The complex zeros of the Riemannn zeta-function are identical to the zeros of the Riemann ξ-function, ξ(s). Thus, if the Riemann hypothesis (RH) is true for the zeta-function, then it is true for ξ(s). Since ξ(s) is entire, the zeros of ξ (s), its derivative, would then also satisfy a Riemann hypothesis. We investigate the pair correlation function of the zeros of ξ (s) under the assumption that RH is true. We then deduce consequences about the size of gaps between these zeros and the proportion of these zeros that are simple.
Assuming the Generalized Riemann Hypothesis (GRH), we show using the asymptotic large sieve that 91% of the zeros of primitive Dirichlet L-functions are simple. This improves on earlier work ofÖzlük which gives a proportion of at most 86%. We further compute q-analogue of the Pair Correlation Function F (α) averaged over all primitive Dirichlet L-functions in the range |α| < 2 . Previously such a result was available only when the average included all the characters χ. As a corollary of our results, we obtain an asymptotic formula for a sum over characters similar to the one encountered in the Barban-Davenport-Halberstam Theorem.
We investigate the zeros of Epstein zeta functions associated with a positive definite quadratic form with rational coefficients in the vertical strip σ 1 < ℜs < σ 2 , where 1/2 < σ 1 < σ 2 < 1. When the class number of the quadratic form is bigger than 1, Voronin gives a lower bound and Lee gives an asymptotic formula for the number of zeros. In this paper, we improve their results by providing a new upper bound for the error term.
In the paper we introduce a new method how to use only an orthonormality relation of coefficients of Dirichlet series defining given L-functions from the Selberg class to prove joint universality.
We study the value distribution of the Riemann zeta function near the line Re s = 1/2. We find an asymptotic formula for the number of a-values in the rectangle 1for fixed h 1 , h 2 > 0 and 0 < θ < 1/13. To prove it, we need an extension of the valid range of Lamzouri, Lester and Radziwiłł's recent results on the discrepancy between the distribution of ζ(s) and its random model. We also propose the secondary main term for the Selberg's central limit theorem by providing sharper estimates on the line Re s = 1/2 + 1/(log T ) θ .
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