),a (p),...,a (p)) F 0,0,...,0,k F,1 F,2 F,n where S (x ,x ,...,x ) is the Schur polynomial (see [7]). By applying the second λ ,λ ,...,λ 1 2 n 1 2 n Jacobi-Trudi formula, (3.10) S = det[e t ] 0,0,...,0,k λ −i+j where (3.11) λ = S{j ∶ λ ≥ i}S, e are the symmetric polynomials of i-variables. Note that λ − i + j ≤ 2n. By (3.8) and (3.11), (3.12) e t ≤ (2n) . λ −i+j i 10 Jiseong Kim Therefore, by (3.10) and (3.12), (3.13) S (a (p),...,a (p)) ≤ n!(2n) . 0,0,...,k F,1 F,n (3.9) and (3.13) give the upper bound of this lemma. In order to apply Lemma 2.1, we also need to show that for arbitrarily small fixed ǫ > 0, (3.14) A (m, 1,..., 1) ≪ m . F ǫ By the Euler product (3.7) and GRP, A (m, 1,..., 1) ≪ d (m) F n −s n where d (m) is the n-th divisor function (the coefficients of m of ζ(s) ). In addition, by logm (log 2+o(1)) log log m d (m) ≤ d (m) and d (m) ≤ e , n 2 2 we get (3.14). By Lemma 2.1 with the bound A (p, 1,..., 1) ≪ n, 4 n −1 (3.15) SA (m, 1,..., 1)S ≪ X(log X) . m=1 3.3. Main proposition. This subsection is devoted to prove the following proposition. 1−ǫ (log X) Proposition 3.2. Assume GRP and GVK. Let X be sufficiently large. Let e ≪ 1−ǫ h ≤ h ≪ X . Then 1 2 ǫ x+h x+h 2X 1 2 2 1 1 1 − +ǫ (3.16) U A (m, 1,..., 1) − A (m, 1,..., 1)U dx ≪ (log X) . Q Q S F F F,ǫ X X h h 1 2 m=x m=x In order to prove Proposition 3.2, we need the following lemmas. Lemma 3.3. For sufficiently large X, X 2 X SA (p, 1,..., 1)S 1 (3.17) = + O (1). Q Q F p p p=1 p=1 Proof. The proof of this basically follows from [6, Theorem 5.13]. Let F be the dual of F. The Rankin-Selberg L-function ∞ 2 SA (n, 1,..., 1)S (3.18) L (s) = ˜ Q F ×F n=1 is meromorphic with a simple pole at s = 1. And for some constant d , (3.19) L (σ + it) ≠ 0 F ×F for σ > 1 − with StS ≫ 1 (see [5]). Therefore, by [6, Theorem 5.13], (log StS+3) SA (p, 1,..., 1)S Λ(p) = x + o (x) F F p=1 where Λ is the von-Mangoldt function. Therefore, partial summation over p gives (3.17). n −1 Lemma 3.4. Assume GRP and GVK. Let (log X) ≪ h = o(X). Then x+h 2X − +ǫ (3.20) U A (m, 1,..., 1)1 ′ (m)U dx ≪ X(log X) . F S F,ǫ X h m=x Zero-free regions and averages over short intervals 11 Proof. By the Cauchy-Schwarz inequality, x+h 2X A (m, 1,...1)1 (m) 2 ∑ F S m=x T T dx X h x+h x+h 2X ≪ T 1 ′(m)TT SA (m, 1,...1)1 ′ (m)S Tdx Q Q (3.21) S S F S h X m=x m=x 1 1 x+h x+h 2X 2X 2 2 2 2 ′ ′ ≪ Š ‰ Q 1 (m)Ž dx� Š ‰ Q SA (m, 1,...1)1 (m)S Ž dx� . S F S S S h X X m=x m=x By squaring out the inner sum over m and Lemma 2.2, x+h h 2X 2X+h 2X ′ ′ ′ ′ ‰ 1 (m)Ž dx ≪ hŠ 1 (m)1 (m + l) + 1 (m)� Q S Q Q S S Q S m=x l=1 m=X m=X (3.22) log P log P 2 2 ≪ h X( ) + hX log Q log Q 2 − +2ǫ ≪ h X(log X) . In addition, by squaring out the last integrand in (3.21), x+h 2X+h 2X 2 4 ‰ SA (m, 1,...1)1 ′ (m)S Ž dx ≪ h SA (m, 1,...1)1 ′ (m)S Q Q F S F S m=x m=X (3.23) h 2X 2 2 ′ ′ + hQ Q SA (m, 1,...1)1 (m)S SA (m + l, 1,..., 1)1 (m + l)S . F S F S m=X l=1 By Lemma 2.1 and 3.3, we get h 2X 2 2 2 − +2ǫ h SA (m, 1,...1)1 ′ (m)S SA (m+l, 1,..., 1)1 ′ (m+l)S ≪ h (2X +h)(log X) . Q Q F S F S F,ǫ l=1 m=X n −1 Hence, by the assumption (log X) ≪ h = o(X) and (3.15), x+h 2X 2 2 − +ǫ T A (m, 1,...1) 1 ′(m)T dx ≪ h X(log X) . F S F,ǫ m=x Therefore, we get (3.20). Remark 3.5. In order to get some results similar to (3.20), one may avoid assuming GRP by some large sieve techniques if h is bigger than X . But the upper bounds will be weaker than (3.20) (see [6, Theorem 7.14]). 1−ǫ (log X) Lemma 3.6. (Parseval’s bound) Let X and A be sufficiently large. Let e ≪ h ≤ ǫ 1 1−ǫ h = O (X ). Then 2 ǫ x+h x+h 1 2 2X 2 1 1 1 (3.24) U A (m, 1,..., 1)1 (m) − A (m, 1,..., 1)1 (m)U dx Q Q S F S F S X X h h 1 2 m=x m=x −1 2X Xh 2 i A (m, 1,..., 1)1 (m) F S ≪ maxŠ U Q U dt+ 2ǫ 1+it (logX) i=1.2 e m m=X −1 2X 2T 2 Xh A (m, 1,..., 1)1 (m) F S max U U dt� 1+it −1 T >Xh T T m i m=X −A + (log X) . Proof. The proof of this basically follows from [15, Lemma 14]. The 1-bounded condition in [15, Lemma 14] is easily removed (see [10, Lemma 2.1]).  12 Jiseong Kim Let K be the set of integers m ∈ S such that p ∤ m for all p ∈ [P,Q] ∩ P, and let K = {m ∈ S ∶ m ∉ K}. Then (3.25) 2X 2X 2X A (m, 1,..., 1)1 (m) A (m, 1,..., 1)1 (m) A (m, 1,..., 1)1 (m) F S F S F S = + . Q Q Q 1+it 1+it 1+it m m m m=X, m=X, m=X m∈K m∈K The first summand of the right hand side of (3.25) is represented as 2X Q p A (p, 1,..., 1) A (m , 1,..., 1) 1 F F 1 (3.26) Q Q 1+it 1+it p m S{q ∈ [P,Q] ∩ P ∶ qSm }S + 1 X 1 (p,m )=1 p=P 1 1 m = , pm ∈K (see [6, section 17.3], or [15, (9)]). Lemma 3.7. Assume GRP. Let X,T be sufficiently large. Then n −1 2X T 2 A (m, 1,..., 1)1 (m) T (log X) F S U U dt ≪ ( + 1) . S 1 1+it 0 m X m=X, 2 m∈K Proof. By Lemma 2.3 and the Cauchy-Schwarz inequality, (3.27) 2X A (m, 1,..., 1)1 (m) (T + O(X)) F S U U dt ≪ SA (m, 1,..., 1)S 1 (m) Q Q F S 1+it 2 0 m X m=X m∈K m∈K 1 1 2X (T + O(X)) 2 2 ≪ Š SA (m, 1,..., 1)S � Š 1 (m)� Q Q F S m=X m∈K By the definition of K , 2X 1 (m) ≪ 1 Q Q Q m∈K p=P m= , pSm (3.28) Q ≪ X p=P ≪ . Therefore, by combining (3.15) and (3.28), the last term in (3.27) is bounded by n −1 T (log X) ≪ ( + 1) . 