Skip to main content Accessibility help
×
×
Home

On a Li-type criterion for zero-free regions of certain Dirichlet series with real coefficients

  • Alina Bucur (a1), Anne-Maria Ernvall-Hytönen (a2), Almasa Odžak (a3) and Lejla Smajlović (a4)

Abstract

The Li coefficients $\unicode[STIX]{x1D706}_{F}(n)$ of a zeta or $L$ -function $F$ provide an equivalent criterion for the (generalized) Riemann hypothesis. In this paper we define these coefficients, and their generalizations, the $\unicode[STIX]{x1D70F}$ -Li coefficients, for a subclass of the extended Selberg class which is known to contain functions violating the Riemann hypothesis such as the Davenport–Heilbronn zeta function. The behavior of the $\unicode[STIX]{x1D70F}$ -Li coefficients varies depending on whether the function in question has any zeros in the half-plane $\text{Re}(z)>\unicode[STIX]{x1D70F}/2.$ We investigate analytically and numerically the behavior of these coefficients for such functions in both the $n$ and $\unicode[STIX]{x1D70F}$ aspects.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      On a Li-type criterion for zero-free regions of certain Dirichlet series with real coefficients
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      On a Li-type criterion for zero-free regions of certain Dirichlet series with real coefficients
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      On a Li-type criterion for zero-free regions of certain Dirichlet series with real coefficients
      Available formats
      ×

Copyright

References

Hide All
1. Balanzario, E. P. and Sánchez-Ortiz, J., ‘Zeros of the Davenport–Heilbronn counterexample’, Math. Comp. 76 (2007) 20452049.
2. Bombieri, E. and Ghosh, A., ‘Around Davenport–Heilbronn function’, Uspekhi Mat. Nauk 66 (2011) 1566 (Russian); Russian Math. Surveys 66 (2011) 221–270 (English).
3. Bombieri, E. and Lagarias, J. C., ‘Complements to Li’s criterion for the Riemann hypothesis’, J. Number Theory 77 (1999) 274287.
4. Bucur, A., Ernvall-Hytönen, A.-M., Odžak, A., Roditty-Gershon, E. and Smajlović, L., ‘On 𝜏-Li coefficients for Rankin–Selberg L-functions’, Women in numbers Europe , Association for Women in Mathematics Series 2 (ed. Bucur, A. et al. ; Springer International, Switzerland, 2015) 167190.
5. Davenport, H. and Heilbronn, H., ‘On the zeros of certain Dirichlet series (second paper)’, J. Lond. Math. Soc. (2) 11 (1936) 307312.
6. Droll, A. D., ‘Variations of Li’s criterion for an extension of the Selberg class’, PhD Thesis, Queen’s University Ontario, Canada, 2012; available at http://qspace.library.queensu.ca/jspui/bitstream/1974/7352/1/Droll_Andrew_D_201207_PhD.pdf.
7. Ernvall-Hytönen, A.-M., Odžak, A., Smajlović, L. and Sušić, M., ‘On the modified Li criterion for a certain class of L-functions’, J. Number Theory 156 (2015) 340367.
8. Johansson, F., ‘Arb: a C library for ball arithmetic’, ACM Commun. Comput. Algebra 47 (2013) no. 4, 166169.
9. Kaczorowski, J., ‘Axiomatic theory of L functions: the Selberg class’, Analytic number theory, C.I.M.E. Summer School, Cetraro, Italy, 2002 , Lecture Notes in Mathematics 1891 (eds Perelli, A. and Viola, C.; Springer, 2006) 133209.
10. Kaczorowski, J. and Perelli, A., ‘On the structure of the Selberg class, I: 0⩽d⩽1’, Acta Math. 182 (1999) 207241.
11. Lagarias, J. C., ‘Li coefficients for automorphic L-functions’, Ann. Inst. Fourier 57 (2007) 16891740.
12. Maslanka, K., ‘Li’s criterion for the Riemann hypothesis — numerical approach’, Opuscula Math. 24 (2004) no. 1, 103114.
13. Mazhouda, K., ‘On the 𝜏-Li coefficients for automorphic L-functions’, Rocky Mountain J. Math., to appear.
14. Omar, S., Ouni, R. and Mazhouda, K., ‘On the zeros of Dirichlet L-functions’, LMS J. Comput. Math. 14 (2011) 140154.
15. Proskurin, N. V., ‘On the cubic L-function’, St. Petersburg Math. J. 24 (2013) no. 2, 353370.
16. Selberg, A., ‘Old and new conjectures and results about a class of Dirichlet series’, Proceedings of Amalfi Conference on Analytic Number Theory (ed. Bombieri, E. et al. ; Università di Salerno, 1992) 367385.
17. Smajlović, L., ‘On Li’s criterion for the Riemann hypothesis for the Selberg class’, J. Number Theory 130 (2010) 828851.
18. Titchmarsh, E. C., The theory of the Riemann zeta-function (Clarendon Press, Oxford, 1951).
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

LMS Journal of Computation and Mathematics
  • ISSN: -
  • EISSN: 1461-1570
  • URL: /core/journals/lms-journal-of-computation-and-mathematics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed