Algebra & Number Theory

The sixth moment of automorphic $L$-functions

Vorrapan Chandee and Xiannan Li

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Abstract

We consider the L-functions L(s,f), where f is an eigenform for the congruence subgroup Γ1(q). We prove an asymptotic formula for the sixth moment of this family of automorphic L-functions.

Article information

Source
Algebra Number Theory, Volume 11, Number 3 (2017), 583-633.

Dates
Received: 16 February 2016
Revised: 25 July 2016
Accepted: 16 December 2016
First available in Project Euclid: 19 October 2017

Permanent link to this document
https://projecteuclid.org/euclid.ant/1508431774

Digital Object Identifier
doi:10.2140/ant.2017.11.583

Mathematical Reviews number (MathSciNet)
MR3649362

Zentralblatt MATH identifier
06722476

Subjects
Primary: 11M41: Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72}
Secondary: 11F11: Holomorphic modular forms of integral weight 11F67: Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols

Keywords
moment of $L$-functions automorphic $L$-functions $\Gamma_1(q)$

Citation

Chandee, Vorrapan; Li, Xiannan. The sixth moment of automorphic $L$-functions. Algebra Number Theory 11 (2017), no. 3, 583--633. doi:10.2140/ant.2017.11.583. https://projecteuclid.org/euclid.ant/1508431774


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References

  • V. Chandee and X. Li, “The eighth moment of Dirichlet $L$-functions”, Adv. Math. 259 (2014), 339–375.
  • J. B. Conrey and A. Ghosh, “A conjecture for the sixth power moment of the Riemann zeta-function”, Internat. Math. Res. Notices 15 (1998), 775–780.
  • J. B. Conrey and S. M. Gonek, “High moments of the Riemann zeta-function”, Duke Math. J. 107:3 (2001), 577–604.
  • J. B. Conrey, D. W. Farmer, J. P. Keating, M. O. Rubinstein, and N. C. Snaith, “Integral moments of $L$-functions”, Proc. London Math. Soc. $(3)$ 91:1 (2005), 33–104.
  • J. B. Conrey, H. Iwaniec, and K. Soundararajan, “The sixth power moment of Dirichlet $L$-functions”, Geom. Funct. Anal. 22:5 (2012), 1257–1288.
  • A. Diaconu, D. Goldfeld, and J. Hoffstein, “Multiple Dirichlet series and moments of zeta and $L$-functions”, Compositio Math. 139:3 (2003), 297–360.
  • G. Djanković, “The sixth moment of the family of $\Gamma_1(q)$-automorphic $L$-functions”, Arch. Math. $($Basel$)$ 97:6 (2011), 535–547.
  • I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, 7th ed., Elsevier, Amsterdam, 2007.
  • A. J. Harper, “Sharp conditional bounds for moments of the Riemann zeta function”, preprint, 2013.
  • A. Ivić, “On the ternary additive divisor problem and the sixth moment of the zeta-function”, pp. 205–243 in Sieve methods, exponential sums, and their applications in number theory (Cardiff, 1995), edited by G. R. H. Greaves et al., London Math. Soc. Lecture Note Ser. 237, Cambridge University Press, 1997.
  • H. Iwaniec and X. Li, “The orthogonality of Hecke eigenvalues”, Compos. Math. 143:3 (2007), 541–565.
  • N. M. Katz and P. Sarnak, Random matrices, Frobenius eigenvalues, and monodromy, American Mathematical Society Colloquium Publications 45, American Mathematical Society, Providence, RI, 1999.
  • J. P. Keating and N. C. Snaith, “Random matrix theory and $\zeta(1/2+it)$”, Comm. Math. Phys. 214:1 (2000), 57–89.
  • F. Oberhettinger, Tables of Bessel transforms, Springer, Berlin, 1972.
  • Z. Rudnick and K. Soundararajan, “Lower bounds for moments of $L$-functions”, Proc. Natl. Acad. Sci. USA 102:19 (2005), 6837–6838.
  • K. Soundararajan, “Moments of the Riemann zeta function”, Ann. of Math. $(2)$ 170:2 (2009), 981–993.
  • E. C. Titchmarsh, The theory of the Riemann zeta-function, 2nd ed., Oxford University Press, 1986.
  • G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, 1944.