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…
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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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