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Identities Involving Coefficients of Automorphic L-Functions

Authors:

O M Fomenko

Institution:

(1) St.Petersburg Department, Steklov Mathematical Institute, St.Petersburg, Russia

Abstract:

Let f(z) be a holomorphic Hecke eigenform of weight k with respect to SL(2, ℤ) and let

$L(s,\operatorname{sym} ^2 f) = \sum\limits_{n = 1}^\infty {c_n n^{ - s} } ,\quad \operatorname{Re} s > 1,$

denote the symmetric square L-function of f. A Voronoi type formula for

$C(x) = \sum\limits_{n \leqslant x} {c_n }$

and the relation

$C(x) = \Omega _ \pm (x^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}} )$

are proved. Heuristic approaches to estimation of exponential sums arising in this connection are considered. Bibliography: 9 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 314, 2004, pp. 247–256.

Keywords:

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