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)
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)
A linear combination L(s) of two Dirichlet L-functions has infinitely many complex zeros in Re s < 0. In this note we prove an infinity of complex zeros of L (k) (s) in the same region.
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)
We investigate the zeros of degree one L-functions from the extended Selberg class off the critical line.