# 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} }

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

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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

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- 2020

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- 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

- Mathematics, Computer Science
- 2022

The Littlewood’s lemma on the rectangle R ǫ along with the Hadamard product of ξ ( s ) and the asymptotic for the logarithmic derivative of ζ ( s) are used to prove that as T → ∞, κ is the proportion of zeros on the critical line and it is proved as a consequence that κ = 1.

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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…

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### Almost all the non-trivial zeros of ζ ( s ) are on the critical line

- Mathematics, Computer Science
- 2022

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

- Mathematics, Philosophy
- 2022

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 .…

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### UNEXPECTED AVERAGE VALUES OF GENERALIZED VON MANGOLDT FUNCTIONS IN RESIDUE CLASSES

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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…

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