Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-24T12:07:50.377Z Has data issue: false hasContentIssue false
  • English
  • Français

Moments of the Dedekind zeta function and other non-primitive L-functions

Part of: Zeta and $L$-functions: analytic theory Algebraic number theory: global fields

Published online by Cambridge University Press:  15 November 2019

WINSTON HEAP
WINSTON HEAP*
Affiliation:
Department of Mathematics, University of York, York, YO10 5DD, U.K. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We give a conjecture for the moments of the Dedekind zeta function of a Galois extension. This is achieved through the hybrid product method of Gonek, Hughes and Keating. The moments of the product over primes are evaluated using a theorem of Montgomery and Vaughan, whilst the moments of the product over zeros are conjectured using a heuristic method involving random matrix theory. The asymptotic formula of the latter is then proved for quadratic extensions in the lowest order case. We are also able to reproduce our moments conjecture in the case of quadratic extensions by using a modified version of the moments recipe of Conrey et al. Generalising our methods, we then provide a conjecture for moments of non-primitive L-functions, which is supported by some calculations based on Selberg’s conjectures.

MSC classification

Primary: 11M06: $zeta (s)$ and $L(s, chi)$
Secondary: 11R42: Zeta functions and $L$-functions of number fields 11M50: Relations with random matrices
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 170 , Issue 1 , January 2021 , pp. 191 - 219
Copyright
© Cambridge Philosophical Society 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Bruggeman, R. W. and Motohashi, Y.. Fourth power moment of Dedekind zeta functions of real quadratic number fields with class number one. Functiones et Approximatio 29 (2001), 41–79.CrossRefGoogle Scholar
Bruggeman, R. W. and Motohashi, Y.. Sum formula for Kloosterman sums and fourth moment of the Dedekind zeta-function over the Gaussian number field. Functiones et Approximatio 31 (2003), 23–92.CrossRefGoogle Scholar
Bombieri, E. and Hejhal, D. A.. On the distribution of zeros of linear combinations of Euler products. Duke Math. J. 80 (1995), 821–862.CrossRefGoogle Scholar
Chandrasekharan, K. and Narasimhan, R.. The approximate functional equation for a class of zeta functions. Math. Ann. 152 (1963), 30–64.CrossRefGoogle Scholar
Conrey, J. B. and Farmer, D. W.. Mean values of L-functions and symmetry. Internat. Math. Res. Notices 17 (2000), 883–908.CrossRefGoogle Scholar
Conrey, J. B., Farmer, D. W., Keating, J. P., Rubinstein, M. O. and Snaith, N. C.. Integral moments of L-functions. Proc. London Math. Soc. (3) 91 (2005), 33–104.CrossRefGoogle Scholar
Conrey, J. B. and Ghosh, A.. A conjecture for the sixth power moment of the Riemann zeta-function. Int. Math. Res. Not. 15 (1998), 775–780.CrossRefGoogle Scholar
Conrey, J. B. and Ghosh, A.. On the Selberg class of Dirichlet series: small degrees. Duke Math. J. 72 (1993), 673–693.CrossRefGoogle Scholar
Conrey, J. B. and Gonek, S.. High moments of the Riemann zeta function, Duke Math. J. 107 (2001), 577–604.CrossRefGoogle Scholar
Fomenko, O. M.. Mean values connected with the Dedekind zeta function. J. Math. Sci. (3) 150 (2008), 2115–2122.CrossRefGoogle Scholar
Gonek, S., Hughes, C. P. and Keating, J. P.. A Hybrid Euler–Hadamard product for the Riemann Zeta Function. Duke Math J. 136 (3) (2007), 507–549.Google Scholar
Heap, W.. The twisted second moment of the Dedekind zeta function of a quadratic field. Preprint available at arxiv: 1211.2182.Google Scholar
Iwaniec, H. and Kowalski, E.. Analytic Number Theory. American Math. Soc. Vol. 53 (Colloquium Publications, 2004).CrossRefGoogle Scholar
Keating, J. P. and Snaith, N. C.. Random matrix theory and ζ(1/2 + it), Comm. Math. Phys. 214 (2000), 57–89.CrossRefGoogle Scholar
Liu, J. and Ye, Y.. Superposition of zeros of distinct L-functions. Forum Math. 14 (2002), 419–455.CrossRefGoogle Scholar
Montgomery, H. and Vaughan, R.. Hilbert’s inequality. J. London Math. Soc. (2) 8 (1974), 73–82.CrossRefGoogle Scholar
Montgomery, H. and Vaughan, R.. Multiplicative number theory I: classical theory. Camb. stud. Adv. Math. 97 (Cambridge university press, 2006).CrossRefGoogle Scholar
Motohashi, Y.. A note on the mean value of the Dedekind zeta function of the quadratic field. Math. Ann. 188 (1970), 123–127.CrossRefGoogle Scholar
Müller, W.. The mean square of the Dedekind zeta function in quadratic number fields. Math. Proc. Cam. Phil. Soc. 106 (1989), 403–417.CrossRefGoogle Scholar
Neukirch, J.. Algebraic Number Theory. (Springer–Verlag Berlin Heidelberg, 1999).CrossRefGoogle Scholar
Rosen, M.. A generalization of Mertens’ theorem. J. Ramanujan Math. Soc. 14 (1999), 1–19.Google Scholar
Ramachandra, K.. Application of a theorem of Montgomery and Vaughan to the zeta function. J. London Math. Soc. (2) 10 (1975), 482–486.CrossRefGoogle Scholar
Rudnick, Z. and Sarnak, P.. Zeros of principal L-functions and random matrix theory. Duke J. of Math. 81 (1996), 269–322.CrossRefGoogle Scholar
Sarnak, P.. Fourth moments of Grossencharakteren zeta functions. Comms. Pure and App. Math. 38 (1985), 167–178.CrossRefGoogle Scholar
Sarnak, P.. Quantum chaos, symmetry and zeta functions. Curr. Dev. Math. (1997), 84–115.CrossRefGoogle Scholar
Selberg, A.. Old and new conjectures and results about a class of Dirichlet series, Collected papers, vol. 2 (Springer-Verlag, Berlin Heidelberg New York, 1991).Google Scholar