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The fourth moment of quadratic Dirichlet L-functions over function fields

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Abstract

We obtain an asymptotic formula for the fourth moment of quadratic Dirichlet L-functions over \({\mathbb{F}_q[x]}\), as the base field \({\mathbb{F}_q}\) is fixed and the genus of the family goes to infinity. According to conjectures of Andrade and Keating, we expect the fourth moment to be asymptotic to \({q^{2g+1} P(2g+1)}\) up to an error of size \({o(q^{2g+1})}\), where P is a polynomial of degree 10 with explicit coefficients. We prove an asymptotic formula with the leading three terms, which agrees with the conjectured result.

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References

  1. Andrade J.C., Keating J.P.: The mean value of \({L(\frac12,\chi)}\) in the hyperelliptic ensemble. Journal of Number Theory 132(12), 2793–2816 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  2. Andrade J.C., Keating J.P.: Conjectures for the integral moments and ratios of L-functions over function fields. Journal of Number Theory 142, 102–148 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  3. Andrade J.C., Keating J.P.: Mean value theorems for L-functions over prime polynomials for the rational function field. Acta Arithmetica 161(4), 371–385 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. BucurA. Diaconu A.: Moments of quadratic Dirichlet L-functions over rational function fields. Moscow Mathematical Journal 10(3), 485–517 (2010)

    MathSciNet  MATH  Google Scholar 

  5. Carneiro E., Chandee V.: Bounding \({\zeta(s)}\) in the critical strip. Journal of Number Theory 131(3), 363–384 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Carneiro E., Littmann F., Vaaler J.D.: Gaussian subordination for the Beurling-Selberg extremal problem. Transactions of the American Mathematical Society 365(7), 3493–3534 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chandee V.: On the correlation of shifted values of the Riemann zeta function. The Quarterly Journal of Mathematics 62(3), 545–572 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. V. Chandee and K. Soundararajan. Bounding \({\vert \zeta(\frac12+it)\vert}\) on the Riemann hypothesis. Bulletin of the London Mathematical Society, (2)43 (2011), 243–250.

  9. J.B. Conrey, D.W. Farmer, J.P. Keating, M.O. Rubinstein, and N.C. Snaith. Integral moments of L-functions. Proceedings of the London Mathematical Society (3), (1)91 (2005), 33–104.

  10. Diaconu A., Goldfeld D., Hoffstein J.: Multiple Dirichlet series and moments of zeta and L-functions. Compositio Mathematica 139(3), 297–360 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  11. Faifman D., Rudnick Z.: Statistics of the zeros of zeta functions in families of hyperelliptic curves over a finite field. Compositio Mathematica 146(1), 81–101 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. A. Florea. Improving the error term in the mean value of \({L(1/2,\chi_D)}\) in the hyperelliptic ensemble. International Mathematics Research Notices, (2016).

  13. A. Florea. The second and third moment of \({L(1/2,\chi_D)}\) in the hyperelliptic ensemble. Forum Mathematicum, (2016).

  14. Hardy G.H., Littlewood J.E.: Contributions to the theory of the Riemann zeta-function and the theory of the distribution of primes. Acta Mathematica 41(1), 119–196 (1916)

    Article  MathSciNet  MATH  Google Scholar 

  15. A. Harper. Sharp conditional bounds for moments of the Riemann zeta function. preprint, (2013).

  16. Hayes D.R.: The expression of a polynomial as a sum of three irreducibles. Acta Arithmetica 11, 461–488 (1966)

    MathSciNet  MATH  Google Scholar 

  17. Ingham A.E.: Mean-value theorems in the theory of the Riemann zeta-function. Proceedings of the London Mathematical Society 27, 273–300 (1926)

    MathSciNet  MATH  Google Scholar 

  18. M. Jutila. On the mean value of \({L({1\over 2},\,\chi )}\) for real characters. Analysis, (2)1 (1981), 149–161.

  19. N.M. Katz and P. Sarnak. Random matrices, Frobenius eigenvalues, and monodromy, vol. 45 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI (1999).

  20. Katz N.M., Sarnak P.: Zeroes of zeta functions and symmetry. Bulletin of the American Mathematical Society (N.S.) 36(1), 1–26 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  21. Keating J.P., Snaith N.C.: Random matrix theory and L-functions at \({s=1/2}\). Communications in Mathematical Physics 214(1), 91–110 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  22. Keating J.P., Snaith N.C.: Random matrix theory and \({\zeta(1/2+it)}\). Communications in Mathematical Physics 214(1), 57–89 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  23. M. Rosen. Number Theory in Function Fields, vol. 210 of Graduate Texts in Mathematics. Springer, New York (2002).

  24. M.O. Rubinstein and K. Wu. Moments of zeta functions associated to hyperelliptic curves over finite fields. Philosophical transactions of the Royal Society A, (2040)373 (2015), 20140307.

  25. K. Soundararajan. Nonvanishing of quadratic Dirichlet L-functions at \({s=\frac12}\). Annals of Mathematics (2), (2)152 (2000), 447–488.

  26. Soundararajan K., Young M.P.: The second moment of quadratic twists of modular L-functions. Journal of the European Mathematical Society 12(5), 1097–1116 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  27. Soundararajan K.: Moments of the Riemann zeta function. Annals of Mathematics (2) 170(2), 981–993 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  28. A. Weil. Sur les courbes algébriques et les variétés qui s’en déduisent. Actualités Sci. Ind., no. 1041, Publ. Inst. Math. Univ. Strasbourg 7. Hermann et Cie., Paris (1948).

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  1. Department of Mathematics, Stanford University, Stanford, CA, 94305, USA

    Alexandra Florea

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  1. Alexandra Florea
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Florea, A. The fourth moment of quadratic Dirichlet L-functions over function fields. Geom. Funct. Anal. 27, 541–595 (2017). https://doi.org/10.1007/s00039-017-0409-8

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  • DOI: https://doi.org/10.1007/s00039-017-0409-8

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