1−ǫ Lemma 3.8. Assume GRP. Let X be sufficiently large. Let 2 ≤ H ≪ X . Let v ∈ I ∶= [H log P − 1,H log Q]. Let v+1 A (p, 1,..., 1) N (s) = Q v,H p=e (3.29) 2Xe A (m , 1,..., 1) 1 F 1 R (s) = . v,H v m S{q ∈ [P,Q] ∩ P ∶ qSm }S + 1 m =Xe 1 Zero-free regions and averages over short intervals 13 Then (3.30) 2X T 2 A (m, 1,..., 1)1 (m) T + X 2 F S n −1 U U dt ≪ (log X) Q F,ǫ 2ǫ 1+it (log X) e m HX m=X T + X Q n −1 2 2 + (log X) + H log SN (1 + it)S SR (1 + it)S dt. j,H j,H 1 S 2ǫ (log X) P e XP j∈I Proof. The proof of this basically follows from [14, Lemma 12] . By (3.25), (3.26), 2X A (m, 1,..., 1)1 (m) b(m) F S Q = QN (1 + it)R (1 + it) + Q j,H j.H 1+it 1+it m m m=X j∈I 2X≤m≤2Xe (3.31) 2X 2X A (m, 1,..., 1)1 (m) c(m) F S + + Q Q Q 1+it 1+it m m m=X, m=X, p=P m∈K p Sm for some b(m) ≪ SA (m, 1,..., 1)S, c(m) ≪ SA (p, 1,.., 1)A (m , 1,.., 1)S for some p ∈ F F F 1 [P,Q],pm = m. These b(n), c(n) come from the difference 2X A (m, 1,...1)1 (m) F S − N (1 + it)R (1 + it). Q Q j,H j.H 1+it m=X, j∈I m∈K By Lemma 2.1, 1 1 H H 2Xe 2Xe A (m, 1,..., 1) e 2 2 S S ≪ SA (m, 1,..., 1)S Q Q F,ǫ F m X m=2X m=2X e X 2 n −1 ≪ (log X) . F,ǫ X H Therefore, 1 1 H H 2Xe 2Xe 2 2 b(m) b(m) U U dt ≪ (T + X) U U Q Q 2ǫ 1+it (logX) e m m m=2X m=2X 2Xe (3.32) A (m, 1,..., 1) ≪ (T + X) U U m=2X T + X n −1 ≪ (log X) . F,ǫ HX By (3.15) and Lemma 2.3, 2X 2X c(m) T + X U U dt ≪ Š Tc(m)T � Q Q Q 2ǫ 1+it 2 (log X) e m X m=X, m=1, p=P 2 2 p Sm p Sm for some p∈[P,Q] 2X T + X X 1 4 2 (3.33) T T ≪ ( ) Š Q A (m, 1,..., 1) � F F X P m=1 n −1 T + X X(log X) ≪ . F,ǫ P 14 Jiseong Kim Finally, by the Cauchy-Schwarz inequality, U N (1 + it)R (1 + it)U dt j,H j.H 2ǫ (logX) j∈I (3.34) 2 2 ≪ (H logQ − H log P ) SN (1 + it)S SR (1 + it)S dt. j,H j,H 2ǫ (logX) j∈I Let’s consider 2 2 (3.35) Q SN (1 + it)S SR (1 + it)S dt. j,H j,H 2ǫ (logX) j∈I 2ǫ (log X) −A For e ≪ T ≪ X, by Lemma 1.2, SN (1 + it)S ≪ (log X) . Therefore, (3.35) is j,H F,A bounded by −2A 2 (log X) SR (1 + it)S dt j,H 2ǫ (log X) j∈I 2ǫ (log X) for e ≪ T ≪ X. By Lemma 2.2, Lemma 2.3 and (3.17), 2 − SR (1 + it)S dt ≪ (T + Xe )( ). j,H F v 2ǫ (log X) e H Xe Therefore, (3.35) is bounded by Q TQ −2A (3.36) H log (log X) ( + 1). P X 3.4. Proof of Proposition 3.2. Proof. Let’s split the integrand in (3.16) by x+h x+h 1 2 1 1 Š Q A (m, 1,..., 1)1 (m) − Q A (m, 1,..., 1)1 (m)� F S F S h h 1 2 m=x m=x x+h x+h 1 2 1 1 + Š A (m, 1,..., 1)1 ′ (m) − A (m, 1,..., 1)1 ′ (m)�. Q Q F S F S h h 1 2 m=x m=x By Lemma 3.6, x+h x+h 1 2 2X 2 1 1 1 (3.37) U A (m, 1,..., 1)1 (m) − A (m, 1,..., 1)1 (m)U dx Q Q S F S F S X X h h 1 2 m=x m=x is bounded by −1 2X Xh i A (m, 1,..., 1)1 (m) F S ≪ max Š U U dt 2ǫ 1+it (logX) i=1,2 e m m=X −1 2X 2T (3.38) Xh A (m, 1,..., 1)1 (m) F S + max U U dt� 1+it −1 T T m T >Xh m=X −A + (log X) . Zero-free regions and averages over short intervals 15 By Lemma 3.8, (3.36) and choosing H = (log X) , (3.39) n −1 −1 2X Xh 2 i A (m, 1,..., 1)1 (m) 1 3A Q (log X) F S −ǫ− 3 2 U U dt ≪ (log X) Š + 1� + Q F,ǫ 2ǫ 1+it (log X) e m h m=X, i 2 m∈K n −1 (log X) − +3 ≪ (log X) F,ǫ for i = 1, 2, and sufficiently large A. By the similar argument in (3.39), −1 2X 2T Xh A (m, 1,..., 1)1 (m) F S i − +3 max U U dt ≪ (log X) F,ǫ 1+it −1 T T m Xh <T ≤X m=X, m∈K for i = 1, 2. In addition, by applying the results of the average of SA (m, 1,.., 1)S , −1 2X −1 2X 2T 2 Xh h A (m, 1,..., 1)1 (m) T + X F S i i 2 max U U dt ≪ max SA (m, 1,..., 1)S Q Q S F,ǫ F 1+it T ≥X T ≥X T T m T X m=X, m=X m∈K −1 ≪ h F,ǫ for i = 1, 2. Therefore, (3.38) is bounded by (3.40) (log X) . By Lemma 3.4, x+h x+h 1 2 2X 2 1 1 1 U A (m, 1,..., 1)1 ′ (m) − A (m, 1,..., 1)1 ′ (m)U dx Q Q F S F S X X h h (3.41) 1 2 m=x m=x − +ǫ ≪ (log X) . F,ǫ Combining (3.40) and (3.41) we get (3.16). 3.5. Proof of Theorem 3.10. Remark 3.9. In [17], it is shown that for sufficiently large X, X 2 n −n +ǫ n +1 (3.42) A (m, 1,..., 1) ≪ X . F F,ǫ m=1 Combining Proposition 3.2 and (3.42) we obtain the following theorem. 1−ǫ (log X) Theorem 3.10. Assume GRP and GVK. Let X be sufficiently large, and let e ≪ 1−ǫ h ≪ X . Then x+h 1 1 − +ǫ (3.43) A (m, 1,..., 1) ≪ (log X) F F,ǫ m=x for almost all x ∈ [X, 2X]. Proof. By the Chebyshev inequality and Proposition 3.2, 1−ǫ x+h x+X 1 1 − +ǫ A (m, 1,..., 1) − A (m, 1,..., 1) ≪ (log X) Q Q F F F,ǫ 1−ǫ h X m=x m=x 16 Jiseong Kim for almost all x ∈ [X, 2X]. By (3.42), 1−ǫ x+X 2 n −n −1+ǫ n +1 A (m, 1,..., 1) ≪ X F F,ǫ 1−ǫ m=x − +ǫ ≪ (log X) . F,ǫ 3.6. Proof of Corollary 1.3. In this subsection, we prove Corollary 1.3 by applying Proposition 3.2. Proof. For convenience we assume that X ∈ N. Let’s consider 2X x+h 1 1 (3.44) U A (m, 1,..., 1)U . Q Q X h x=X m=x By squaring out the inner sums and applying the upper bound A (m, 1,..., 1) ≪ m , F ǫ (3.44) is h 2X 1 1 −1 Q Sh − lSŠ Q A (m, 1,..., 1)A (m + l, 1,..., 1)� + O ( ) + O (hX ). F F F F,ǫ Xh h l=−h m=X l≠0 By Proposition 3.2 and (3.42), (3.44) is bounded by − +ǫ ≪ (log X) . F,ǫ ǫ 2 1 −1 − +ǫ 2 3 Since ,hX ≪ (log X) , the proof is complete. 3.7. Proof of Corollary 1.6. 1−ǫ ′ (log X) Proof. Let h = e . By (1.23), we have ′ −1 x+h X h (log X) U{x ∈ [ ,X − h ) ∶ SA (m, 1,..., 1)S > }U (3.45) 2 log log X m=x ≫ X. By combining the above inequality with Remark 1.5, x+h ′ θ+ǫ−1 A (m, 1,..., 1) ≪ h (log X) Q F F,ǫ m=x −1 h (log X) (3.46) = o Š � F,ǫ (log log X) x+h = o Š SA (m, 1,..., 1)S� F,ǫ F m=x 1−ǫ X (log X) X for almost all x ∈ [ ,X −e ). Therefore, the number of sign changes is ≫ . F,ǫ ′ 2 h 4. Acknowledgements The author would like to thank his advisor Professor Xiaoqing Li for her helpful advice on some zero-free regions and the generalized Ramanujan-Petersson conjecture. The author wishes to express his thanks to Professor Andrew Granville and Professor Kaisa Matomaki for their helpful comments. Zero-free regions and averages over short intervals 17 References [1] W. Casselman and J. Shalika. The unramified principal series of p-adic groups. II. The Whittaker