## Dirichlet series associated to periodic arithmetic functions and the zeros of Dirichlet $$L$$-functions.(English)Zbl 1035.11041

Dubickas, A. (ed.) et al., Analytic and probabilistic methods in number theory. Proceedings of the third international conference in honour of J. Kubilius, Palanga, Lithuania, September 24–28, 2001. Vilnius: TEV (ISBN 9955-491-30-2/hbk). 282-296 (2002).
Let $$f:\mathbb{Z}\to\mathbb{C}$$ and $$f(n+q)=f(n)$$ for $$n\in\mathbb{N}$$. The associated Dirichlet series is $$L(s, f)=\sum_{n=1}^{\infty}f(n)/n^s.$$ It can be continued analytically throughout $$\mathbb{C}$$ with at most one simple pole at $$s=1$$. Let $$\rho_\chi=\beta_\chi+i\gamma_\chi$$ denote the nontrivial zeros of the Dirichlet $$L$$-function $$L(s, \chi)$$ attached to a primitive Dirichlet character $$\chi\bmod Q$$ or of the Riemann zeta-function $$\zeta(s)= L(s, \chi\bmod1)$$. Continuing his researches [Abh. Math. Semin. Univ. Hamb. 71, 113–121 (2001; Zbl 1010.11049), Ramanujan J. 6, 295–306 (2002; Zbl 1069.11038)] the author proves that for $$T\to\infty$$ $\sum_{| \gamma_\chi| \leq T}L(\rho_\chi, f)=\left(f(1)-{1\over\phi([q,Q])} \sum _{\substack{ a\bmod [q,Q]\\ (a, [q,Q])=1}} (\overline{\chi}f)(a)\right) {T\over \pi}\log T +O(T),$ where $$\varphi(n)$$ is Euler’s totient and $$[q, Q]$$ denotes the least common multiple of $$q$$ and $$Q$$. From this he derives a necessary condition for the quotient $$L(s, f)/L(s, \chi)$$ to be an entire function. For some cases entire quotients $$L(s, f)/L(s, \chi)$$ are calculated explicitly.
For the entire collection see [Zbl 1006.00014].

### MSC:

 11M41 Other Dirichlet series and zeta functions 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$

### Citations:

Zbl 1010.11049; Zbl 1069.11038