Oscillations of coefficients of symmetric square L-functions over primes

Abstract

Let L(s, sym²f) be the symmetric-square L-function associated to a primitive holomorphic cusp form f for SL(2,\mathbf{\mathbb{Z}}), with t_f(n,1) denoting the nth coefficient of the Dirichlet series for it. It is proved that, for N≥2 and any α ∈ ℝ, there exists an effective positive constant c such that {\displaystyle {\sum }_{n\le N}\mathrm{\Lambda }(n)t_f(n,\mathrm{1})e(n\alpha )\ll N\exp (-c\sqrt{\log N})}, where Λ(n) is the von Mangoldt function, and the implied constant only depends on f. We also study the analogue of Vinogradov’s three primes theorem associated to the coefficients of Rankin-Selberg L-functions.

Keywords

symmetric-square L-function / primitive holomorphic cusp form / Fourier coefficient

References

[1] Davenport H. Multiplicative Number Theory. 3rd ed. Berlin: Springer-Verlag, 2000

[2] Deligne P. La conjecture de Weil I. Publ Math Inst Hautes Études Sci, 1974, 43: 273−307

[3] Fouvry É, Ganguly S. Strong orthogonality between the Möbius function, additive characters, and Fourier coefficients of cusp forms. Compos Math, 2014, 150: 763−797

[4] Goldfeld D. Automorphic Forms and L-Functions for the Group GL(n, R). Cambridge: Cambridge Univ Press, 2006

[5] Hoffstein J, Lockhart P. Coefficients of Maass forms and the Siegel zero. Ann Math, 1994, 140: 161−181

[6] Iwaniec H, Kowalski E. Analytic Number Theory. Amer Math Soc Colloq Publ, Vol 53. Providence: Amer Math Soc, 2004

[7] Lau Y K, Lü G. Sums of Fourier coefficients of cusp forms. Q J Math, 2011, 62: 687−716

[8] Liu J, Ye Y. Perron’s formula and the prime number theorem for automorphic L-functions. Pure Appl Math Q, 2007, 3: 481−497

[9] Perelli A. On some exponential sums connected with Ramanujan’s τ -function. Mathematika, 1984, 31: 150−158

[10] Stephen M D. Cancellation in additively twisted sums on GL(n). Amer J Math, 2006, 128: 699−729

[11] Vaughan R C. An elementary method in prime number theory. Acta Arith, 1980, 37: 111−115

[12] Vinogradov I M. Some theorems concerning the theory of primes. Mat Sb (N S), 1937, 2: 179−195

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Higher Education Press and Springer-Verlag Berlin Heidelberg