function. Compositio Math., 41(2):207–231, 1980. [2] Étienne Fouvry and Satadal Ganguly. Strong orthogonality between the Möbius function, additive characters and Fourier coefficients of cusp forms. Compos. Math., 150(5):763–797, 2014. [3] Dorian Goldfeld. Automorphic forms and L-functions for the group GL(n,R), volume 99 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2015. With an appendix by Kevin A. Broughan, Paperback edition of the 2006 original [ MR2254662]. [4] Andrew Granville, Adam J. Harper, and K. Soundararajan. A new proof of Halász’s theorem, and its consequences. Compos. Math., 155(1):126–163, 2019. [5] Peter Humphries and Farrell Brumley. Standard zero-free regions for Rankin-Selberg L-functions via sieve theory. Math. Z., 292(3-4):1105–1122, 2019. [6] Henryk Iwaniec and Emmanuel Kowalski. Analytic number theory, volume 53 of American Mathemat- ical Society Colloquium Publications. American Mathematical Society, Providence, RI, 2004. [7] Jesse Jääsaari. On short sums involving Fourier coefficients of Maass forms. J. Théor. Nombres Bor- deaux, 32(3):761–793, 2020. [8] Yujiao Jiang, Guangshi Lü, and Zhiwei Wang. Exponential sums with multiplicative coefficients with- out the Ramanujan conjecture. Math. Ann., 379(1-2):589–632, 2021. [9] Matti Jutila. The additive divisor problem and its analogs for Fourier coefficients of cusp forms. I. Math. Z., 223(3):435–461, 1996. [10] Jiseong Kim. On Hecke eigenvalues of cusp forms in almost all short intervals. International Journal of Number Theory, 0(0):1–14, 0. [11] Guangshi Lü. Sums of absolute values of cusp form coefficients and their application. J. Number Theory, 139:29–43, 2014. [12] Alexander P. Mangerel. Divisor-bounded multiplicative functions in short intervals, 2021. [13] Kaisa Matomäki and Maksym Radziwiłł. A note on the liouville function in short intervals. arXiv, [14] Kaisa Matomäki and Maksym Radziwiłł. Sign changes of Hecke eigenvalues. Geom. Funct. Anal., 25(6):1937–1955, 2015. [15] Kaisa Matomäki and Maksym Radziwiłł. Multiplicative functions in short intervals. Ann. of Math. (2), 183(3):1015–1056, 2016. [16] Kaisa Matomäki, Maksym Radziwiłł, and Terence Tao. Correlations of the von Mangoldt and higher divisor functions II: divisor correlations in short ranges. Math. Ann., 374(1-2):793–840, 2019. [17] Jaban Meher and M. Ram Murty. Oscillations of coefficients of Dirichlet series attached to automorphic forms. Proc. Amer. Math. Soc., 145(2):563–575, 2017. [18] Hugh L. Montgomery. Topics in multiplicative number theory. Lecture Notes in Mathematics, Vol. 227. Springer-Verlag, Berlin-New York, 1971. [19] Hugh L. Montgomery and Robert C. Vaughan. Multiplicative number theory. I. Classical theory, vol- ume 97 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, [20] R. W. K. Odoni. Solution of a generalised version of a problem of Rankin on sums of powers of cusp-form coefficients. Acta Arith., 104(3):201–223, 2002. [21] Takuro Shintani. On an explicit formula for class-1 “Whittaker functions” on GL over P -adic fields. Proc. Japan Acad., 52(4):180–182, 1976. Email address: [email protected] University at Buffalo, Department of Mathematics 244 Mathematics Building Buffalo, NY 14260-2900 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematics arXiv (Cornell University)

Applications of zero-free regions on averages and shifted convolution sums of Hecke eigenvalues

Kim, Jiseong
Mathematics , Volume 2024 (2110) – Oct 13, 2021
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Abstract

APPLICATIONS OF ZERO-FREE REGIONS ON AVERAGES AND SHIFTED CONVOLUTION SUMS OF HECKE EIGENVALUES JISEONG KIM Abstract. By assuming the Vinogradov-Korobov type zero-free regions and the gener- alized Ramanujan-Petersson conjecture, we give some nontrivial upper bounds of almost all short sums of Fourier coefficients of Hecke-Maass cusp forms for SL(n,Z). As an application, we obtain the nontrivial upper bounds for the averages of shifted sums for SL(n,Z) × SL(n,Z). We also give some results on sign changes by assuming some zero- free regions and the generalized Ramanujan-Petersson conjecture. Keywords. Automorphic form; Hecke eigenvalue; Shifted sum; Sign changes; Zero-free region 1. Introduction There are many interesting results in the behavior of multiplicative functions which are concerned about mean values in short intervals and their applications. For a recent account of the results, we refer the reader to [15], [4]. Hecke relations reveal that the normalized Fourier coefficients {A (m, 1,..., 1)} of Hecke-Maass forms for SL(n,Z) are multiplicative, and it is conjectured that these are divisor-bounded (generalized Ramanujan-Petersson conjecture. see subsection 3.2). In addition, there have been many investigations on an- alytic properties of automorphic L-functions which were useful to study the behavior of {A (m, 1,..., 1)}. Therefore, it is natural to try to apply the methods in multiplicative num- ber theory to automorphic L-functions. In this paper, we show that the simple methods in [13], [15] work on {A (m, 1,..., 1)} under the generalized Ramanujan-Petersson conjecture and SL(n,Z) Vinogradov-Korobov zero free regions. As applications, we get some non- trivial upper bounds of the shifted convolution sums and sign-changes of {A (m, 1,..., 1)}. Before we give some results on Hecke-Maass cusp forms for SL(n,Z), we look over the case of holomorphic Hecke cusp forms. 