Spectral decomposition of convolutions

A. I. Vinogradov

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 12, Tome 212 (1994), pp. 71-90

Résumé

We obtain a spectral decomposition for number-theoretic convolutions of the form $\tau(n)\times\tau(n\pm l;\chi_q)$ where the function $\tau$ is the number of divisors, $\chi_q$ is the quadratic Dirichlet character of module $q$, $l$ is a fixed shift, and $n$ is the summation parameter. This is done by using the shortened functional equation for the convolution, obtained by the author (Zap. Nauchn. Semin. POMI, 211, 104–119 (1994)). A presentation using the vector-matrix language is conducted for two convolutions with shifts $\pm l$ simultaneously, which simplifies symbolic writing of the spectral decompositions. Bibliography: 5 titles.