Optimising Selberg's method for critical zeros

Abstract

We revisit Zhuravlev's 1974 quantitative form of Selberg's sign-change method for the zeros of the Riemann zeta function on the critical line. Zhuravlev's work, originally in Russian and little known in the West, established an explicit positive proportion of such zeros. Using modern techniques we optimise the arithmetic mean value at the heart of the method, which has an exact closed form for reciprocal-square-root coefficients and is improved further by a positive-semidefinite family of mollifiers. We thereby obtain the strongest result yet known from Selberg's method, proving that at least $7\%$ of the non-trivial zeros of the Riemann zeta function lie on the critical line.