1.1. Holomorphic Hecke cusp forms. Let H = {z = x + iy ∶ x ∈ R,y ∈ (0,∞)},G = a b −1 SL(2,Z). Define j (z) = (cz + d) where γ = Œ ‘. When a holomorphic function c d f ∶ H → C satisfies f(γz) = j (z) f(z) for all γ ∈ SL(2,Z), it is called a modular form of weight k. It is well-known that f(z) has a Fourier expansion at the cusp ∞ (1.1) f(z) = b e(nz) n=0 2πiz where e(z) = e , and the normalized Fourier coefficients a(n) of f(z) are defined by k−1 (1.2) a(n) ∶= b n . The set of all modular forms of fixed weight k is a vector space, and it is denoted M , and arXiv:2110.06855v3 [math.NT] 18 May 2022 2 Jiseong Kim the set of all modular forms which have zero constant term is denoted S . Let T be the k n n-th Hecke operator on S such that 1 az + b (T f)(z) ∶= √ f( ). n Q Q n d ad=n b(mod d) It is known that there is an orthonormal basis of S which consists of eigenfunctions for all Hecke operators T , and these are called Hecke cusp forms. When f is a Hecke cusp form, the eigenvalues λ (n) of the n-th Hecke operator satisfy a(n) = a(1)λ (n), and λ (n) are multiplicative. For Hecke cusp forms f, it is known that +ǫ (1.3) λ (n) ≪ X f f,ǫ n=1 for sufficiently large X (see [17]). Therefore, when x ∈ [X, 2X], x+h x+h x λ (n) ≪ S λ (n)S + S λ (n)S Q Q Q f f f (1.4) n=x n=1 n=1 +ǫ ≪ x . f,ǫ for 1 ≤ h ≤ X. Also for any small fixed ǫ > 0 and h ≫ X , x+h x+h λ (n) ≪ Sλ (n)S Q Q f f n=x n=x (1.5) Sλ (p)S − 1 ≪ h (1 + ) p=1 p∈P by Shiu’s theorem (see Lemma 2.1), where P is the set of all primes. By the Sato-Tate law, Sλ (p)S − 1 8 −1 3π (1 + ) ≪ (log X) p=1 for sufficiently large X (see [11], [20]. − 1 ∼ −0.1511...). 3π In [9], Jutila proved that for any fixed ǫ > 0, +ǫ (1.6) λ (n)λ (n + l) ≪ X , f f f,ǫ n=1 where 1 ≤ l ≤ X . We now apply the above bounds to get the following theorem. Theorem 1.1. Let ǫ > 0 be small fixed, and let X be sufficiently large. Let 1 ≤ h ≪ X . Then x+h 1 1 − − +ǫ 2 6 U λ (n)U ≪ max Šh ,X � f f,ǫ n=x for almost all x ∈ [X, 2X]. Proof. For convenience we assume that X ∈ N. Consider 2X x+h 1 1 (1.7) U λ (n)U . Q Q X h n=x x=X By squaring out the inner sums, (1.7) is Zero-free regions and averages over short intervals 3 2X h x+h−l (2 − δ (l)) λ (n)λ (n + l) Q Q Q 0 f f Xh n=x x=X l=0 where δ (l) = 1 for l = 0, and 0 otherwise. The diagonal terms (l = 0) of (1.7) are 2X x+h (1.8) λ (n) . Q Q Xh n=x x=X x 2 1 By using the fact that ∑ λ (n) ∼ c X for some constant c , (1.8) is bounded by O ( ). f f f f n=1 The off-diagonal terms (1 ≤ l ≤ h) of (1.7) are h 2X x+h−l (1.9) λ (n)λ (n + l), Q Q Q f f Xh n=x l=1 x=X and by Deligne’s bound λ (n) ≪ n . Therefore, (1.9) is bounded by h 2X−h+l h 2 ǫ Š (h − l) λ (n)λ (n + l) + (h − l) X �. Q Q Q f f Xh l=1 x=X+h−l l=1 By (1.6), h 2X−h+l 2 +ǫ (1.10) (h − l) λ (n)λ (n + l) ≪ h X . Q Q f f f,ǫ l=1 x=X+h−l Therefore, (1.9) is bounded by − +ǫ ǫ−1 (1.11) O ŠX + hX �. f,ǫ By the assumption on the range of h, (1.8) is bounded by −1 − +ǫ O max Šh ,X �. f,ǫ By the standard application of the Chebyshev inequality, the proof is complete. Note that the upper bound in Theorem 1.1 is better than the trivial bound from (1.5) when h is bigger than some power of log X, and it is better than the trivial bound from +ǫ (1.4) when h ≪ X . Note that the upper bound of shifted sums was crucial for the proof of Theorem 1.1. 1.2. Zero-free regions and shifted convolution sum. To generalize Theorem 1.1 for Fourier coefficients A (m, 1,..., 1) of Hecke-Maass cusp forms F for SL(n,Z), it suffices to have some upper bounds of shifted convolution sums A (m, 1,..., 1)A (m + l, 1,..., 1) F F m=1 for some range of l, where A (m + l, 1,..., 1) is the complex conjugate of A (m+l, 1,..., 1). F F Getting nontrivial upper bounds or asymptotics of the above shifted sum has many appli- cations. But as far as the author is aware, it does not yet exist in the literature for the cases of n ≥ 3. As an alternative, let us consider the successful approaches in multiplicative number theory. Let s = σ + it where σ,t ∈ R. Define A (m, 1,..., 1) L (s) ∶= . m=1 The generalized Riemann hypothesis claims that there are no nontrivial zeros of L (s) for σ > . The present state of knowledge of the zero-free region is, there are no nontrivial zeros of L (s) where σ > 1 − log St + 3S 4 Jiseong Kim for some constant c > 0 (this c being dependent on the conductor of F ). But for the F F Riemann zeta function ζ(s), it is known that there are no nontrivial zeros where σ > 1 − (log St + 3S) log log St + 3S and StS ≫ 1 for some constant c > 0. This zero-free region (the power of the logarithm in the denominator is ) is called the Vinogradov-Korobov zero-free region, and this gives that for any A > 0, −A A+2 ≪ (log X) when t > (log X) 1+it p=P +ǫ (log X) 3 for e ≪ P ≤ Q ≪ X, any small fixed ǫ > 0. By applying the above inequality, the authors of [13] show that the nontrivial results of the Liouville function can be obtained by the method which is relatively simpler than the methods in [15] (note that the methods in [15] are sophisticate, general, and give better results when one does not have some wide zero-free regions). This motivates us to prove the analog of the above upper bound by assuming the Ramanujan-Petersson conjecture (see (3.8)) and the existence of c such that L (s) has the SL(n,Z) Vinogradov-Korobov zero-free region (1.12) σ > 1 − . (log St + 3S) log log St + 3S Lemma 1.2. Let ǫ > 0 be small fixed, and let A be sufficiently large. Assume the zero-free 2ǫ (log X) region (1.12) and the generalized Ramanujan-Petersson conjecture. Then for e ≪ t ≪ X, A (p, 1,..., 1) −A (1.13) Q ≪ (log X) 1+it p=P p∈P +ǫ 1−ǫ (log X) (log X) for all e < P < Q < e . 