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On two pages in his lost notebook, Ramanujan recorded several theorems involving the modified Bessel function K ν (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(More)
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 zeta-function, 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(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¨Ozlük 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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