Spectral methods in arithmetic problems

Combining ideas of convolution due to Rankin with spectral considerations of Selberg, the author proposes a new approach to obtaining mean values for certain number-theoretic functions $f(n)$. This approach is illustrated for the examples of functions $f(n)=\tau(Mn^2+N)$, $\tau(n)\tau(Mn+N)$, where $\tau(n)$ is the number of divisors of $n$.