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We study properties of zeros of the derivatives of the Riemann zeta function ζ(s). Levinson and Montgomery [8] achieved several important theorems for the behavior of zeros of ζ(s) (m = 1, 2, 3, · · · ). If we assume the Riemann hypothesis, ζ ′(s) has no non-real zero in Re s < 1 2 and ζ(s) (m > 1) has at most finitely many zeros in Re s < 1 2 .(More)
On two pages in his lost notebook, Ramanujan recorded several theorems involving the modified Bessel function Kν(z). These include Koshliakov’s formula and Guinand’s formula, both connected with the functional equation of nonanalytic Eisenstein series, and both discovered by these authors several years after Ramanujan’s death. Other formulas, including one(More)
Abstract. The complex zeros of the Riemannn zeta-function are identical to the zeros of the Riemann xi-function, ξ(s). Thus, if the Riemann Hypothesis is true for the zetafunction, 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(More)
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 of Özlük which gives a proportion of at most 86%. We further compute an q-analogue of the Pair Correlation Function F (α) averaged over all primitive Dirichlet(More)
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