## Journal of the Mathematical Society of Japan

### The universality of symmetric power L-functions and their Rankin-Selberg L-functions

#### Abstract

We establish the universality theorem for the first four symmetric power $L$-functions of automorphic forms and their associated Rankin-Selberg $L$-functions. This generalizes some results of Laurinčikas and Matsumoto and of Matsumoto, respectively.

#### Article information

Source
J. Math. Soc. Japan, Volume 59, Number 2 (2007), 371-392.

Dates
First available in Project Euclid: 1 October 2007

https://projecteuclid.org/euclid.jmsj/1191247592

Digital Object Identifier
doi:10.2969/jmsj/05920371

Mathematical Reviews number (MathSciNet)
MR2325690

Zentralblatt MATH identifier
1159.11017

#### Citation

LI, Hongze; WU, Jie. The universality of symmetric power L -functions and their Rankin-Selberg L -functions. J. Math. Soc. Japan 59 (2007), no. 2, 371--392. doi:10.2969/jmsj/05920371. https://projecteuclid.org/euclid.jmsj/1191247592

#### References

• B. Bagchi, The statistical behaviour and universality properties of the Riemann zeta-function and other allied Dirichlet series, Ph.,D. thesis, Indian Statistical Institute, Calcutta, 1981.
• K.-L. Chung, A course in probability theory, Third edition, Academic Press, Inc., San Diego, CA, 2001, xviii+419 pp.
• J. Cogdell and P. Michel, On the complex moments of symmetric power $L$-functions at $s=1$, IMRN, 31 (2004), 1561–1618.
• S. Gelbart and H. Jacquet, A relation between automorphic representations of $GL(2)$ and $GL(3)$, Ann. Sci. École Norm. Sup. (4), 11 (1978), 471–542.
• H. Iwaniec, Topics in Classical Automorphic Forms, Graduate Studies in Mathematics, 17, American Mathematical Society, Providence, Rhode Island, 1997.
• A. Kač\.enas and A. Laurinčikas, On Dirichlet series related to certain cusp forms, Liet. Mat. Rink., 38 (1998), 82–97 = Lithuanian Math. J., 38 (1998), 64–76, MR1663828 (99j:11049).
• H. Kim, Functoriality for the exterior square of $GL_4$ and symmetric fourth of $GL_2$, Appendix 1 by Dinakar Ramakrishnan, Appendix 2 by Henry H. Kim and Peter Sarnak, J. Amer. Math. Soc., 16 (2003), 139–183.
• H. Kim and F. Shahidi, Functorial products for $GL_2\times GL_3$ and functorial symmetric cube for $GL_2$ (with an appendix by C. J. Bushnell and G. Henniart), Ann. of Math., 155 (2002), 837–893.
• H. Kim and F. Shahidi, Cuspidality of symmetric power with applications, Duke Math. J., 112 (2002), 177–197.
• Y.-K. Lau and J. Wu, A density theorem on automorphic $L$-functions and some applications, Trans. Amer. Math. Soc., 358 (2006), 441–472.
• A. Laurinčikas, Limit Theorems for the Riemann Zeta-Function, Kluwer, Dordrecht, 1996.
• A. Laurinčikas, On limit distribution of the Matsumoto zeta-function II, Liet. Mat. Rink., 36 (1996), 464–485 = Lithuanian Math. J., 36 (1996), 371–387.
• A. Laurinčikas and K. Matsumoto, The universality of zeta-function attached to certain cusp forms, Acta Arith., 98 (2001), 345–359.
• A. Laurinčikas and K. Matsumoto, The joint universality of twisted automorphic $L$-functions, J. Math. Soc. Japan, 56 (2004), 923–939.
• A. Laurinčikas, K. Matsumoto and J. Steuding, The universality of $L$-function associated with new forms, Izv. Ross. Akad. Nauk Ser. Mat., 67 (2003), 83–98 = Izv. Math., 67 (2003), 77–90.
• K. Matsumoto, The mean values and the universality of Rankin-Selberg $L$-functions, In: Number theory, Turku, 1999, de Gruyter, Berlin, 2001, pp.,201–221.
• A. Perelli, General $L$-functions, Ann. Mat. Pura Appl. (4), 130 (1982), 287–306.
• R. A. Rankin, An $\Omega$-result for the coefficients of cusp forms, Math. Ann., 283 (1973), 239–250.
• W. Rudin, Functional Analysis, Second edition, International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991, xviii+424 pp.
• Z. Rudnick and P. Sarnak, Zeros of principal $L$-functions and random matrix theory, Duke Math. J., 81 (1996), 269–322.
• J.-P. Serre, Cours d'arithmétiques, Deuxième édition revue et coorigée, Le Mathématicien, No.,2. Presses Universitaires de France, Paris, 1977, p.,188.
• G. Tenenbaum, Introduction to analytic and probabilistic number theory, Translated from the second French edition (1995) by C. B. Thomas, Cambridge Studies in Advanced Mathematics 46, Cambridge University Press, Cambridge, 1995, xvi+448 pp.
• S. M. Voronin, A theorem on the “universality” of the Riemann zeta-function, Izv. Akad. Nauk SSSR Ser. Mat., 39 (1975), 475–486 = Math. USSR-Izv., 9 (1975), 443–453.