Additive divisor problem and Applications

Additive Divisor Problem (ADP) is concerned with finding an asymptotic formula for the sum $\sum_{n<X}d(n)d(n+a)$, where $d(n)=\sum_{d|n}1$ is the divisor function. Surprisingly, the ADP arises naturally in quite different problems of number theory. For example, it is related to the investigation of the 4th moment of the Riemann zeta-function, the second moment of automorphic $L$-functions and the mean values of the length of continued fractions. In the talk, I will describe the ADP and its applications.