Simple Zeros of Primitive Dirichlet L-Functions and the Asymptotic Large Sieve

Abstract

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.

Pair correlation estimates for the zeros of the zeta function via semidefinite programming

“…Results for zeros of Dirichlet L-functions. To obtain averaged bounds for the percentage of simple zeros of primitive Dirichlet L-functions we use the framework established by Chandee, Lee, Liu and Radziwi l l [16]. Let Φ be a real-valued smooth function supported in the interval [a, b] with 0 < a < b < ∞.…”

“…Using the asymptotic large sieve, Chandee, Lee, Liu and Radziwi l l [16] showed the following asymptotic formula under GRH…”

Hilbert spaces and the pair correlation of zeros of the Riemann zeta-function

“…As was suggested in Montgomery's original paper [35], it is interesting to study the pair correlation of zeros of the family of Dirichlet L-functions in q-aspect. Montgomery had in mind improving the analogue of (2.7) for this family of L-functions (see [9,40]), and so it is not surprising that the analogue of Theorems 1 and 2 can also be improved. In this section, we indicate such an improvement.…”

A Note on Simple Zeros of Primitive Dirichlet -Functions

“…The weight emphasises the nontrivial zeros near the real axis. Using his formula, he proved that at least 11/12 of the zeros of c 2015 Australian Mathematical Publishing Association Inc. 0004-9727/2015 $16.00 K. Sono [2] Dirichlet L-functions, averaged over characters (both primitive and nonprimitive ones) and conductors, are simple in a proper sense. Recently Conrey et al [4] developed a new method named the asymptotic large sieve.…”

“…Using their techniques, several remarkable results on primitive L-functions have been obtained. One such example is the main theorem of Chandee et al [2]. Throughout their paper (and this paper), the GRH is assumed.…”