Skip to main content
SpringerLink
Log in

The twisted mean square and critical zeros of Dirichlet L-functions

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

In this work, we obtain an asymptotic formula for the twisted mean square of a Dirichlet L-function with a longer mollifier, whose coefficients are also more general than before. As an application we obtain that, for every Dirichlet L-function, more than 41.72% of zeros are on the critical line and more than 40.74% of zeros are simple and on the critical line. These proportions also improve previous results which were proved only for the Riemann zeta-function.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Balasubramanian, R., Conrey, J.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)

    MathSciNet  MATH  Google Scholar 

  2. Bauer, P.: Zeros of Dirichlet \(L\)-series on the critical line. Acta Arith. 93, 37–52 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bettin, S., Gonek, S.: The \(\theta =\infty \) conjecture implies the Riemann hypothesis. Mathematika 63(1), 29–33 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bettin, S., Chandee, V.: Trilinear forms with kloosterman fractions. Adv. Math. 328, 1234–1262 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bettin, S., Chandee, V., Radziwiłł, M.: The mean square of the product of the Riemann zeta-function with Dirichlet polynomials. J. Reine Angew. Math. 720, 51–79 (2017)

    MathSciNet  MATH  Google Scholar 

  6. Bui, H., Conrey, J.B., Young, M.: More than 41% of the zeros of the zeta function are on the critical line. Acta Arith. 150(1), 35–64 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bui, H.: Critical zeros of the Riemann zeta-function. arXiv:1410.2433

  8. Conrey, J.B.: Zeros of derivatives of the Riemann’s \(\xi \)-function on the critical line. J. Number Theory 16, 49–74 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  9. Conrey, J.B., Ghosh, A.: A simpler proof of Levinson’s theorem. Math. Proc. Camb. Philos. Soc. 97, 385–395 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  10. Conrey, J.B., Ghosh, A., Gonek, S.M.: Large gaps between zeros of the zeta-function. Mathematika 33(2), 212–238 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  11. Conrey, J.B.: More than two fifths of the zeros of the Riemann zeta function are on the critical line. J. Reine Angew. Math. 339, 1–26 (1989)

    MathSciNet  MATH  Google Scholar 

  12. Conrey, J.B., Iwaniec, H., Soundararajan, K.: Critical zeros of Dirichlet \(L\)-function. J. Reine Angew. Math. 681, 175–198 (2013)

    MathSciNet  MATH  Google Scholar 

  13. Conrey, J.B., Iwaniec, H., Soundararajan, K.: Asymptotic large sieve. arXiv:1105.1176

  14. Deshouillers, J.-M., Iwaniec, H.: Kloosterman sums and Fourier coefficients of cusp forms. Invent. Math. 70, 219–288 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  15. Deshouillers, J.-M., Iwaniec, H.: Power mean values of the Riemann zeta function II. Acta Arith. 48, 305–312 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  16. Feng, S.: Zeros of the Riemann zeta function on the critical line. J. Number Theory 132(4), 511–542 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Heath-Brown, D.R.: Simple zeros of the Riemann zeta-function on the critical line. Bull. Lond. Math. Soc. 11, 17–18 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  18. Heath-Brown, D.R.: Prime numbers in short intervals and a generalized Vaughan identity. Can. J. Math. 34(6), 1365–1377 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kühn, P., Robles, N., Zeindler, D.: On mean values of mollifiers and \(L\)-functions associated to primitive cusp form. Math. Z. (2018). https://doi.org/10.1007/s00209-018-2099-9

  20. Levinson, N.: More than one third of zeros of Riemann’s zeta-function are on \(\sigma =1/2\). Adv. Math. 13, 383–436 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  21. Li, X., Radziwiłł, M.: The Riemann-zeta function on vertical arithmetic progressions. Int. Math. Res. Not. 2, 325–354 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Pratt, K., Robles, N.: Perturbed moments and a longer mollifier for critical zeros of \(\zeta \). Res. Number Theory 4(1), 26 (2018). Art. 9

    Article  MathSciNet  MATH  Google Scholar 

  23. Radziwiłł, M.: Limitations to mollifying \(\zeta (s)\). arXiv:1207.6583

  24. Robles, N., Roy, A., Zaharescu, A.: Twisted second moments of the Riemann zeta-function and applications. J. Math. Anal. Appl. 434(1), 271–314 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  25. Pratt, K., Robles, N., Zaharescu, A., Zeindler, D.: Conbinatorial applications of autocorrelation ratios. arXiv:1802.10521

  26. Selberg, A.: On the zeros of Riemann’s zeta-function. Skr. Norske Videnskaps-Akad. Oslo I(10), 1–59 (1942)

    MATH  Google Scholar 

  27. Sono, K.: An application of generalized mollifiers to the Riemann zeta-function. Kyushu J. Math. 72, 35–69 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  28. Soundararajan, K.: Mean-values of the Riemann zeta-function. Mathematika 42(1), 158–174 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  29. Wu, X.: Distinct zeros and simple zeros for the family of Dirichlet \(L\)-functions. Quart. J. Math. 67, 757–779 (2016)

    MathSciNet  MATH  Google Scholar 

  30. Young, M.P.: A short proof of Levinson’s theorem. Arch. Math. 95, 539–548 (2010)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We would like to express our heartfelt thanks to the anonymous referee for his careful reading and helpful suggestion.

Author information

Authors and Affiliations

  1. School of Mathematics, Hefei University of Technology, Hefei, 230009, People’s Republic of China

    Xiaosheng Wu

Authors
  1. Xiaosheng Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiaosheng Wu.

Additional information

This work is supported by the National Nature Science Foundation of China (Grant No. 11871187) and the Fundamental Research Funds for the Central Universities.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wu, X. The twisted mean square and critical zeros of Dirichlet L-functions. Math. Z. 293, 825–865 (2019). https://doi.org/10.1007/s00209-018-2209-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-018-2209-8

Keywords

Mathematics Subject Classification

Search

Navigation