1 1 Proof. The proof of this basically follows from the proof of [13, Lemma 2]. Let κ = ,T = log X StS+1 . By Perron’s Formula, Q s s κ+iT A (p, 1,..., 1) 1 Q − P 1 1 = logL (s + 1 + it) ds + O(R ) S F F 1+it p 2πi κ−iT s p=P (1.14) p∈P −1 +O(P ) where 4Q 1 4P SA (m, 1,..., 1)S Q SA (m, 1,..., 1)S P F 1 F 1 κ κ R = max Š ( ) , ( ) � Q Q Q P 1 1 m m m max(1,T S log S) m max(1,T S log S) m=1 m=1 −1 (see [19, Corollary 5.3]). Note that O(P ) comes from SA (p , 1,..., 1)S k(1+it) p , k≥2,p∈[P ,Q ]∩P 1 1 by applying Lemma 3.1 (A (p , 1,..., 1) ≪ O(1)). By applying the upper bound (3.14) F F ǫ c 1 F (A (m, 1,..., 1) ≪ m ), it is easy to show that R ≪ . Let σ = . F ǫ F F.ǫ 0 2 1 (log X) 3 (log log X) 3 Zero-free regions and averages over short intervals 5 By shifting the line of integral in (1.14) to [−σ − iT,−σ + iT ] and changing variables, 0 0 Q −σ +iu −σ +iu 1 0 0 +iT Q − P A (p, 1,..., 1) 1 1 1 = log L (1 − σ + it + iu) du F 0 1+it p 2πi −iT −σ + iu p=P , 0 (1.15) p∈P 1 −1 + O ( ) + O (P ). F,ǫ F Note that a standard application of the Borel-Caratheodory theorem gives (see [2, Section 4]), (1.16) log L (σ + it + iu) ≪ S log(t + u + 2)S F 1 F for σ ∈ [1 − σ , 1 + κ]. Therefore, by applying (1.16) for bounding the integral in (1.15), 1 0 −A (1.13) is bounded by (log X) . In [7], Jesse showed that under the assumption of the generalized Lindelöf hypothesis and a weak version of the Ramanujan-Petersson conjecture, when F is a nontrivial Hecke-Maass cusp form for SL(n,Z) and A (m, 1,..., 1) are normalized Hecke eigenvalues of F, x+h 2X 1− U A (m, 1,..., 1)U dx ∼ B X F F X X m=x 1− +ǫ 1−ǫ for some B , where h ∈ [X ,X ] for some small fixed ǫ > 0. In the following subsec- 1−ǫ (log X) 1−ǫ tion, we consider the cases h ∈ [e ,X ]. 1.3. Main results. The mean-square version of Theorem 3.10. Let ǫ > 0 be small fixed, and let X 1−ǫ (log X) 1−ǫ be sufficiently large. Let e ≪ h ≪ X . Assume the generalized Ramanujan- ǫ ǫ Petersson conjecture on F and the zero-free region (1.12). Then x+h 2X 2 2 − +ǫ U A (m, 1,..., 1)U dx ≪ h (log X) . S F F,ǫ X X m=x This theorem implies the following corollaries. 1−ǫ (log X) Corollary 1.3. Let ǫ > 0 be small fixed, and let X be sufficiently large. Let e ≪ 1−ǫ h ≪ X . Assume the generalized Ramanujan-Petersson conjecture on F and the zero-free region (1.12). Then h 2X 1 2 − +ǫ Sh − lSŠ A (m, 1,..., 1)A (m + l, 1,..., 1)� ≪ (log X) . Q Q F F F,ǫ Xh l=−h m=X l≠0 By using the Rankin-Selberg bound, one can remove the weight Sh−lS at the price of losing some cancellations. The SL(n,Z) Sato-Tate conjecture with the generalized Ramanujan- Petersson conjecture implies that for sufficiently large X, SA (p, 1,..., 1)S − 1 κ −1 (1.17) (1 + ) ≍ (log X) p≤X where iθ iθ iθ iθ iθ 2 1 2 n j κ ∶= Se + e + ... + e S Se − e S dθ dθ ...dθ n M 1 2 n−1 S n−1 n−1 [0,2π) n!(2π) θ +θ +...+θ =0 1≤j<k≤n 1 2 (see [8, (1.16)]). For example, κ ≍ 0.8911,κ ≍ 0.8853, and κ ≍ 0.8863. Therefore, by a 3 4 5 variant of the Shiu’s theorem (2.3), we have h 2X 2κ −2 (1.18) A (m, 1,...., 1)A (m + h, 1,..., 1) ≪ X(log X) Q Q F F l=1 m=X 6 Jiseong Kim under the generalized Ramanujan-Petersson conjecture and (1.17). Assuming the Vinogradov- Korobov zero-free region instead of the Sato-Tate conjecture, we prove the following upper bound which is better than (1.18). 1−ǫ (log X) Corollary 1.4. Let ǫ > 0 be small fixed and let X be sufficiently large. Let e ≪ 1−ǫ h ≪ X . Assume the generalized Ramanujan-Petersson conjecture on F and the zero-free region (1.12). Then h 2X − +ǫ (1.19) Š A (m, 1,..., 1)A (m + l, 1,..., 1)� ≪ (log X) . Q Q F F F,ǫ X(2h − 1) l=−h m=X l≠0 Proof. The left-hand side of (1.19) can be represented as x+h 2X 1 1 ‰ Ž (1.20) Q A (m, 1,...1) A ([x], 1,..., 1)dx. F F X X 2h − 1 m=[x−h] m≠x By the Cauchy-Schwarz inequality and Theorem 3.10, x+h 2X ‰ Ž Q A (m, 1,...1) A ([x], 1,..., 1)dx F F X 2h − 1 m=[x−h] m≠x x+h 2 2X (1.21) 1 2 SA ([x], 1,..., 1)S F 2 ≪ Š T A (m, 1,...1)T + dx� X F S F X 2h − 1 (2h − 1) m=[x−h] − +ǫ ≪ X(log X) . F,ǫ Remark 1.5. If the zero-free region (1.22) σ > 1 − (log StS + 3) holds for some constants c > 0, θ < 1, then it is easy to show that we can replace − +ǫ θ−1+ǫ log P (log X) in Theorem 3.10 with (log X) (the cancellation comes from the log Q +ǫ (log X) truncation in Lemma 3.4. Therefore, we only need to replace P = e with P = θ+ǫ (log X) e in the beginning of Section 2). Since the Vinogradov-Korobov zero-free region is proved for the Riemann zeta function, one could get some similar cancellations on the symmetric square L-functions. As an application of (1.22), let’s talk about the sign-changes of {A (m, 1,..., 1)} . By m=1 assuming the Ramanujan-Petersson conjecture, it is proved that −1 (1.23) SA (m, 1,...1)S ≫ x(log x) . Q F m≤x Essentially, this lower bound comes from Q SA (p, 1,..., 1)S log p ≥ Q SA (p, 1,..., 1)S log p F F p≤x p≤x (1.24) = ‰1 + o(1)Žx (see [8, (1.15)]). When F is self-dual, A (m, 1,..., 1) ∈ R for all m ∈ N. Hence, for the self-dual Hecke-Maass cusp forms F, if x+h x+h A (m, 1,..., 1) = o‰ SA (m, 1,..., 1)SŽ, Q Q F F m=x m=x Zero-free regions and averages over short intervals 7 x+h then there’s at least one sign changes among {A (m, 1,..., 1)} . Therefore, we will prove m=x the following corollary in Section 3. Corollary 1.6. Let ǫ > 0 be sufficiently small and let X be sufficiently large. Let F be a self-dual Hecke-Maass cusp form for