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Symmetric Squares of Hecke L-Functions and Fourier Coefficients of Cusp Forms

Authors:

O. M. Fomenko

Affiliation:

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

Abstract:

Let S_k(

₀(N)) be the space of cusp forms of even weight k for ₀(N), let $mathcal{F}_0$ be the set of all newforms in S_k( Gcy

₀(N)), and let $mathcal{H}_2 (s,f)$ be the symmetric square of the Hecke L-function of a form $f in mathcal{F}_0$ . It is proved that for N=p we have

$sumlimits_{f in mathcal{F}_0 ,mathcal{H}_2 (1/2,f) ne 0} {1 gg N^{1 - varepsilon } ,}$

where the

-constant depends only on epsiv

and k. Let f(z)S_k(0(N)):

$f(x) = sumlimits_{n = 1}^infty {a_f (n)e^{2pi inz} ,{text{ }}a_f (n)n^{ - (k - 1)/2} = b_f (n).}$

The distribution of values of the sums

$sumlimits_{n leqslant X} {b_f (n){text{ and }}sumlimits_{n leqslant X} {b_f (n)^2 } }$

for increasing X and N is studied. Bibliography: 13 titles.

Keywords:

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