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

Abstract

Let $A(s)$ be a general Dirichlet polynomial and $\Phi$ be a smooth function supported in $[1,2]$ with mild bounds on its derivatives. New main terms for the integral $I(\alpha ,\beta )={\int }_{\mathbb{R}}\zeta (\frac{1}{2}+\alpha +it)\zeta (\frac{1}{2}+\beta +it)|A(\frac{1}{2}+it){|}^2\Phi (\frac{t}{T})dt$ are given. For the error term, we show that the length of the Feng mollifier can be increased from $\theta <\frac{17}{33}$ to $\theta <\frac{6}{11}$ 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 $\zeta (s)$ on the critical line.

Fixed error in statement of Theorem 1.2

References in corpus (2)

• Critical zeros of the Riemann zeta-function
• The $\theta =\infty$ conjecture implies the Riemann hypothesis

Cited by in corpus (2)

• Lower bounds for discrete negative moments of the Riemann zeta function
• More than five-twelfths of the zeros of $\zeta$ are on the critical line