Perturbed moments and a longer mollifier for critical zeros of $$\zeta $$ ζ

Perturbed moments and a longer mollifier for critical zeros of $$\zeta $$ ζ Let A(s) be a general Dirichlet polynomial and $$\Phi $$ Φ be a smooth function supported in [1, 2] with mild bounds on its derivatives. New main terms for the integral $$I\left( \alpha ,\beta \right) =\int _{\mathbb {R}} \zeta \left( \frac{1}{2}+\alpha +it\right) \zeta \left( \frac{1}{2}+\beta +it\right) |A\left( \frac{1}{2}+it\right) |^2 \Phi \left( \frac{t}{T}\right) dt$$ I α , β = ∫ R ζ 1 2 + α + i t ζ 1 2 + β + i t | A 1 2 + i t | 2 Φ t T d t are given. For the error term, we show that the length of the Feng mollifier can be increased from $$\theta < \frac{17}{33}$$ θ < 17 33 to $$\theta < \frac{6}{11}$$ θ < 6 11 by decomposing the error into Type I and Type II sums and then studying the resulting sums of Kloosterman sums. As an application, we slightly increase the proportion of zeros of $$\zeta (s)$$ ζ ( s ) on the critical line. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Research in Number Theory Springer Journals

Perturbed moments and a longer mollifier for critical zeros of $$\zeta $$ ζ

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Publisher
Springer Journals
Copyright
Copyright © 2018 by SpringerNature
Subject
Mathematics; Number Theory
eISSN
2363-9555
D.O.I.
10.1007/s40993-018-0103-4
Publisher site
See Article on Publisher Site

Abstract

Let A(s) be a general Dirichlet polynomial and $$\Phi $$ Φ be a smooth function supported in [1, 2] with mild bounds on its derivatives. New main terms for the integral $$I\left( \alpha ,\beta \right) =\int _{\mathbb {R}} \zeta \left( \frac{1}{2}+\alpha +it\right) \zeta \left( \frac{1}{2}+\beta +it\right) |A\left( \frac{1}{2}+it\right) |^2 \Phi \left( \frac{t}{T}\right) dt$$ I α , β = ∫ R ζ 1 2 + α + i t ζ 1 2 + β + i t | A 1 2 + i t | 2 Φ t T d t are given. For the error term, we show that the length of the Feng mollifier can be increased from $$\theta < \frac{17}{33}$$ θ < 17 33 to $$\theta < \frac{6}{11}$$ θ < 6 11 by decomposing the error into Type I and Type II sums and then studying the resulting sums of Kloosterman sums. As an application, we slightly increase the proportion of zeros of $$\zeta (s)$$ ζ ( s ) on the critical line.

Journal

Research in Number TheorySpringer Journals

Published: Feb 6, 2018

References

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