SL(n,Z). Assume the generalized Ramanujan- Petersson conjecture on F, the zero-free region (1.22) for some θ < . Then the number of X X sign changes in {A (m, 1,..., 1)} is ≫ . F F,ǫ 1−ǫ m=1 (log X) Remark 1.7. We use the generalized Ramanujan-Petersson conjecture to give the upper bounds of second moments, shifted convolution sums of Hecke eigenvalue squares over some subsets in long intervals. Remark 1.8. One may apply some methods in [4], [12] to get some results similar to the results in this paper, without assuming the SL(n,Z) Vinogradov-Korobov zero-free regions. The results from the above methods may hold for a wider range of h, but cancellations are smaller than the results in this paper. Specifically, under the generalized Ramanujan- Petersson conjecture, the best possible upper bound for Theorem 3.10 from [12, Theorem 1.7] is (0.1211...) 32+ (log X) . Note that the above bound is not sufficiently small to get a bound better than (1.18). Remark 1.9. When χ is a primitive, non-principal Dirichlet character modulo q, the arguments in the proof of Theorem 1.1 gives x+h 2 1 X Xd(q) h U Q χ(n)U ≪ max ‰ , , Ž h h X n=x q for almost all x ∈ [X, 2X], where 1 ≤ h ≤ X and d(q) is the number of divisors of q. In order to treat the off-diagonal terms, one may use Weil’s bound for Kloosterman sums. 2. Lemmas From now on, ǫ be an arbitrary but small fixed positive number, and the value of ǫ may +ǫ 1−ǫ (log X) (log X) differ from one occurrence to another. Let P = e and Q = e . Let S X 2X be the set of integers in [ , ] having at least one prime factor in [P,Q]. We denote d d ′ X 2X ′ ′ S ∶= {n ∈ [ , ] ∶ n ∉ S }. For convenience we denote S instead of S , S instead of S . d 1 d 1 d d Let P denote the set of all prime numbers. For simplicity of notation, we write Ma , Qa instead of M a , Q a p p p p p p p∈P p∈P for any a , respectively. For abbreviation, let GVK stand for the SL(n,Z) zero-free region (1.12) and let GRP stand for the generalized Ramanujan-Petersson conjecture. We apply the following lemma for the average of multiplicative functions over intervals. Lemma 2.1. (Variants of Shiu’s theorem) Let α,β ≥ 1, ǫ > 0. Let g be a non-negative multiplicative function (arithmetic function) such that v v g(p ) ≤ α for all p ∈ P, v ∈ N, (2.1) 3 g(n) ≤ βn for all n ∈ N. Then for 2 ≤ X ≤ Y ≤ X, 1 ≤ H ≤ X, X+Y X 1 g(p) − 1 (2.2) g(n) ≪ ‰1 + Ž, Q M α,β,ǫ Y p n=X p=1 8 Jiseong Kim H X+Y X g(p) − 1 2 (2.3) g(n)g(n + l) ≪ HY ‰1 + Ž Q Q M α,β,ǫ p=1 l=1 n=X Proof. See [16, Lemma 2.3]. X 2X By Lemma 2.1, it is easy to check that typical numbers in [ , ] are contained in S . By d d (2.2), X logP U{n ∈ [X, 2X] ∶ n ∉ S }U ≪ d log Q (2.4) X 1 − +ǫ ≪ (log X) . Lemma 2.2. Let h ≥ 1. For sufficiently large X, h 2X log P (2.5) 1 ′(n)1 ′(n + l) ≪ hX( ) Q Q S S ǫ log Q n=X l=1 Proof. This comes from a direct application of (2.3) by substituting 1 (n) into g(n). The following lemma will be needed after Lemma 3.6. Lemma 2.3. (Dirichlet mean value theorem) For any {a } ⊂ C, n n∈N 2X 2X T 2 (2.6) U Q U dt ≪ (T + X) Q Sa S . it 0 n n=X n=X Proof. See [18, Theorem 9.2]. 3. Proof of Theorem 3.10, and corollaries n−1 3.1. Hecke-Maass cusp forms for SL(n,Z). Let n ≥ 2, let v = (v ,v ,...,v ) ∈ C 1 2 n−1 n−1 and m = (m ,m ,...,m ,m ) ∈ Z . Let U (Z) be the set of all upper triangular 1 2 n−2 n−1 n−1 matrices of the form ⎡ ⎤ ⎢ 1 ⎥ ⎢ ⎥ ⎢ I ∗ ⎥ (3.1) U (Z) = ⎢ ⎥ n−1 ⎢ ⋱ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ b⎦ with ∑ r = n − 1, I denotes the r × r identity matrix, ∗ denotes arbitrary integer i r i i i=1 i entries (Similarly, U (R) with ∗ ∈ R). Let F(z) be a Hecke-Maass form for SL(n,Z) such that ∞ ∞ A (m ,m ,...,m ) F 1 2 n−1 F(z) = ⋯ Q Q Q Q k(n−k) n−1 m =1 m =1 m ≠0 γ∈U (Z)ÓSL(n−1,Z) 1 n−2 n−1 2 n−1 Sm S (3.2) ∏ k=1 × W (Mγ z;v,ψ n−1 ) Jacquet 1,1,...,1, Sm S n−1 where ⎡ ⎤ m m ⋯m Sm S 1 2 n−2 n−1 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ (3.3) M = ⎢ m m ⎥, 1 2 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ (3.4) γ =  , 1 Zero-free regions and averages over short intervals 9 and W is the Jacquet-Whittaker function of type v associated to a character ψ Jacquet m such that (3.5) W (z;v,ψ ) = I (w ⋅ u ⋅ z)ψ (u)d u. Jacquet m S v n m U (R) where n−1 n−1 b v i,j j I (z) = y , M M i=1 j=1 2πi(m u +m u +...+m u ) 1 1,2 2 2,3 n−1 n−1,n ψ denotes the character of U (R) such that ψ (u) ∶= e m n m where ⎡ ⎤ 1 u u ⋯ u 1,2 1,3 1,n ⎢ ⎥ ⎢ ⎥ 1 u ⋯ u ⎢ 2.3 2,n ⎥ ⎢ ⎥ (3.6) u = ⎢ ⋱ ⋮ ⎥, ⎢ ⎥ ⎢ ⎥ 1 u n−1,n ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ and [ ] ⎡ ⎤ (−1) ⎢ ⎥ ⎢ ⎥ ⎢ 1 ⎥ w = ⎢ ⎥. ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ For more details, we refer the reader to [3, Chapter 5, Chapter 9]. 3.2. Generalized Ramanujan-Petersson conjecture (GRP). In order to apply Lemma 2.1, it is necessary to assume the GRP. L (s) has an Euler product −s −2s L (s) = Š1 − A (p,..., 1)p + A (1,p,..., 1)p F M F F p∈P −1 n−1 −(n−1)s n −ns − ⋯ + (−1) A (1,..., 1,p)p + (−1) p � (3.7) −1 −s = Š1 − a (p)p � M M F,j p∈P j=1 for some a (p) ∈ C. GRP claims F,j (3.8) Sa (p)S = 1. F,j Therefore, we get the following lemma. Lemma 3.1. Assume GRP. Then for any k ∈ N and p ∈ P, A (p , 1, 1..., 1) = O (1). F n Proof. It is known that (see [21], [1]) for any k ∈ N and p ∈ P, (3.9) A (p , 1,..., 1) = S (a (p),a (p),...,a (p)) F 0,0,...,0,k F,1 F,2 F,n where S (x ,x ,...,x ) is the Schur polynomial (see [7]). By applying the second λ ,λ ,...,λ 1 2 n 1 2 n Jacobi-Trudi formula, (3.10) S = det[e t ] 0,0,...,0,k λ −i+j where (3.11) λ = S{j ∶ λ ≥ i}S, e are the symmetric polynomials of i-variables. Note that λ − i + j ≤ 2n. By (3.8) and (3.11), (3.12) e t ≤ (2n) . λ −i+j i 10 Jiseong Kim Therefore, by (3.10) and (3.12), (3.13) S (a (p),...,a (p)) ≤ n!