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JOURNALS // Zapiski Nauchnykh Seminarov POMI // Archive

Zap. Nauchn. Sem. POMI, 1994 Volume 212, Pages 71–90 (Mi znsl5897)

This article is cited in 1 paper

Spectral decomposition of convolutions

A. I. Vinogradov

St. Petersburg Department of V. A. Steklov Institute of Mathematics of the Russian Academy of Sciences

Abstract: 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.

UDC: 511.512

Received: 20.09.1993


 English version:
Journal of Mathematical Sciences, 1997, 83:6, 731–744

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