Advertisement

On mean values of mollifiers and L-functions associated to primitive cusp forms

Article

Abstract

We study the second moment of the L-function associated to a holomorphic primitive cusp form of even weight perturbed by a new family of mollifiers. This family is a natural extension of the mollifers considered by Conrey and by Bui, Conrey and Young. As an application, we improve the current lower bound on critical zeros of holomorphic primitive cusp forms.

Keywords

Dirichlet polynomial Mollifier Zeros on the critical line Ratios conjecture technique Autocorrelation Holomorphic cusp form Modular forms Generalized Möbius functions 

Mathematics Subject Classification

Primary 11M26 Secondary 11M06 11N64 

Notes

Acknowledgements

The first author wishes to acknowledge partial support from SNF grant PP00P2 138906. The second author wishes to thank Keiju Sono for a cordial correspondence while working on similar results. Sono’s results in [31] for the Riemann zeta-function overlap with our computations and these were produced independently of ours. The authors are extremely grateful to the anonymous referees for their comments and suggestions. Their corrections have removed inaccuracies and greatly increased the clarity of the manuscript.

References

  1. 1.
    Abramowitz, M., Stegun, I.A. (eds.): Confluent hypergeometric functions in Ch 13 of handbook of mathematical functions with formulas, graphs, and mathematical tables, 9th printing. Dover, New York (1972)Google Scholar
  2. 2.
    Anderson, R.J.: Simple zeros of the Riemann zeta-function. J. Number Theory 17, 176–182 (1983)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Balasubramanian, R., Conrey, B., Heath-Brown, D.R.: Asymptotic mean square of the product of the Riemann zeta-function and a Dirichlet polynomial. J. Reine Angew. Math. 357, 161–181 (1985)MathSciNetMATHGoogle Scholar
  4. 4.
    Bernard, D.: Modular case of Levinson’s theorem. Acta Arith. 167(3), 201–237 (2015)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Blomer, V.: Shifted convolution sums and subconvexity bounds for automorphic L-functions. Int. Math. Res. Notices 2004(73), 3905–3926 (2004).  https://doi.org/10.1155/S1073792804142505 MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Blomer, V.: Rankin–Selberg \(L\)-functions on the critical line. Manuscr. Math. 117, 111–133 (2005)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Bui, H.M., Conrey, B., Young, M.P.: More than 41% of the zeros of the zeta function are on the critical line. Acta Arith. 150(1), 35–64 (2011)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Chaubey, S., Malik, A., Robles, N., Zaharescu, A.: Zeros of normalized combinations of \(\xi ^{(k)}(s)\) on \({Re} (s)=1/2\). J. Math. Anal. Appl. 461, 1771–1785 (2018)Google Scholar
  9. 9.
    Conrey, J.B.: Zeros of derivatives of the Riemann’s \(\xi \)-function on the critical line. J. Number Theory 16, 49–74 (1983)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Conrey, J.B.: More than two fifths of the zeros of the Riemann zeta function are on the critical line. J. Reine Angew. Math. 399, 1–26 (1989)MathSciNetMATHGoogle Scholar
  11. 11.
    Conrey, J.B., Farmer, D.W., Zirnbauer, M.R.: Autocorrelation of ratios of \(L\)-functions. Comm. Number Theory Phys. 94(2), 593–636 (2008)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Conrey, J.B., Snaith, N.C.: Applications of the \(L\)-functions ratios conjectures. Proc. Lond. Math. Soc. 94(3), 594–646 (2007)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Deligne, P.: La conjecture de Weil. I. Publications Mathématiques de l’IHÉS 43, 273–307 (1974)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Deshouillers, J.M., Iwaniec, H.: Kloosterman sums and Fourier coefficients of cusp forms. Invent. Math. 70, 219–288 (1982)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Deshouillers, J.M., Iwaniec, H.: Power mean values of the Riemann zeta function II. Acta Arith. 48, 305–312 (1984)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Farmer, D.W.: Mean value of Dirichlet series associated with holomorphic cusp forms. J. Number Theory 49, 209–245 (1994)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Feng, S.: Zeros of the Riemann zeta function on the critical line. J. Number Theory 132, 511–542 (2012)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Hafner, J.L.: Zeros on the critical line for Dirichlet series attached to certain cusp forms. Math. Ann. 264, 21–37 (1983)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Hafner, J.L.: Zeros on the critical line for Maass wave form \(L\)-functions. J. Reine Angew. Math. 377, 127–158 (1987)MathSciNetMATHGoogle Scholar
  20. 20.
    Heath-Brown, D.R.: Simple zeros of the Riemann zeta-function on the critical line. Bull. Lond. Math. Soc. 11, 17–18 (1979)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Iwaniec, H., Kowalski, E.: Analytic Number Theory. Amer. Math. Soc, Providence, RI (2004)CrossRefMATHGoogle Scholar
  22. 22.
    Kim, I.I.: Functoriality for the exterior square of \(\operatorname{GL}_4\) and the symmetric fourth of \(\operatorname{GL}_2\) (with appendices by I. Ramakrishnan, I. I. Kim and P. Sarnak). J. Am. Math. Soc. 16, 139–183 (2003)CrossRefGoogle Scholar
  23. 23.
    Kühn, P., Robles, N., Zeindler, D.: On a mollifier of the perturbed Riemann zeta-function. J. Number Theory 174, 274–321 (2017)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Levinson, N.: More than One Third of Zeros of Riemann’s Zeta-Function are on \(\sigma = \tfrac{1}{2}\). Adv. Math. 13, 383–436 (1974)CrossRefMATHGoogle Scholar
  25. 25.
    Pratt, K., Robles, N., Zaharescu, A., Zeindler, D.: Combinatorial applications of autocorrelation ratios. arXiv:1802.10521
  26. 26.
    Pratt, K., Robles, N.: Perturbed moments and a longer mollifier for critical zeros of \(\zeta \). Res. Number Theory 4, 9 (2018).  https://doi.org/10.1007/s40993-018-0103-4 MathSciNetCrossRefGoogle Scholar
  27. 27.
    Rankin, R.A.: Contributions to the theory of Ramanujan’s function \(\tau (n)\) and similar arithmetical functions II. The order of the Fourier coefficients of integral modular forms. Proc. Cambridge Philos. Soc. 35, 351–372 (1939)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Rezvyakova, I.S.: On the zeros of Hecke \(L\)-functions and of their linear combinations on the critical line. Dokl. Akad. Nauk 431, 741–746 (2010). (in Russian)MathSciNetMATHGoogle Scholar
  29. 29.
    Ricotta, G.: Real zeros and size of Rankin-Selberg L-functions in the level aspect. Duke Math. J. 131, 291–350 (2006)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Robles, N., Roy, A., Zaharescu, A.: Twisted second moments of the Riemann zeta-function and applications. J. Math. Anal. Appl. 434, 271–314 (2016)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Sono, K.: An application of generalized mollifiers to the Riemann zeta-function. Kyushu J. Math. 72(1), 35–69 (2018)Google Scholar
  32. 32.
    Young, M.P.: A short proof of Levinson’s theorem. Arch. Math. 95, 539–548 (2010)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Zhang, Q.: Integral mean values of modular \(L\)-functions. J. Number Theory 115, 100–122 (2005)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institut für MathematikUniversität ZürichZurichSwitzerland
  2. 2.Department of MathematicsUniversity of IllinoisUrbanaUSA
  3. 3.Wolfram Research IncChampaignUSA
  4. 4.Department of Mathematics and Statistics, Fylde CollegeLancaster UniversityLancasterUK

Personalised recommendations