On a mollifier of the perturbed Riemann zeta-function

Abstract

The mollification ζ(s) + ζ (s) put forward by Feng is computed by analytic methods coming from the techniques of the ratios conjectures of L-functions. The current situation regarding the percentage of non-trivial zeros of the Riemann zeta-function on the critical line is then clarified.

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

“…Bui et al [6] were able to get 41.05%. Feng [16] claimed a value of 41.27%, though this was contested in [5,19,24], and reduced to 41.07% due to an incomplete claim on the error terms.…”

“…[1,3,7,8,21]. The applications of I are very deep, as one may use asymptotic estimates for I to make sense of the distribution of values of L-functions, the location of their critical zeros, as well as upper and lower bounds for the size of L-functions (see, among many examples, [9][10][11]16,19,23,24]). As often stressed, one key aspect to obtaining good results is to make sure that θ be as large as possible.…”

“…Refinements on the value of θ due to Conrey [8] have increased that percentage to 40.88%. Adding, or refining, the structure of the coefficients a n of the Dirichlet polynomial in (1.1) also leads to improved values of the above mentioned proportion, [6,7,16,19,20,24].…”

“…The main innovations of this paper lie in studying the second case of (1.6), and extending the range of θ for which one may prove an asymptotic formula. The second case is colloquially known as the "Feng" mollifier, since it was first exploited in [16,19].…”

A small improvement in the gaps between consecutive zeros of the Riemann zeta-function

“…It is possible that Feng and Wu's result λ > 3.072 can also be obtained just assuming the Riemann hypothesis. For another application of Feng's mollifier, see [8] and the references therein.…”