Zeta function of the additive divisor problem and the spectral expansion of the automorphic Laplacian

A. I. Vinogradov ; L. A. Takhtadzhyan

Zapiski Nauchnykh Seminarov POMI, Automorphic functions and number theory. Part II, Tome 134 (1984), pp. 84-116

Résumé

The representation for zeta function of the additive divisor problem $\zeta_k(s)=\sum_{n=1}^\infty\frac{\tau(n)\tau(n+k)}{n^s}$, $\operatorname{Re}s>1$, in terms of spectral data of the automorphic Laplacian is presented. With its help the meromorphic continuation of $\zeta_k(s)$ into the whole complex plane is proved and an estimate of the order of $\zeta_k(s)$ in the critical strip $0\operatorname{Re}s\leqslant1$ is obtained. Using the method of complex integration the asymptotic formula $$ \sum_{n\leqslant x}\tau(n)\tau(n+k)=xP_k(\log x)+O(x^{\frac23+\varepsilon}),\quad\varepsilon>0, $$ is derived where $P_k(x)$ is a quadratic polynomial.