Integral power sums of Fourier coefficients of symmetric square 𝐿-functions

He, Xiaoguang

doi:10.1090/proc/14516

Integral power sums of Fourier coefficients of symmetric square $L$-functions
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by Xiaoguang He

Proc. Amer. Math. Soc. 147 (2019), 2847-2856

DOI: https://doi.org/10.1090/proc/14516

Published electronically: March 26, 2019

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Abstract:

Let $f(z)$ be a holomorphic Hecke eigenform of even weight $k$ for $\operatorname {SL}(2,\mathbb {Z})$, and denote $L(s, \mathrm {sym}^2f)$ be the corresponding symmetric square $L$-function associated to $f$. Suppose that $\lambda _{\mathrm {sym}^2f} (n)$ is the $n$th normalized Fourier coefficient of $L(s, \mathrm {sym}^2f)$. In this paper, we investigate the sum $\sum _{n\leq x}\lambda ^j_{\mathrm {sym}^2f}(n)$ for $j=2,3,4$, and get some new results which improve the previous results.

References

J. Bourgain, Decoupling, exponential sums and the Riemann zeta function, J. Amer. Math. Soc. 30 (2017), no. 1, 205–224. MR 3556291, DOI 10.1090/jams/860
Pierre Deligne, La conjecture de Weil. I, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 273–307 (French). MR 340258
O. M. Fomenko, Identities involving coefficients of automorphic L-functions, J. Math. 57 (2005), 494–505.
O. M. Fomenko, Mean value theorems for automorphic $L$-functions, Algebra i Analiz 19 (2007), no. 5, 246–264 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 19 (2008), no. 5, 853–866. MR 2381948, DOI 10.1090/S1061-0022-08-01024-8
Stephen Gelbart and Hervé Jacquet, A relation between automorphic representations of $\textrm {GL}(2)$ and $\textrm {GL}(3)$, Ann. Sci. École Norm. Sup. (4) 11 (1978), no. 4, 471–542. MR 533066
Aleksandar Ivić, Exponent pairs and the zeta function of Riemann, Studia Sci. Math. Hungar. 15 (1980), no. 1-3, 157–181. MR 681438
Henryk Iwaniec and Emmanuel Kowalski, Analytic number theory, American Mathematical Society Colloquium Publications, vol. 53, American Mathematical Society, Providence, RI, 2004. MR 2061214, DOI 10.1090/coll/053
H. Jacquet and J. A. Shalika, On Euler products and the classification of automorphic representations. I, Amer. J. Math. 103 (1981), no. 3, 499–558. MR 618323, DOI 10.2307/2374103
H. Jacquet and J. A. Shalika, On Euler products and the classification of automorphic forms. II, Amer. J. Math. 103 (1981), no. 4, 777–815. MR 623137, DOI 10.2307/2374050
Henry H. Kim, Functoriality for the exterior square of $\textrm {GL}_4$ and the symmetric fourth of $\textrm {GL}_2$, J. Amer. Math. Soc. 16 (2003), no. 1, 139–183. With appendix 1 by Dinakar Ramakrishnan and appendix 2 by Kim and Peter Sarnak. MR 1937203, DOI 10.1090/S0894-0347-02-00410-1
Henry H. Kim and Freydoon Shahidi, Functorial products for $\textrm {GL}_2\times \textrm {GL}_3$ and the symmetric cube for $\textrm {GL}_2$, Ann. of Math. (2) 155 (2002), no. 3, 837–893. With an appendix by Colin J. Bushnell and Guy Henniart. MR 1923967, DOI 10.2307/3062134
Henry H. Kim and Freydoon Shahidi, Cuspidality of symmetric powers with applications, Duke Math. J. 112 (2002), no. 1, 177–197. MR 1890650, DOI 10.1215/S0012-9074-02-11215-0
Huixue Lao, On the fourth moment of coefficients of symmetric square $L$-function, Chinese Ann. Math. Ser. B 33 (2012), no. 6, 877–888. MR 2996556, DOI 10.1007/s11401-012-0746-8
Huixue Lao, Mark McKee, and Yangbo Ye, Asymptotics for cuspidal representations by functoriality from $GL(2)$, J. Number Theory 164 (2016), 323–342. MR 3474392, DOI 10.1016/j.jnt.2016.01.008
Huixue Lao and Ayyadurai Sankaranarayanan, The average behavior of Fourier coefficients of cusp forms over sparse sequences, Proc. Amer. Math. Soc. 137 (2009), no. 8, 2557–2565. MR 2497466, DOI 10.1090/S0002-9939-09-09855-4
Huixue Lao and Ayyadurai Sankaranarayanan, The distribution of Fourier coefficients of cusp forms over sparse sequences, Acta Arith. 163 (2014), no. 2, 101–110. MR 3200164, DOI 10.4064/aa163-2-1
Y.-K. Lau, G.-S. Lü, and J. Wu, Integral power sums of Hecke eigenvalues, Acta Arith. 150 (2011), no. 2, 193–207. MR 2836386, DOI 10.4064/aa150-2-7
Kohji Matsumoto, The mean values and the universality of Rankin-Selberg $L$-functions, Number theory (Turku, 1999) de Gruyter, Berlin, 2001, pp. 201–221. MR 1822011
R. M. Nunes, On the subconvexity estimate for self-dual $GL(3)$ $L$-functions in the $t$-aspect, arXiv preprint arXiv:1703.04424v1 (2017).
Alberto Perelli, General $L$-functions, Ann. Mat. Pura Appl. (4) 130 (1982), 287–306. MR 663975, DOI 10.1007/BF01761499
Zeév Rudnick and Peter Sarnak, Zeros of principal $L$-functions and random matrix theory, Duke Math. J. 81 (1996), no. 2, 269–322. A celebration of John F. Nash, Jr. MR 1395406, DOI 10.1215/S0012-7094-96-08115-6
Freydoon Shahidi, On certain $L$-functions, Amer. J. Math. 103 (1981), no. 2, 297–355. MR 610479, DOI 10.2307/2374219
Freydoon Shahidi, Third symmetric power $L$-functions for $\textrm {GL}(2)$, Compositio Math. 70 (1989), no. 3, 245–273. MR 1002045
Hengcai Tang, Estimates for the Fourier coefficients of symmetric square $L$-functions, Arch. Math. (Basel) 100 (2013), no. 2, 123–130. MR 3020126, DOI 10.1007/s00013-013-0481-8