(2n) . 0,0,...,k F,1 F,n (3.9) and (3.13) give the upper bound of this lemma. In order to apply Lemma 2.1, we also need to show that for arbitrarily small fixed ǫ > 0, (3.14) A (m, 1,..., 1) ≪ m . F ǫ By the Euler product (3.7) and GRP, A (m, 1,..., 1) ≪ d (m) F n −s n where d (m) is the n-th divisor function (the coefficients of m of ζ(s) ). In addition, by logm (log 2+o(1)) log log m d (m) ≤ d (m) and d (m) ≤ e , n 2 2 we get (3.14). By Lemma 2.1 with the bound A (p, 1,..., 1) ≪ n, 4 n −1 (3.15) SA (m, 1,..., 1)S ≪ X(log X) . m=1 3.3. Main proposition. This subsection is devoted to prove the following proposition. 1−ǫ (log X) Proposition 3.2. Assume GRP and GVK. Let X be sufficiently large. Let e ≪ 1−ǫ h ≤ h ≪ X . Then 1 2 ǫ x+h x+h 2X 1 2 2 1 1 1 − +ǫ (3.16) U A (m, 1,..., 1) − A (m, 1,..., 1)U dx ≪ (log X) . Q Q S F F F,ǫ X X h h 1 2 m=x m=x In order to prove Proposition 3.2, we need the following lemmas. Lemma 3.3. For sufficiently large X, X 2 X SA (p, 1,..., 1)S 1 (3.17) = + O (1). Q Q F p p p=1 p=1 Proof. The proof of this basically follows from [6, Theorem 5.13]. Let F be the dual of F. The Rankin-Selberg L-function ∞ 2 SA (n, 1,..., 1)S (3.18) L (s) = ˜ Q F ×F n=1 is meromorphic with a simple pole at s = 1. And for some constant d , (3.19) L (σ + it) ≠ 0 F ×F for σ > 1 − with StS ≫ 1 (see [5]). Therefore, by [6, Theorem 5.13], (log StS+3) SA (p, 1,..., 1)S Λ(p) = x + o (x) F F p=1 where Λ is the von-Mangoldt function. Therefore, partial summation over p gives (3.17). n −1 Lemma 3.4. Assume GRP and GVK. Let (log X) ≪ h = o(X). Then x+h 2X − +ǫ (3.20) U A (m, 1,..., 1)1 ′ (m)U dx ≪ X(log X) . F S F,ǫ X h m=x Zero-free regions and averages over short intervals 11 Proof. By the Cauchy-Schwarz inequality, x+h 2X A (m, 1,...1)1 (m) 2 ∑ F S m=x T T dx X h x+h x+h 2X ≪ T 1 ′(m)TT SA (m, 1,...1)1 ′ (m)S Tdx Q Q (3.21) S S F S h X m=x m=x 1 1 x+h x+h 2X 2X 2 2 2 2 ′ ′ ≪ Š ‰ Q 1 (m)Ž dx� Š ‰ Q SA (m, 1,...1)1 (m)S Ž dx� . S F S S S h X X m=x m=x By squaring out the inner sum over m and Lemma 2.2, x+h h 2X 2X+h 2X ′ ′ ′ ′ ‰ 1 (m)Ž dx ≪ hŠ 1 (m)1 (m + l) + 1 (m)� Q S Q Q S S Q S m=x l=1 m=X m=X (3.22) log P log P 2 2 ≪ h X( ) + hX log Q log Q 2 − +2ǫ ≪ h X(log X) . In addition, by squaring out the last integrand in (3.21), x+h 2X+h 2X 2 4 ‰ SA (m, 1,...1)1 ′ (m)S Ž dx ≪ h SA (m, 1,...1)1 ′ (m)S Q Q F S F S m=x m=X (3.23) h 2X 2 2 ′ ′ + hQ Q SA (m, 1,...1)1 (m)S SA (m + l, 1,..., 1)1 (m + l)S . F S F S m=X l=1 By Lemma 2.1 and 3.3, we get h 2X 2 2 2 − +2ǫ h SA (m, 1,...1)1 ′ (m)S SA (m+l, 1,..., 1)1 ′ (m+l)S ≪ h (2X +h)(log X) . Q Q F S F S F,ǫ l=1 m=X n −1 Hence, by the assumption (log X) ≪ h = o(X) and (3.15), x+h 2X 2 2 − +ǫ T A (m, 1,...1) 1 ′(m)T dx ≪ h X(log X) . F S F,ǫ m=x Therefore, we get (3.20). Remark 3.5. In order to get some results similar to (3.20), one may avoid assuming GRP by some large sieve techniques if h is bigger than X . But the upper bounds will be weaker than (3.20) (see [6, Theorem 7.14]). 1−ǫ (log X) Lemma 3.6. (Parseval’s bound) Let X and A be sufficiently large. Let e ≪ h ≤ ǫ 1 1−ǫ h = O (X ). Then 2 ǫ x+h x+h 1 2 2X 2 1 1 1 (3.24) U A (m, 1,..., 1)1 (m) − A (m, 1,..., 1)1 (m)U dx Q Q S F S F S X X h h 1 2 m=x m=x −1 2X Xh 2 i A (m, 1,..., 1)1 (m) F S ≪ maxŠ U Q U dt+ 2ǫ 1+it (logX) i=1.2 e m m=X −1 2X 2T 2 Xh A (m, 1,..., 1)1 (m) F S max U U dt� 1+it −1 T >Xh T T m i m=X −A + (log X) . Proof. The proof of this basically follows from [15, Lemma 14]. The 1-bounded condition in [15, Lemma 14] is easily removed (see [10, Lemma 2.1]).  12 Jiseong Kim Let K be the set of integers m ∈ S such that p ∤ m for all p ∈ [P,Q] ∩ P, and let K = {m ∈ S ∶ m ∉ K}. Then (3.25) 2X 2X 2X A (m, 1,..., 1)1 (m) A (m, 1,..., 1)1 (m) A (m, 1,..., 1)1 (m) F S F S F S = + . Q Q Q 1+it 1+it 1+it m m m m=X, m=X, m=X m∈K m∈K The first summand of the right hand side of (3.25) is represented as 2X Q p A (p, 1,..., 1) A (m , 1,..., 1) 1 F F 1 (3.26) Q Q 1+it 1+it p m S{q ∈ [P,Q] ∩ P ∶ qSm }S + 1 X 1 (p,m )=1 p=P 1 1 m = , pm ∈K (see [6, section 17.3], or [15, (9)]). Lemma 3.7. Assume GRP. Let X,T be sufficiently large. Then n −1 2X T 2 A (m, 1,..., 1)1 (m) T (log X) F S U U dt ≪ ( + 1) . S 1 1+it 0 m X m=X, 2 m∈K Proof. By Lemma 2.3 and the Cauchy-Schwarz inequality, (3.27) 2X A (m, 1,..., 1)1 (m) (T + O(X)) F S U U dt ≪ SA (m, 1,..., 1)S 1 (m) Q Q F S 1+it 2 0 m X m=X m∈K m∈K 1 1 2X (T + O(X)) 2 2 ≪ Š SA (m, 1,..., 1)S � Š 1 (m)� Q Q F S m=X m∈K By the definition of K , 2X 1 (m) ≪ 1 Q Q Q m∈K p=P m= , pSm (3.28) Q ≪ X p=P ≪ . Therefore, by combining (3.15) and (3.28), the last term in (3.27) is bounded by n −1 T (log X) ≪ ( + 1) . 