On the Rankin-Selberg problem in families. (English) Zbl 1523.11075
Summary: In this paper, we investigate the Rankin-Selberg problem over short intervals in families of holomorphic modular forms and Hecke-Maass cusp forms. We assume a Lindelöf-on-average bound for holomorphic modular forms. On the other hand, we make no assumptions for Hecke-Maass cusp forms.
MSC:
| 11F25 | Hecke-Petersson operators, differential operators (one variable) |
| 11F66 | Langlands \(L\)-functions; one variable Dirichlet series and functional equations |
| 11N37 | Asymptotic results on arithmetic functions |
| 11N25 | Distribution of integers with specified multiplicative constraints |
| 11N36 | Applications of sieve methods |
Software:
GL(n)packReferences:
| [1] | Blomer, V.; Buttcane, J.; Raulf, N., A Sato-Tate law for \(\operatorname{GL}(3)\), Comment. Math. Helv., 89, 895-919 (2014) · Zbl 1317.11053 |
| [2] | Goldfeld, D., Automorphic Forms and L-Functions for the Group \(\operatorname{GL}(n, \operatorname{R})\), Cambridge Studies in Advanced Mathematics, vol. 99 (2015), Cambridge University Press: Cambridge University Press Cambridge, with an appendix by Kevin A. Broughan, paperback edition of the 2006 original [MR2254662] · Zbl 1320.11034 |
| [3] | Goldfeld, D.; Stade, E.; Woodbury, M., An asymptotic orthogonality relation for \(\operatorname{GL}(n, \mathbb{R})\), arXiv math.NT (2023) |
| [4] | Gorodetsky, O.; Mangerel, A.; Rodgers, B., Squarefrees are Gaussian in short intervals, J. Reine Angew. Math., 795, 1-44 (2023) · Zbl 1522.11093 |
| [5] | Gorodetsky, O.; Matomäki, K.; Radziwiłł, M.; Rodgers, B., On the variance of squarefree integers in short intervals and arithmetic progressions, Geom. Funct. Anal., 31, 111-149 (2021) · Zbl 1473.11180 |
| [6] | Harman, G., Prime-Detecting Sieves, London Mathematical Society Monographs Series, vol. 33 (2007), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 1220.11118 |
| [7] | Hoffstein, J.; Lockhart, P., Coefficients of Maass forms and the Siegel zero, Ann. Math. (2), 140, 161-181 (1994), with an appendix by Dorian Goldfeld, Hoffstein and Daniel Lieman · Zbl 0814.11032 |
| [8] | Huang, B., On the Rankin-Selberg problem, Math. Ann., 381, 1217-1251 (2021) · Zbl 1483.11098 |
| [9] | Ivić, A., Large values of certain number-theoretic error terms, Acta Arith., 56, 135-159 (1990) · Zbl 0659.10053 |
| [10] | Ivić, A., On the Rankin-Selberg problem in short intervals, Mosc. J. Comb. Number Theory, 2, 3-17 (2012) · Zbl 1284.11125 |
| [11] | Iwaniec, H.; Kowalski, E., Analytic Number Theory, American Mathematical Society Colloquium Publications, vol. 53 (2004), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 1059.11001 |
| [12] | Khan, R.; Young, M. P., Moments and hybrid subconvexity for symmetric-square L-functions, J. Inst. Math. Jussieu, 1-45 (2021) |
| [13] | Kim, H. H., Functoriality for the exterior square of \(\operatorname{GL}_4\) and the symmetric fourth of \(\operatorname{GL}_2\), J. Am. Math. Soc., 16, 139-183 (2003), with appendix 1 by Dinakar Ramakrishnan and appendix 2 by Kim and Peter Sarnak · Zbl 1018.11024 |
| [14] | Kim, J., On the asymptotics of the shifted sums of Hecke eigenvalue squares, Forum Math., 35, 297-328 (2023) · Zbl 1523.11079 |
| [15] | Mangerel, A. P., Divisor-bounded multiplicative functions in short intervals, Res. Math. Sci., 10, Article 12 pp. (2023) · Zbl 1527.11072 |
| [16] | Matomäki, K.; Radziwiłł, M.; Tao, T., Correlations of the von Mangoldt and higher divisor functions II: divisor correlations in short ranges, Math. Ann., 374, 793-840 (2019) · Zbl 1416.11138 |
| [17] | Montgomery, H. L., Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, CBMS Regional Conference Series in Mathematics, vol. 84 (1994), American Mathematical Society: American Mathematical Society Providence, RI, Published for the Conference Board of the Mathematical Sciences, Washington, DC · Zbl 0814.11001 |
| [18] | Rankin, R. A., Contributions to the theory of Ramanujan’s function \(\tau(n)\) and similar arithmetical functions. I. The zeros of the function \(\sum_{n = 1}^\infty \tau(n) / n^s\) on the line \(\operatorname{Re} s = 13 / 2\). II. The order of the Fourier coefficients of integral modular forms, Proc. Camb. Philos. Soc., 35, 351-372 (1939) · JFM 65.0353.01 |
| [19] | Selberg, A., Bemerkungen über eine Dirichletsche Reihe, die mit der Theorie der Modulformen nahe verbunden ist, Arch. Math. Naturvidensk., 43, 47-50 (1940) · JFM 66.0377.01 |
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.
