Abstract
This paper consists of three parts. In the first part, we extend a classical result of Bohr and Courant about values of the Riemann zeta-function.We actually establish a joint denseness result about values of Dirichlet L-functions and automorphic L-functions for SL(2, ℤ) by using a weak version of Selberg’s orthogonality and a probabilistic limit theorem. The second part concerns the discrepancy D(N, f) of the Hecke eigenvalues λf (p) with respect to the Sato–Tate measure, where f is a holomorphic primitive form of SL(2, ℤ), for which the Sato–Tate conjecture was proved. We estimate the discrepancy D(N, f) toward Akiyama–Tanigawa’s conjecture, assuming the generalized Riemann hypothesis for all the symmetric power L-functions attached to f. In the third part, we give an application of discrepancy to a proof of joint universality of L-functions.
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References
S. Akiyama and Y. Tanigawa, Calculation of values of L-functions associated to elliptic curves, Math. Comput., 68:1201–1231, 1999.
B. Bagchi, The Statistical Behaviour and Universality Properties of the Riemann Zeta-Function and Other Allied Dirichlet Series, PhD thesis, Indian Statistical Institute, Calcutta, 1981.
B. Bagchi, A joint universality theorem for Dirichlet L-functions, Math. Z., 181:319–334, 1982.
T. Barnet-Lamb, D. Geraghty, M. Harris, and R. Taylor, A family of Calabi–Yau varieties and potential automorphy. II, Publ. Res. Inst. Math. Sci., 47:29–98, 2011.
H. Bohr and R. Courant, Neue Anwendungen der Theorie der DiophantischenApproximationen auf die Riemannsche Zetafunktion, J. Reine Angew. Math., 144:249–274, 1914.
D. Bump, Automorphic Forms and Representations, Cambridge University Press, 1998.
S.M. Gonek, Analytic Properties of Zeta and L-Functions, PhD thesis, University of Michigan, 1979.
J. González and J. Jiménez-Urroz, The Sato–Tate distribution and the values of Fourier coefficients of modular newforms, Exp. Math., 21:84–102, 2012.
A.M. Güloğlu, Low-lying zeroes of symmetric power L-functions, Int. Math. Res. Not., 9:517–550, 2005.
H. Iwaniec, Topics in Classical Automorphic Forms, Grad. Stud. Math., Vol. 17, AMS, Providence, RI, 1997.
H. Iwaniec and E. Kowalski, Analytic Number Theory, Colloq. Publ., Am. Math. Soc., Vol. 53, AMS, Providence, RI, 2004.
R. Kačinskaitė and A. Laurinčikas, Joint value distribution of the Matsumoto zeta-functions, Integral Transforms Spec. Funct., 16:415–422, 2005.
A.A. Karatsuba and S.M. Voronin, The Riemann Zeta-Function, de Gruyter Expo. Math., Vol. 5, Walter de Gruyter, Berlin, 1992.
L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, Dover Publications, Mineola, NY, 2006.
A. Laurinčikas, Limit Theorems for the Riemann Zeta-Function, Kluwer Academic Publishers, Dordrecht, 1996.
A. Laurinčikas, On the zeros of linear combinations of the Matsumoto zeta-functions, Lith. Math. J., 38:144–159, 1998.
A. Laurinčikas and K. Matsumoto, The universality of zeta-functions attached to certain cusp forms, Acta Arith., 98:345–359, 2001.
A. Laurinčikas and K. Matsumoto, The joint universality of twisted automorphic L-functions, J. Math. Soc. Japan, 56:923–939, 2004.
J. Liu, Y. Wang, and Y. Ye, A proof of Selberg’s orthogonality for automorphic L-functions, Manuscr. Math., 118:135–149, 2005.
B. Mazur, Finding meaning in error terms, Bull. Am. Math. Soc., 45:185–228, 2008.
H. Mishou, Joint universality theorems for pairs of automorphic zeta functions, Math. Z., 277:1113–1154, 2014.
H. Mishou and H. Nagoshi, Functional distribution of L(s, χd) with real characters and denseness of quadratic class numbers, Trans. Am. Math. Soc., 358:4343–4366, 2006.
H. Mishou and H. Nagoshi, The joint universality for pairs of zeta functions in the Selberg class, preprint, 2015.
M. Ram Murty, Oscillations of Fourier coefficients of modular forms, Math. Ann., 262:431–446, 1983.
M. Ram Murty, Recent developments in the Langlands program, C. R. Math. Acad. Sci., Soc. R. Can., 24:33–54, 2002.
M. Ram Murty and V. Kumar Murty, Non-vanishing of L-functions and Applications, Birkhäuser, Basel, 1997.
M. Ram Murty and C. S. Rajan, Stronger multiplicity one theorems for forms of general type on GL2, in Analytic Number Theory, Vol. 2, Prog. Math., Vol. 139, Birkhäuser, Boston, MA, 1996, pp. 669–683.
V. Kumar Murty, Explicit formulae and the Lang–Trotter conjecture, Rocky Mt. J. Math., 15:535–551, 1985.
H. Nagoshi, On the universality for L-functions attached to Maass forms, Analysis, 25:1–22, 2005.
H. Nagoshi, Value-distribution of Rankin–Selberg L-functions, in New Directions in Value-Distribution Theory of Zeta and L-Functions, Shaker Verlag, Aachen, 2009, pp. 275–287.
H. Nagoshi and J. Steuding, Universality for L-functions in the Selberg class, Lith. Math. J., 50:293–311, 2010.
H. Niederreiter, The distribution of values of Kloosterman sums, Arch. Math., 56:270–277, 1991.
W. Rudin, Real and Complex Analysis, 3rd ed., McGraw-Hill, New York, 1987.
J. Steuding, Value Distribution of L-Functions, Lect. Notes Math., Vol. 1877, Springer-Verlag, Berlin, Heidelberg, 2007.
R. Šleževičien˙e, The joint universality for twists of Dirichlet series with multiplicative coefficients by characters, in Analytic and Probabilistic Methods in Number Theory. Proceedings of the Third International Conference in Honour of J. Kubilius, Palanga, Lithuania, September 24–28, 2001, TEV, Vilnius, 2002, pp. 303–319.
E.C. Titchmarsh, The Theory of the Riemann Zeta-function, 2nd ed., Clarendon Press, Oxford, 1986.
S.M. Voronin, Theorem on the “universality” of the Riemann zeta-function, Math. USSR, Izv., 9:443–453, 1975.
S.M. Voronin, Analytic properties of Dirichlet generating functions of arithmetic objects, Math. Notes, 24:966–969, 1979.
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* This work was supported by JSPS KAKENHI grant No. 25400005
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Nagoshi, H. Joint value-distribution of L-functions and discrepancy of Hecke eigenvalues∗ . Lith Math J 56, 325–356 (2016). https://doi.org/10.1007/s10986-016-9322-3
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DOI: https://doi.org/10.1007/s10986-016-9322-3