TY - JOUR
AU - Levinson, N.
AB - By N. LEVINSONt [Received 27 July 1964] To J. E. LITTLEWOOD on his 80th birthday 1. As usual, let A(n) = logp for n = p , p prime, and be zero for other n. Let t/j(x) = JA(n), n ^ x. An elementary proof will be given for the following THEOREM. For 0 < a ^ \ , ^ -°). (1.1) 1 (r If a ^ log log #/log JC the term O(a; - ) is O(ax). The inequality (1.1) may be compared with A. Selberg's famous inequality which can be written as tjj{x)\ogx+ £ A{n)ijj(x/n) = 2a;logo; + 0(a;). (1.2) In (1.1), ijj(x/n) is weighted more heavily for small n than in (1.2). On the other hand, the weighting also involves A(n) more heavily for small n. The formula (1.1) has a slightly larger error term than (1.2). The proof of (1.1) given here is much longer than proofs of (1.2). The derivation of (1.1) can be extended to yield *Kx)+ S Try S ^ (1.3) where P _ is a polynomial of degree m — 2 and m 2 JA^O'). (1.4) ij = n An easy induction shows that (1.5) di n The derivation of (1.1) 
TI - A Variant of the Selberg Inequality
JF - Proceedings of the London Mathematical Society
DO - 10.1112/plms/s3-14A.1.191
DA - 1965-01-01
UR - https://www.deepdyve.com/lp/wiley/a-variant-of-the-selberg-inequality-XBiM36AFT6
SP - 191
EP - 198
VL - s3-14A
IS - 1
DP - DeepDyve
ER -