Fractional moments of automorphic 𝐿-functions

Fomenko, O.

doi:10.1090/S1061-0022-2011-01143-4

Fractional moments of automorphic $L$-functions
HTML articles powered by AMS MathViewer

by O. M. Fomenko
Translated by: G. V. Kuz′mina and O. M. Fomenko

St. Petersburg Math. J. 22 (2011), 321-335

DOI: https://doi.org/10.1090/S1061-0022-2011-01143-4

Published electronically: February 8, 2011

PDF | Request permission

Abstract:

Upper and lower bounds for fractional moments of automorphic $L$- functions are found.

References

D. R. Heath-Brown, Fractional moments of the Riemann zeta function, J. London Math. Soc. (2) 24 (1981), no. 1, 65–78. MR 623671, DOI 10.1112/jlms/s2-24.1.65
M. Jutila, On the value distribution of the zeta function on the critical line, Bull. London Math. Soc. 15 (1983), no. 5, 513–518. MR 705532, DOI 10.1112/blms/15.5.513
R. W. K. Odoni, A problem of Rankin on sums of powers of cusp-form coefficients, J. London Math. Soc. (2) 44 (1991), no. 2, 203–217. MR 1136435, DOI 10.1112/jlms/s2-44.2.203
Stephen Gelbart and Hervé Jacquet, A relation between automorphic representations of $\textrm {GL}(2)$ and $\textrm {GL}(3)$, Ann. Sci. École Norm. Sup. (4) 11 (1978), no. 4, 471–542. MR 533066
Daniel Bump and David Ginzburg, Symmetric square $L$-functions on $\textrm {GL}(r)$, Ann. of Math. (2) 136 (1992), no. 1, 137–205. MR 1173928, DOI 10.2307/2946548
Eduard Wirsing, Das asymptotische Verhalten von Summen über multiplikative Funktionen, Math. Ann. 143 (1961), 75–102 (German). MR 131389, DOI 10.1007/BF01351892
R. M. Gabriel, Some results concerning the integrals of moduli of regular functions along certain curves, J. London Math. Soc. 2 (1927), 112–117.
H. L. Montgomery and R. C. Vaughan, Hilbert’s inequality, J. London Math. Soc. (2) 8 (1974), 73–82. MR 337775, DOI 10.1112/jlms/s2-8.1.73
Anton Good, The square mean of Dirichlet series associated with cusp forms, Mathematika 29 (1982), no. 2, 278–295 (1983). MR 696884, DOI 10.1112/S0025579300012377
A. Laurinčikas, On moments of zeta-functions associated to certain cusp forms, Chebyshevskiĭ Sb. 5 (2004), no. 3(11), 138–152. MR 2280027
Carlos Julio Moreno, The Hoheisel phenomenon for generalized Dirichlet series, Proc. Amer. Math. Soc. 40 (1973), 47–51. MR 327682, DOI 10.1090/S0002-9939-1973-0327682-0
A. Sankaranarayanan, Fundamental properties of symmetric square $L$-functions. I, Illinois J. Math. 46 (2002), no. 1, 23–43. MR 1936073
Masao Koike, Higher reciprocity law, modular forms of weight $1$ and elliptic curves, Nagoya Math. J. 98 (1985), 109–115. MR 792775, DOI 10.1017/S0027763000021401
O. M. Fomenko, Mean values associated with the Dedekind zeta function, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 350 (2007), no. Analitcheskaya Teoriya Chisel i Teoriya Funktsiĭ. 22, 187–198, 203 (Russian, with Russian summary); English transl., J. Math. Sci. (N.Y.) 150 (2008), no. 3, 2115–2122. MR 2722976, DOI 10.1007/s10958-008-0126-9
D. R. Heath-Brown, The twelfth power moment of the Riemann-function, Quart. J. Math. Oxford Ser. (2) 29 (1978), no. 116, 443–462. MR 517737, DOI 10.1093/qmath/29.4.443
M. Jutila, Lectures on a method in the theory of exponential sums, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 80, Published for the Tata Institute of Fundamental Research, Bombay; by Springer-Verlag, Berlin, 1987. MR 910497
S. Zamarys, On fractional moments of Dirichlet $L$-functions. II, Liet. Mat. Rink. 46 (2006), no. 4, 606–621 (English, with English and Lithuanian summaries); English transl., Lithuanian Math. J. 46 (2006), no. 4, 494–510. MR 2320367, DOI 10.1007/s10986-006-0045-8