On the general additive divisor problem

Aleksandar Ivic (1) , Jie Wu (2)

1 Katedra Matematike
2 IECN - Institut Élie Cartan de Nancy

We obtain a new upper bound for $\sum_{h\le H}\Delta_k(N,h)$ for $1\le H\le N$, $k\in \N$, $k\ge3$, where $\Delta_k(N,h)$ is the (expected) error term in the asymptotic formula for $\sum_{N < n\le2N}d_k(n)d_k(n+h)$, and $d_k(n)$ is the divisor function generated by $\zeta(s)^k$. When $k=3$ the result improves, for $H\ge N^{1/2}$, the bound given in the recent work \cite{[1]} of Baier, Browning, Marasingha and Zhao, who dealt with the case $k=3$.

Mots clés: Ramanujan sum; power moments; Riemann zeta-function; general additive divisor problem; divisor function

Domaines: Théorie des nombres [math.NT]

Aleksandar Ivic, Jie Wu. On the general additive divisor problem. Proceedings of the Steklov Institute of Mathematics, 2012, 276, pp.140-148. ⟨hal-00602899v2⟩