# 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} }

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

- Mathematics
- 2014

In this unpublished note, we sketch an idea of using a three-piece mollifier to slightly improve the known percentages of zeros and simple zeros of the Riemann zeta-function on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

- Mathematics
- 2016

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

- Mathematics
- 2014

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

- Mathematics
- 2014

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

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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
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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

- Mathematics
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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

- Mathematics
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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
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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.

- Mathematics
- 1989

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

- Mathematics
- 1993

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

- Mathematics
- 1995

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

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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…