Short mollifiers of the Riemann zeta-function

2025/08/14 by Conrey, J. Brian, Farmer, David W., Kwan, Chung-Hang +2
#11M06 #11M26 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2508.11108

Abstract

We apply the calculus of variations to construct a new sequence of linear combinations of derivatives of the Riemann ζ-function adapted to Levinson's method, which yield a positive proportion of zeros of the ζ-function on the critical line, regardless of how short the mollifier is. Our construction extends readily to modular L-functions. Even with Levinson's original choice of mollifier, our method more than doubles the proportions of zeros on the critical line for modular L-functions previously obtained by Bernard and Kühn--Robles--Zeindler, while relying on the same arithmetic inputs. This indicates that optimizing the linear combinations, an approach that has received relatively little attention, has a more pronounced effect than refining the mollifier when it is short. Curiously, our linear combinations provide non-trivial smooth approximations of Siegel's \mathfrakf-function in the celebrated Riemann--Siegel formula.

Citations

• More than five-twelfths of the zeros of ζ are on the critical line
• On mean values of mollifiers and L-functions associated to primitive\n cusp forms
• Modular case of Levinson's theorem

Related

• On a choice of the mollified function in the Levinson-Conrey method
• On a mollifier of the perturbed Riemann zeta-function
• Critical zeros of the Riemann zeta-function
• More than five-twelfths of the zeros of ζ are on the critical line
• Limitations to mollifying ζ(s)