A variant of the Linnik-Sprindzuk theorem for simple zeros of Dirichlet L-functions
William D. Banks
TL;DR
The paper develops a Linnik–Sprindžuk–type framework for simple zeros of Dirichlet $L$-functions by introducing RH_sim^\dagger and showing it is controlled by the generalized Lindelöf hypothesis LH^*. Under LH^*, RH_sim^\dagger for one primitive character implies RH_sim for all Dirichlet $L$-functions, hence all simple zeros lie on the critical line and Siegel zeros are ruled out. Central to the approach is the auxiliary Dirichlet series $D(s,\chi)=\frac{L'(s,\chi)^2}{L(s,\chi)}$, which encodes simple zeros as poles and interacts with twisted sums involving the arithmetical function $\ell(n)=(\Lambda*\log)(n)$. The authors establish equivalences between RH_sim, smoothed prime-sum criteria, and twisted-sum identities, and they derive explicit main-term formulas for twisted sums under RH_sim^\star and LH^\star, enabling a transfer of simple-zero information across characters. The results yield a robust criterion for the nonexistence of Siegel zeros and provide a mechanism to propagate information about simple zeros across the Dirichlet $L$-function family, with potential implications for zero-density estimates and GRH-related goals.
Abstract
For a primitive Dirichlet character $X$, a new hypothesis $RH_{sim}^\dagger[X]$ is introduced, which asserts that (1) all simple zeros of $L(s,X)$ in the critical strip are located on the critical line, and (2) these zeros satisfy some specific conditions on their vertical distribution. We show that $RH_{sim}^\dagger[X]$ (for any $X$) is a consequence of the generalized Riemann hypothesis. Assuming only the generalized Lindelöf hypothesis, we show that if $RH_{sim}^\dagger[X]$ holds for one primitive character $X$, then it holds for every such $X$. If this occurs, then for every character $χ$ (primitive or not), all simple zeros of $L(s,χ)$ in the critical strip are located on the critical line. In particular, Siegel zeros cannot exist in this situation.