Multiple dirichlet series and moments of zeta and L-functions

Compositio Mathematica

Volumn 139, Issue 3, 2003, Pages 297-360

  Diaconu, Adrian ^a   Goldfeld, Dorian ^a   Hoffstein, Jeffrey ^b  

^a Columbia University ^*  (United States)

^b BROWN UNIVERSITY   (United States)

Author keywords

L functions; Moments; Multiple dirichlet series; Twists; Zeta functions

Indexed keywords

EID: 4544331452     PISSN: 0010437X     EISSN: None     Source Type: Journal    
DOI: 10.1023/B:COMP.0000018137.38458.68     Document Type: Article

References (34)

1. [A] Atkinson, F. V.: The mean value of the Riemann zeta function, Acta Math. 81 (1949), 353-376.
2. [B-H] Březin, E. and Hlikami, S.: Characteristic polynomials of random matrices, Comm. Math. Phys. 214 (2000), 111-135.
3. [B-F-H-1] Bump, D., Friedberg, S. and Hoffstein, J.: Sums of twisted GL(3) automorphic L-functions, Preprint.
4. [B-F-H-2] Bump, D., Friedberg, S. and Hoffstein, J.: On some applications of automorphic forms to number theory, Bull. Amer. Math. Soc. 33 (1996), 157-175. (Pubitemid 126344271)
5. [C-F] Conrey, J. B. and Farmer, D.W.: Mean values of L-functions and symmetry, Internat. Math. Res. Notices 17 (2000), 883-908.
6. [CFKRS] Conrey, J. B., Farmer, D. W., Keating, J. P., Rubenstein, M. O. and Smith, N. C.: Moments of Zeta and L-functions.
7. [C-G] Conrey, J. B. and Gonek, S. M.: High moments of the Riemann zeta-function, Duke Math. J. 107 (2001), 577-604.
8. [C-Gh-1] Conrey, J. B. and Ghosh, A.: A conjecture for the sixth power moment of the Riemann zeta-function, Internat. Math. Res. Notices 15 (1998), 775-780.
9. [C-Gh-2] Conrey, J. B. and Ghosh, A.: On mean values of the zeta-function, Internat. Mathematica 31 (1984), 159-161.
10. [C-N] Chandrasekharan, K. and Narasimhan, R.: Functional equations with multiple gamma factors and the average order of arithmetical functions, Ann. of Math. 76 (1962), 93-136.
11. Davenport, H.: Multiplicative Number Theory, 3rd edn, Grad. Texts in Math. 74, Springer-Verlag, New York, 2000.
12. [F-F] Fisher, B. and Friedberg, S.: Double Dirichlet Series Over Function Fields.
13. [F-H] Friedberg, S. and Hoffstein, J.: Nonvanishing theorems for automorphic L-functions on GL(2), Ann. of Math. 142 (1995), 385-423.
14. [G] Good, A.: The convolution method for Dirichlet series. In: The Selberg Trace Formula and Related Topics (Brunswick, Maine, 1984), Contemp. Math. 53, Amer. Math. Soc., Providence, RI, 1986, pp. 207-214.
15. [G-H] Goldfeld, D. and Hoffstein, J.: Eisenstein series of 1/2-integral weight and the mean value of real Dirichlet L-series, Invent. Math. 80 (1985), 185-208.
16. [H] Hoffstein, J.: Eisenstein series and theta functions on the metaplectic group, In: M. Ram Murty (ed.), Theta Functions: from the Classical to the Modern, CRM Proc. Lect. Notes 1, Amer. Math. Soc., Providence, RI, 1993, 65-104.
17. [H-B] Heath-Brown, D. R.: A mean value estimate for real character sums, Acta Arith. 72 (1995), 235-275. (Pubitemid 126010025)
18. [H-L] Hardy, G. H. and Littlewood, J. E.: Contributions to the theory of the Riemann zeta-function and the theory of the distributions of primes, Acta Math. 41 (1918), 119-196.
19. [Ho] Hörmander, L.: An Introduction to Complex Analysis in Several Variables, Van Nostrand, Princeton, N.J., 1966.
20. [H-R] Hoffstein, J. and Rosen, M.: Average values of L-series in function fields, J. Reine Angew. Math. 426 (1992), 117-150.
21. Ingham, A. E.: Mean-value theorems in the theory of the Riemann zeta-function, Proc. London Math. Soc. 27 (1926), 273-300.
22. [J] Jutila, M.: On the mean value of L(1/2, χ) for real characters, Analysis 1 (1981), 149-161.
23. [K-S] Katz, N. M. and Sarnak, P.: Random Matrices, Frobenius Eigenvalues, and Monodromy, Amer. Math. Soc. Colloq. Publ., 45, Amer. Math, Soc., Providence, RI, 1999.
24. [Ke-Sn-I] Keating, J. P. and Snaith, N. C.: Random matrix theory and ((1/2 + it), Comm. Math. Phys. 214 (2000), 57-89.
25. [Ke-Sn-2] Keating, J. P. and Snaith, N. C.: Random matrix theory and L-function at s = 1/2, Comm. Math. Phys. 214 (2000), 91-110.
26. [K] Kubota, T.: On Automorphic Forms and the Reciprocity Law in a Number Field, Kinokuniya Book Store Co., Tokyo, 1969.
27. [Mot1] Motohashi, Y.: An explicit formula for the fourth power mean of the Riemann zeta-function, Acta Math. 170 (1993), 181-220.
28. [Mot2] Motohashi, Y.: A relation between the Riemann zeta-function and the hyperbolic Laplacian, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 22(4) (1995), 299-313.
29. [Mot3] Motohashi, Y.: Spectral Theory of the Riemann Zeta Function, Cambridge Univer-sity Press, Cambridge, 1997.
30. [S] Siegel, C. L.: Die Funktionalgleichungen einiger Dirichlestscher Reithen, Math. Zeit. 63 (1956), 363-373.
31. [So] Soundarajan, K.: Nonvanishing of quadratic Dirichlet L-functions at s = 1/2, Ann. of Math. 152 (2000), 447-488.
32. [St] Stark, H. M.: Unpublished notes.
33. [T] Titchmarsh, E. C.: In: D. R. Heath-Brown (ed.). The Theory of the Riemann Zeta-function, 2nd edn, The Clarendon Press, Oxford University Press, New York, 1986.
34. [Z] Zavorotny, N. I.: Automorphic Functions and Number Theory, Parts I, II (Russian), Akad. Nauk SSSR, Dal'nevostochn. Otdel., Vladivostok (1989), 69-124a, 254.