Short mollifiers of the Riemann zeta-function

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 $\mathfrak{f}$-function in the celebrated Riemann--Siegel formula.