1−ǫ Lemma 3.8. Assume GRP. Let X be sufficiently large. Let 2 ≤ H ≪ X . Let v ∈ I ∶= [H log P − 1,H log Q]. Let v+1 A (p, 1,..., 1) N (s) = Q v,H p=e (3.29) 2Xe A (m , 1,..., 1) 1 F 1 R (s) = . v,H v m S{q ∈ [P,Q] ∩ P ∶ qSm }S + 1 m =Xe 1 Zero-free regions and averages over short intervals 13 Then (3.30) 2X T 2 A (m, 1,..., 1)1 (m) T + X 2 F S n −1 U U dt ≪ (log X) Q F,ǫ 2ǫ 1+it (log X) e m HX m=X T + X Q n −1 2 2 + (log X) + H log SN (1 + it)S SR (1 + it)S dt. j,H j,H 1 S 2ǫ (log X) P e XP j∈I Proof. The proof of this basically follows from [14, Lemma 12] . By (3.25), (3.26), 2X A (m, 1,..., 1)1 (m) b(m) F S Q = QN (1 + it)R (1 + it) + Q j,H j.H 1+it 1+it m m m=X j∈I 2X≤m≤2Xe (3.31) 2X 2X A (m, 1,..., 1)1 (m) c(m) F S + + Q Q Q 1+it 1+it m m m=X, m=X, p=P m∈K p Sm for some b(m) ≪ SA (m, 1,..., 1)S, c(m) ≪ SA (p, 1,.., 1)A (m , 1,.., 1)S for some p ∈ F F F 1 [P,Q],pm = m. These b(n), c(n) come from the difference 2X A (m, 1,...1)1 (m) F S − N (1 + it)R (1 + it). Q Q j,H j.H 1+it m=X, j∈I m∈K By Lemma 2.1, 1 1 H H 2Xe 2Xe A (m, 1,..., 1) e 2 2 S S ≪ SA (m, 1,..., 1)S Q Q F,ǫ F m X m=2X m=2X e X 2 n −1 ≪ (log X) . F,ǫ X H Therefore, 1 1 H H 2Xe 2Xe 2 2 b(m) b(m) U U dt ≪ (T + X) U U Q Q 2ǫ 1+it (logX) e m m m=2X m=2X 2Xe (3.32) A (m, 1,..., 1) ≪ (T + X) U U m=2X T + X n −1 ≪ (log X) . F,ǫ HX By (3.15) and Lemma 2.3, 2X 2X c(m) T + X U U dt ≪ Š Tc(m)T � Q Q Q 2ǫ 1+it 2 (log X) e m X m=X, m=1, p=P 2 2 p Sm p Sm for some p∈[P,Q] 2X T + X X 1 4 2 (3.33) T T ≪ ( ) Š Q A (m, 1,..., 1) � F F X P m=1 n −1 T + X X(log X) ≪ . F,ǫ P 14 Jiseong Kim Finally, by the Cauchy-Schwarz inequality, U N (1 + it)R (1 + it)U dt j,H j.H 2ǫ (logX) j∈I (3.34) 2 2 ≪ (H logQ − H log P ) SN (1 + it)S SR (1 + it)S dt. j,H j,H 2ǫ (logX) j∈I Let’s consider 2 2 (3.35) Q SN (1 + it)S SR (1 + it)S dt. j,H j,H 2ǫ (logX) j∈I 2ǫ (log X) −A For e ≪ T ≪ X, by Lemma 1.2, SN (1 + it)S ≪ (log X) . Therefore, (3.35) is j,H F,A bounded by −2A 2 (log X) SR (1 + it)S dt j,H 2ǫ (log X) j∈I 2ǫ (log X) for e ≪ T ≪ X. By Lemma 2.2, Lemma 2.3 and (3.17), 2 − SR (1 + it)S dt ≪ (T + Xe )( ). j,H F v 2ǫ (log X) e H Xe Therefore, (3.35) is bounded by Q TQ −2A (3.36) H log (log X) ( + 1). P X 3.4. Proof of Proposition 3.2. Proof. Let’s split the integrand in (3.16) by x+h x+h 1 2 1 1 Š Q A (m, 1,..., 1)1 (m) − Q A (m, 1,..., 1)1 (m)� F S F S h h 1 2 m=x m=x x+h x+h 1 2 1 1 + Š A (m, 1,..., 1)1 ′ (m) − A (m, 1,..., 1)1 ′ (m)�. Q Q F S F S h h 1 2 m=x m=x By Lemma 3.6, x+h x+h 1 2 2X 2 1 1 1 (3.37) U A (m, 1,..., 1)1 (m) − A (m, 1,..., 1)1 (m)U dx Q Q S F S F S X X h h 1 2 m=x m=x is bounded by −1 2X Xh i A (m, 1,..., 1)1 (m) F S ≪ max Š U U dt 2ǫ 1+it (logX) i=1,2 e m m=X −1 2X 2T (3.38) Xh A (m, 1,..., 1)1 (m) F S + max U U dt� 1+it −1 T T m T >Xh m=X −A + (log X) . Zero-free regions and averages over short intervals 15 By Lemma 3.8, (3.36) and choosing H = (log X) , (3.39) n −1 −1 2X Xh 2 i A (m, 1,..., 1)1 (m) 1 3A Q (log X) F S −ǫ− 3 2 U U dt ≪ (log X) Š + 1� + Q F,ǫ 2ǫ 1+it (log X) e m h m=X, i 2 m∈K n −1 (log X) − +3 ≪ (log X) F,ǫ for i = 1, 2, and sufficiently large A. By the similar argument in (3.39), −1 2X 2T Xh A (m, 1,..., 1)1 (m) F S i − +3 max U U dt ≪ (log X) F,ǫ 1+it −1 T T m Xh <T ≤X m=X, m∈K for i = 1, 2. In addition, by applying the results of the average of SA (m, 1,.., 1)S , −1 2X −1 2X 2T 2 Xh h A (m, 1,..., 1)1 (m) T + X F S i i 2 max U U dt ≪ max SA (m, 1,..., 1)S Q Q S F,ǫ F 1+it T ≥X T ≥X T T m T X m=X, m=X m∈K −1 ≪ h F,ǫ for i = 1, 2. Therefore, (3.38) is bounded by (3.40) (log X) . By Lemma 3.4, x+h x+h 1 2 2X 2 1 1 1 U A (m, 1,..., 1)1 ′ (m) − A (m, 1,..., 1)1 ′ (m)U dx Q Q F S F S X X h h (3.41) 1 2 m=x m=x − +ǫ ≪ (log X) . F,ǫ Combining (3.40) and (3.41) we get (3.16). 3.5. Proof of Theorem 3.10. Remark 3.9. In [17], it is shown that for sufficiently large X, X 2 n −n +ǫ n +1 (3.42) A (m, 1,..., 1) ≪ X . F F,ǫ m=1 Combining Proposition 3.2 and (3.42) we obtain the following theorem. 1−ǫ (log X) Theorem 3.10. Assume GRP and GVK. Let X be sufficiently large, and let e ≪ 1−ǫ h ≪ X . Then x+h 1 1 − +ǫ (3.43) A (m, 1,..., 1) ≪ (log X) F F,ǫ m=x for almost all x ∈ [X, 2X]. Proof. By the Chebyshev inequality and Proposition 3.2, 1−ǫ x+h x+X 1 1 − +ǫ A (m, 1,..., 1) − A (m, 1,..., 1) ≪ (log X) Q Q F F F,ǫ 1−ǫ h X m=x m=x 16 Jiseong Kim for almost all x ∈ [X, 2X]. By (3.42), 1−ǫ x+X 2 n −n −1+ǫ n +1 A (m, 1,..., 1) ≪ X F F,ǫ 1−ǫ m=x − +ǫ ≪ (log X) . F,ǫ 3.6. Proof of Corollary 1.3. In this subsection, we prove Corollary 1.3 by applying Proposition 3.2. Proof. For convenience we assume that X ∈ N. Let’s consider 2X x+h 1 1 (3.44) U A (m, 1,..., 1)U . Q Q X h x=X m=x By squaring out the inner sums and applying the upper bound A (m, 1,..., 1) ≪ m , F ǫ (3.44) is h 2X 1 1 −1 Q Sh − lSŠ Q A (m, 1,..., 1)A (m + l, 1,..., 1)� + O ( ) + O (hX ). F F F F,ǫ Xh h l=−h m=X l≠0 By Proposition 3.2 and (3.42), (3.44) is bounded by − +ǫ ≪ (log X) . F,ǫ ǫ 2 1 −1 − +ǫ 2 3 Since ,hX ≪ (log X) , the proof is complete. 3.7. Proof of Corollary 1.6. 1−ǫ ′ (log X) Proof. Let h = e . By (1.23), we have ′ −1 x+h X h (log X) U{x ∈ [ ,X − h ) ∶ SA (m, 1,..., 1)S > }U (3.45) 2 log log X m=x ≫ X. By combining the above inequality with Remark 1.5, x+h ′ θ+ǫ−1 A (m, 1,..., 1) ≪ h (log X) Q F F,ǫ m=x −1 h (log X) (3.46) = o Š � F,ǫ (log log X) x+h = o Š SA (m, 1,..., 1)S� F,ǫ F m=x 1−ǫ X (log X) X for almost all x ∈ [ ,X −e ). Therefore, the number of sign changes is ≫ . F,ǫ ′ 2 h 4. Acknowledgements The author would like to thank his advisor Professor Xiaoqing Li for her helpful advice on some zero-free regions and the generalized Ramanujan-Petersson conjecture. The author wishes to express his thanks to Professor Andrew Granville and Professor Kaisa Matomaki for their helpful comments. Zero-free regions and averages over short intervals 17 References [1] W. Casselman and J. Shalika. The unramified principal series of p-adic groups. II. The Whittaker function. Compositio Math., 41(2):207–231, 1980. [2] Étienne Fouvry and Satadal Ganguly. 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Published: Oct 13, 2021

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