Frontiers of Mathematics in China >
Oscillations of coefficients of symmetric square L-functions over primes
Received date: 30 Jul 2014
Accepted date: 05 Dec 2014
Published date: 12 Oct 2015
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Let L(s, sym2f) be the symmetric-square L-function associated to a primitive holomorphic cusp form f for SL(2,), with tf(n,1) denoting the nth coefficient of the Dirichlet series for it. It is proved that, for N≥2 and any α ∈ , there exists an effective positive constant c such that , 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.
Fei HOU . Oscillations of coefficients of symmetric square L-functions over primes[J]. Frontiers of Mathematics in China, 2015 , 10(6) : 1325 -1341 . DOI: 10.1007/s11464-015-0442-6
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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