# Zeros of the Riemann zeta function on the critical line

@article{Feng2010ZerosOT,
title={Zeros of the Riemann zeta function on the critical line},
author={Shaoji Feng},
journal={arXiv: Number Theory},
year={2010}
}
• S. Feng
• Published 27 February 2010
• Mathematics
• arXiv: Number Theory
it is proved that at least 41.28% zeros of the Riemann zeta function are on the critical line
51 Citations
More than 41% of the zeros of the zeta function are on the critical line
• Mathematics
• 2011
We prove that at least 41.05% of the zeros of the Riemann zeta function are on the critical line.
On large spacing of the zeros of the Riemann zeta-function
• Mathematics
• 2013
Abstract Assuming the Generalized Riemann Hypothesis (GRH), we show that infinitely often consecutive non-trivial zeros of the Riemann zeta-function differ by at least 3.072 times the average spacing.
On gaps between zeros of the Riemann zeta function
• Mathematics
• 2010
Assuming the Riemann Hypothesis, we show that infinitely often consecutive non-trivial zeros of the Riemann zeta-function differ by at least 2.7327 times the average spacing and infinitely often they
On the zeros of the Epstein zeta function
• Mathematics
• 2011
In this article, we count the number of consecutive zeros of the Epstein zeta-function, associated to a certain quadratic form, on the critical line with ordinates lying in $[0,T], T$ sufficiently
Critical zeros of the Riemann zeta-function
In this unpublished note, we sketch an idea of using a three-piece mollifier to slightly improve the known percentages of zeros and simple zeros of the Riemann zeta-function on the critical line.
A computational history of prime numbers and Riemann zeros
• Mathematics
• 2018
We give an informal survey of the historical development of computations related to prime number distribution and zeros of the Riemann zeta function.
On the zeros of Riemann's zeta-function on the critical line
Abstract We combine the mollifier method with a zero detection method of Atkinson to prove in a new way that a positive proportion of the nontrivial zeros of the Riemann zeta-function ζ ( s ) are on
Some notes on Euler products
We focus on a well-known convergence phenomenon, the fact that the $\zeta$ zeros are the universal singularities of certain Euler products.
Modular case of Levinson's theorem
We evaluate the integral mollified second moment of L-functions of primitive cusp forms and we obtain, for such L-function, an explicit positive proportion of zeros which lie on the critical line.

## References

SHOWING 1-10 OF 29 REFERENCES
More than 41% of the zeros of the zeta function are on the critical line
• Mathematics
• 2011
We prove that at least 41.05% of the zeros of the Riemann zeta function are on the critical line.
Simple zeros of the Riemann zeta-function
• Mathematics
• 1993
Assuming the Riemann Hypothesis, Montgomery and Taylor showed that at least 67.25% of the zeros of the Riemann zeta-function are simple. Using Montgomery and Taylor's argument together with an
Zeros of derivatives of Riemann's xi-function of the critical line. II
Abstract Explicit lower bounds for the proportion of zeros of the derivatives of Riemann's xi-function which are on the critical line and simple are given. These lead to upper bounds for the
On the zeros of ?'(s near the critical line
Let ρ = β ′ + i γ ′ denote the zeros of ζ (s), s = σ + i t . It is shown that there is a positive proportion of the zeros of ζ (s) in 0 < t < T satisfyingβ ′ − 1/2 (logT)−1. Further results relying
The Theory of the Riemann Zeta-Function
• Mathematics
• 1987
The Riemann zeta-function embodies both additive and multiplicative structures in a single function, making it our most important tool in the study of prime numbers. This volume studies all aspects
More than two fifths of the zeros of the Riemann zeta function are on the critical line.
In this paper we show that at least 2/5 of the zeros of the Riemann zeta-function are simple and on the critical line. Our method is a refinement of the method Levinson [11] used when he showed that
Long mollifiers of the Riemann Zeta-function
The best current bounds for the proportion of zeros of ζ( s ) on the critical line are due to Conrey [C], using Levinson's method [Lev]. This method can also be used to detect simple zeros on the
Mean-values of the Riemann zeta-function
Let Asymptotic formulae for I k ( T ) have been established for the cases k =1 (Hardy-Littlewood, see [13]) and k = 2 (Ingham, see [13]). However, the asymptotic behaviour of I k ( T ) remains
Selberg's Work on the Zeta-Function
Publisher Summary This chapter presents an account of Selberg's work on the zeta-function. Selberg's publications concerning the Riemann zeta-function appeared in Norwegian journals at a time when