Selberg's orthonormality conjecture and joint universality of L-functions

In the paper we introduce the new approach how to use an orthonormality relation of coefficients of Dirichlet series defining given L-functions from the Selberg class to prove joint universality

Simple zeros of primitive Dirichlet LL-functions and the asymptotic large sieve

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 \"{O}zl\"{u}k which gives a proportion of at most 86%. We further compute an $q$-analogue of the Pair Correlation Function $F(\alpha)$ averaged over all primitive Dirichlet $L$-functions in the range $|\alpha| < 2$ . Previously such a result was available only when the average included all the characters $\chi$.Comment: This work was initiated during the Arithmetic Statistics MRC program at Snowbird, Utah. Corollary 3 and Section 7 are adde

Omega results for cubic field counts via lower-order terms in the one-level density

In this paper, we obtain a precise formula for the one-level density of L-functions attached to non-Galois cubic Dedekind zeta functions. We find a secondary term which is unique to this context, in the sense that no lower-order term of this shape has appeared in previously studied families. The presence of this new term allows us to deduce an omega result for cubic field counting functions, under the assumption of the Generalised Riemann Hypothesis. We also investigate the associated L-functions Ratios Conjecture and find that it does not predict this new lower-order term. Taking into account the secondary term in Roberts???s conjecture, we refine the Ratios Conjecture to one which captures this new term. Finally, we show that any improvement in the exponent of the error term of the recent Bhargava???Taniguchi???Thorne cubic field counting estimate would imply that the best possible error term in the refined Ratios Conjecture is s ???????? (??????? 1/3 +????). This is in opposition with all previously studied families in which the expected error in the Ratios Conjecture prediction for the one-level density is ???????? (??????? 1/2 +????)