# More than five-twelfths of the zeros of $$\zeta$$ are on the critical line

@article{Pratt2018MoreTF,
title={More than five-twelfths of the zeros of \$\$\zeta \$\$ are on the critical line},
author={Kyle Pratt and Nicolas Robles and Alexandru Zaharescu and Dirk Zeindler},
journal={arXiv: Number Theory},
year={2018}
}
• Published 28 February 2018
• Mathematics
• arXiv: Number Theory
The second moment of the Riemann zeta-function twisted by a normalized Dirichlet polynomial with coefficients of the form $(\mu \star \Lambda_1^{\star k_1} \star \Lambda_2^{\star k_2} \star \cdots \star \Lambda_d^{\star k_d})$ is computed unconditionally by means of the autocorrelation of ratios of $\zeta$ techniques from Conrey, Farmer, Keating, Rubinstein and Snaith (2005), Conrey, Farmer and Zirnbauer (2008) as well as Conrey and Snaith (2007). This in turn allows us to describe the…
13 Citations
Explicit zero density estimate for the Riemann zeta-function near the critical line
• A. Simonič
• Mathematics
Journal of Mathematical Analysis and Applications
• 2020
Some explicit and unconditional results on gaps between zeroes of the Riemann zeta-function
• Mathematics
• 2020
We make explicit an argument of Heath-Brown concerning large and small gaps between nontrivial zeroes of the Riemann zeta-function, $\zeta(s)$. In particular, we provide the first unconditional
UNEXPECTED AVERAGE VALUES OF GENERALIZED VON MANGOLDT FUNCTIONS IN RESIDUE CLASSES
• Mathematics
Journal of the Australian Mathematical Society
• 2020
Abstract In order to study integers with few prime factors, the average of $\unicode[STIX]{x1D6EC}_{k}=\unicode[STIX]{x1D707}\ast \log ^{k}$ has been a central object of research. One of the more
Zeros of $\mathrm{GL}_2$ $L$-functions on the critical line
• Mathematics
• 2020
We use Levinson's method and the work of Blomer and Harcos on the $\mathrm{GL}_2$ shifted convolution problem to prove that at least 6.96% of the zeros of the L-function of any holomorphic or Maass
100% of the zeros of $\zeta(s)$ are on the critical line
It is proved that as T → ∞, if κ is the proportion of zeros of ζ ( s ) on the critical line κ lim inf T →∞ T then it is proved as a consequence that κ = 1.
A note on log-type GCD sums and derivatives of the Riemann zeta function
In [22], we defined the so-called “log-type" GCD sums and proved the lower bounds Γ (l) 1 (N )≫l ( loglog N )2+2l . In this note, we obtain the upper bounds Γ 1 (N )≪l ( loglog N )2+2l by two
Maxima of log-correlated fields: some recent developments
• Mathematics
Journal of Physics A: Mathematical and Theoretical
• 2022
We review recent progress relating to the extreme value statistics of the characteristic polynomials of random matrices associated with the classical compact groups, and of the Riemann zeta-function
On a positivity property of the real part of logarithmic derivative of the Riemann $\xi$-function
• Mathematics
• 2022
In this paper we investigate the positivity property of the real part of logarithmic derivative of the Riemann ξ -function for 1/2 < σ < 1 and sufficiently large t . We give an explicit upper and
Almost all the non-trivial zeros of ζ ( s ) are on the critical line
The Cauchy’s argument principle on ξ ( s ) on the rectangle R ǫ along with properties of Riemann zeta function and Riem Mann xi function are used and it is proved that as T → ∞ Also if the authors have then the number of zeros ρ of ζ ( s) is 12 and as a consequence that κ = 1.
100% of the non-trivial zeros of ζ ( s ) are on the critical line
In this manuscript we denote by N ( T ) the number of zeros ρ of ζ ( s ) such that 0 < ℑ ( ρ ) < T . Denote by N 0 ( T ) the number of zeros ρ of ζ ( s ) such that ℜ ( ρ ) = 12 and 0 < ℑ ( ρ ) < T .

## References

SHOWING 1-10 OF 65 REFERENCES
Perturbed moments and a longer mollifier for critical zeros of $$\zeta$$ζ
• Mathematics
• 2017
Let A(s) be a general Dirichlet polynomial and $$\Phi$$Φ be a smooth function supported in [1, 2] with mild bounds on its derivatives. New main terms for the integral I\left( \alpha ,\beta \right)
The mean square of the product of $\zeta(s)$ with Dirichlet polynomials
• Mathematics
• 2014
Improving earlier work of Balasubramanian, Conrey and Heath-Brown, we obtain an asymptotic formula for the mean-square of the Riemann zeta-function times an arbitrary Dirichlet polynomial of length
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
UNEXPECTED AVERAGE VALUES OF GENERALIZED VON MANGOLDT FUNCTIONS IN RESIDUE CLASSES
• Mathematics
Journal of the Australian Mathematical Society
• 2020
Abstract In order to study integers with few prime factors, the average of $\unicode[STIX]{x1D6EC}_{k}=\unicode[STIX]{x1D707}\ast \log ^{k}$ has been a central object of research. One of the more
Zeroes of zeta functions and symmetry
• Mathematics
• 1999
Hilbert and Polya suggested that there might be a natural spectral interpretation of the zeroes of the Riemann Zeta function. While at the time there was little evidence for this, today the evidence
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
Prime Numbers in Short Intervals and a Generalized Vaughan Identity
1. Introduction. Many problems involving prime numbers depend on estimating sums of the form ΣΛ(n)f(n), for appropriate functions f(n), (here, as usual, Λ(n) is the von Mangoldt function). Three
The twisted mean square and critical zeros of Dirichlet L-functions
In this work, we obtain an asymptotic formula for the twisted mean square of a Dirichlet L-function with a longer mollifier, whose coefficients are also more general than before. As an application we