On the non-vanishing of certain Dirichlet series

Given $k\in\mathbb N$, we study the vanishing of the Dirichlet series $$D_k(s,f):=\sum_{n\geq1} d_k(n)f(n)n^{-s}$$ at the point $s=1$, where $f$ is a periodic function modulo a prime $p$. We show that if $(k,p-1)=1$ or $(k,p-1)=2$ and $p\equiv 3\mod 4$, then there are no odd rational-valued functions $f\not\equiv 0$ such that $D_k(1,f)=0$, whereas in all other cases there are examples of odd functions $f$ such that $D_k(1,f)=0$. As a consequence, we obtain, for example, that the set of values $L(1,\chi)^2$, where $\chi$ ranges over odd characters mod $p$, are linearly independent over $\mathbb Q$.Comment: 16 page

The first moment of twisted Hecke LL-functions with unbounded shifts

We compute the first moment of twisted Hecke $L$-functions of prime power level going to infinity, uniformly in the conductor of the twist and in the vertical shift.Comment: 8 page

Statistical distribution of the Stern sequence

We prove that the Stern diatomic sequence is asymptotically distributed according to a normal law, on a logarithmic scale. This is obtained by studying complex moments, and the analytic properties of a transfer operator.Comment: 13 page

An optimal choice of Dirichlet polynomials for the Nyman-Beurling criterion

We give a conditional result on the constant in the B\'aez-Duarte reformulation of the Nyman-Beurling criterion for the Riemann Hypothesis. We show that assuming the Riemann hypothesis and that $\sum_{\rho}\frac{1}{|\zeta'(\rho)|^2}\ll T^{3/2-\delta}$, for some $\delta>0$, the value of this constant coincides with the lower bound given by Burnol.Comment: 9 page