Spectral decomposition of convolutions

Abstract

We obtain a spectral decomposition for number-theoretic convolutions of the form r(n) • r (n • l, Xq), where the function 7" is the number of divisors, X 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)

Convolutions. Comparison of spectral expansions at different vertices

“…In order to reduce the pair (0.1) to the convolutions from [2], we observe that the first convolution in (0.1) coincides with the first convolution from [2]. Therefore, it remains to reduce the second convolution in (0.1) to the second convolution from [2].…”

“…and show that this pair may be reduced to the convolutions from [2,3] up to regular summands in the critical strip.…”

“…Here we preserve the notation of [1][2][3] and carry out a final step in this direction; namely, we compare two spectral expaxmions of convolutions. Our aim in this paper is to express the space of a fixed eigenvalue An in terms of the remaining spectrum, which leads to the Petersson conjecture for the zero weight and to the Gauss-Hasse conjecture [4] for n = 1.…”

“…In [1][2][3], spectral expansions of convolutions are obtained at two different vertices of the fundamental domain Fq of the group P0(q), pr ~ that q ~ 1(4) is prime. Here we preserve the notation of [1][2][3] and carry out a final step in this direction; namely, we compare two spectral expaxmions of convolutions.…”

“…In [2,3], different convolutions convenient for individual vertices are considered. Therefore, before comparing their expansions, we need to prove that they are equal up to regular summands.…”