Dirichlet twists of ${\mathrm{GL}}_n$-automorphic $L$-functions and hyper-Kloosterman Dirichlet series

Van Order, Jeanine

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 30 (2021) no. 3, pp. 633-703.

Nous calculons les valuers moyennes des fonctions $L$ automorphes sur ${\mathrm{GL}}_n$ tordues par des caractères de Dirichlet primitifs et pairs, du conducteur une puissance d’un nombre premier, à des points arbitraires dans la bande critique, en dérivant des formules de sommation spéciales du type Voronoi. Notre calcul est nouveau car la somme est exprimé en termes de la moyenne elle-même, et aussi qu’il voit la dérivation de diverses nouvelles formules de sommation dans le regime des puissances d’un nombre premier. Une conséquence, comme nous l’expliquons, est de montrer les prolongations analytiques et des formules de sommation additive pour les séries de Dirichlet hyper-Kloosterman associées aux fonctions $L$ automorphes sur ${\mathrm{GL}}_n$.

We calculate mean values of ${\mathrm{GL}}_n$-automorphic $L$-functions twisted by primitive even Dirichlet characters of prime-power conductor, at arbitrary points within the critical strip, by derivation of special Voronoi summation formulae. Our calculation is novel in that the twisted sum can be expressed in terms of the average itself, and also that it sees the derivation of various new summation formulae in the setting of prime-power modulus. One consequence, as we explain, is to show the analytic continuation and additive summation formulae for hyper-Kloosterman Dirichlet series associated to ${\mathrm{GL}}_n$-automorphic $L$-functions.

[1] Brumley, Farrell; Blomer, Valentin Estimations élémentaires des sommes de Kloosterman multiples, Le spectre des surfaces hyperboliques (Savoirs actuels), CNRS Éditions, 2011 (Appendix)

[2] Goldfeld, Dorian; Li, Xiaoqing Voronoi formulas on $\mathrm{GL}\left(n\right)$, Int. Math. Res. Not., Volume 2006 (2006) no. 10, 86295, 25 pages | MR | Zbl

[3] Goldfeld, Dorian; Li, Xiaoqing The Voronoi formula for $\mathrm{GL}(n,ℝ)$, Int. Math. Res. Not., Volume 2008 (2008), rnm144, 39 pages | Zbl

[4] Ichino, Atsushi; Templier, Nicolas On the Voronoi formula for $\mathrm{GL}\left(n\right)$, Am. J. Math., Volume 135 (2013) no. 1, pp. 65-101 | DOI | MR | Zbl

[5] Iwaniec, Henryk; Kowalski, Emmanuel Analytic Number Theory, Colloquium Publications, 53, American Mathematical Society, 2004 | MR | Zbl

[6] Kim, Henry H.; Sarnak, Peter Refined estimates towards the Ramanujan and Selberg Conjectures, J. Am. Math. Soc., Volume 16 (2003) no. 1, pp. 139-183 Appendix to H. Kim, “Functoriality for the exterior square of $\mathrm{GL}\left(4\right)$ and symmetric fourth of $\mathrm{GL}\left(2\right)$”

[7] Kıral, Eren Mehmet; Zhou, Fan The Voronoi Formula and double Dirichlet series, Algebra Number Theory, Volume 10 (2016) no. 10, pp. 2267-2286 | DOI | MR | Zbl

[8] Lavrik, A. F. Approximate functional equations of Dirichlet functions, Izv. Akad. Nauk SSSR, Ser. Mat., Volume 32 (1966), pp. 134-185 | MR

[9] Li, Xiannan Upper Bounds on $L$-Functions at the Edge of the Critical Strip, Int. Math. Res. Not., Volume 2010 (2010) no. 4, pp. 727-755 | MR | Zbl

[10] Luo, Wenzhi; Rudnick, Zeév; Sarnak, Peter On Selberg’s Eigenvalue Conjecture, Geom. Funct. Anal., Volume 5 (1995) no. 2, pp. 387-401 | MR

[11] Miller, Stephen D.; Schmid, Wilfried A general Voronoi summation formula for $\mathrm{GL}(n,\mathbf{Z})$, Geometric analysis: Present and future (Harvard University, Cambridge, USA, 2008) (Advanced Lectures in Mathematics), Volume 18, International Press, 2008 | Zbl

[12] Miller, Stephen D.; Zhou, Fan The balanced Voronoi formulas for $\mathrm{GL}\left(n\right)$, Int. Math. Res. Not., Volume 2019 (2019) no. 11, pp. 3473-3484 | DOI | MR | Zbl

[13] Molteni, Giuseppe Upper and lower bounds at $s=1$ for certain Dirichlet series with Euler product, Duke Math. J., Volume 111 (2002) no. 1, pp. 133-158 | DOI | MR | Zbl

[14] Schneider, Peter; Teitelbaum, Jeremy $p$-adic Fourier Theory, Doc. Math., Volume 6 (2001), pp. 447-481 | MR

[15] Selberg, Atle On the estimation of Fourier coefficients of modular forms, Theory of numbers (California Institute of Technology, Pasadena, USA, 1963) (Proceedings of Symposia in Pure Mathematics), Volume 8, American Mathematical Society, 1965, pp. 1-15 | Zbl

[16] Voronoĭ, Georgiĭ Sur une fonction transcendante et ses applications à la sommation de quelques séries, Ann. Sci. Éc. Norm. Supér., Volume 21 (1904), pp. 207-267 | DOI | Numdam | MR | Zbl