A shortened equation for convolutions

A. I. Vinogradov

Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 3, Tome 211 (1994), pp. 104-119

Résumé

The zeta functions of convolutions are Dirichlet series of the general form $\sum^\infty a_n\cdot n^{-s}$ therefore, they are well convergent in the right half-plane $\operatorname{Re}s>1$. In the critical strip $\operatorname{re}s\in(0,1)$ the convolutions can be represented in terms of the Linnik–Selberg zeta functions whose coefficients are Kloosterman sums. In the present paper, these two representations are combined into a single representation in the same way as the shortened equation for the classical Riemann zeta function. Bibliography: 10 titles.