On the error term in a mixed moment of \(L\)-functions. (English) Zbl 07738283
In this paper under review, the authors study (see Theorem 1.1) the error term in the mixed moment asymptotic of
\[
\frac{1}{\varphi^{*}(q)}\sum_{\chi\bmod q} ^{\star}L\left((\frac{1}{2},f\otimes\chi\right)\overline{L\left(\frac{1}{2},\chi\right)}^{2}
\]
for a prime \(q\) and \(f\) a holomorphic Hecke-cusp form or a Hecke Maass-cusp form, for \(SL_{2}(\mathbb{Z})\). In fact, they make a more significant improvement to that established by V. Blomer et al. in their paper: [Am. J. Math. 139, No. 3, 707–768 (2017; Zbl 1476.11081)].
Reviewer: Kamel Mazhouda (Monastir)
MSC:
| 11M06 | \(\zeta (s)\) and \(L(s, \chi)\) |
| 11M41 | Other Dirichlet series and zeta functions |
| 11F11 | Holomorphic modular forms of integral weight |
| 11F12 | Automorphic forms, one variable |
Citations:
Zbl 1476.11081References:
| [1] | V.Blomer, É.Fouvry, E.Kowalski, P.Michel, and D.Milićević, On moments of twisted L‐functions, Amer. J. Math.139 (2017), no. 3, 707-768. · Zbl 1476.11081 |
| [2] | V.Blomer, É.Fouvry, E.Kowalski, P.Michel, and D.Milićević, Some applications of smooth bilinear forms with Kloosterman sums, Tr. Mat. Inst. Steklova296 (2017), no. Analiticheskaya i Kombinatornaya Teoriya Chisel, 24-35, English version published in Proc. Steklov Inst. Math. 296 (2017), no. 1, 18-29. · Zbl 1376.11063 |
| [3] | V.Blomer, É.Fouvry, E.Kowalski, P.Michel, D.Milićević, and W.Sawin, The Second moment theory of families of L‐functions: the case of twisted Hecke L‐functions, Mem. Amer. Math. Soc.282 (2023), no. 1394. · Zbl 1519.11001 |
| [4] | É.Fouvry, K.Emmanuel, and P.Michel, Algebraic trace functions over the primes, Duke Math. J.163 (2014), no. 9, 1683-1736. · Zbl 1318.11103 |
| [5] | D. R.Heath‐Brown, The fourth power moment of the Riemann zeta function, Proc. London Math. Soc. (3)38 (1979), no. 3, 385-422. · Zbl 0403.10018 |
| [6] | H.Iwaniec, The spectral growth of automorphic L‐functions, J. Reine Angew. Math.428 (1992), 139-159. · Zbl 0746.11024 |
| [7] | H.Iwaniec, Biblioteca de la Revista Matemática Iberoamericana [Library of the Revista Matemática Iberoamericana], Revista Matemática Iberoamericana, Madrid, 1995. · Zbl 0847.11028 |
| [8] | H.Iwaniec, Topics in classical automorphic forms, Graduate Studies in Mathematics, vol. 17, American Mathematical Society, Providence, RI, 1997. · Zbl 0905.11023 |
| [9] | H. H.Kim, Functoriality for the exterior square of GL_4 and the symmetric fourth of GL_2, J. Amer. Math. Soc.16 (2003), no. 1, 139-183, With appendix 1 by D. Ramakrishnan and appendix 2 by H. Kim and P. Sarnak. · Zbl 1018.11024 |
| [10] | E.Kowalski, P.Michel, and W.Sawin, Bilinear forms with Kloosterman sums and applications, Ann. of Math. (2)186 (2017), no. 2, 413-500. · Zbl 1441.11194 |
| [11] | I. E.Shparlinski, On sums of Kloosterman and Gauss sums, Trans. Amer. Math. Soc.371 (2019), no. 12, 8679-8697. · Zbl 1453.11105 |
| [12] | I. E.Shparlinski and T.Zhang, Cancellations amongst Kloosterman sums, Acta Arith.176 (2016), no. 3, 201-210. · Zbl 1368.11092 |
| [13] | X.Wu, The fourth moment of Dirichlet L‐functions at the central value, Math. Ann.https://doi.org/10.1007/s00208‐022‐02483‐9. · Zbl 1534.11107 · doi:10.1007/s00208‐022‐02483‐9 |
| [14] | M. P.Young, The fourth moment of Dirichlet L‐functions, Ann. of Math. (2)173 (2011), no. 1, 1-50. · Zbl 1296.11112 |
| [15] | R.Zacharias, Mollification of the fourth moment of Dirichlet L‐functions, Acta Arith.191 (2019), no. 3, 201-257. · Zbl 1454.11159 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
