## Journal of the Mathematical Society of Japan

### A converse theorem for double Dirichlet series and Shintani zeta functions

#### Abstract

A converse theorem for double Dirichlet series is established. As an application, we show that certain zeta functions introduced by Shintani are actually Weyl group multiple Dirichlet series associated to metaplectic Eisenstein series on $GL(2)$.

#### Article information

Source
J. Math. Soc. Japan, Volume 66, Number 2 (2014), 449-477.

Dates
First available in Project Euclid: 23 April 2014

https://projecteuclid.org/euclid.jmsj/1398258180

Digital Object Identifier
doi:10.2969/jmsj/06620449

Mathematical Reviews number (MathSciNet)
MR3201822

Zentralblatt MATH identifier
1311.11043

#### Citation

DIAMANTIS, Nikolaos; GOLDFELD, Dorian. A converse theorem for double Dirichlet series and Shintani zeta functions. J. Math. Soc. Japan 66 (2014), no. 2, 449--477. doi:10.2969/jmsj/06620449. https://projecteuclid.org/euclid.jmsj/1398258180

#### References

• B. Brubaker, D. Bump, G. Chinta, S. Friedberg and J. Hoffstein, Weyl group multiple Dirichlet series. I, In: Multiple Dirichlet Series, Automorphic Forms, and Analytic Number Theory, Proceedings of the Bretton Woods Workshop on Multiple Dirichlet Series, Bretton Woods, NH, July 11–14, 2005, (eds. S. Friedberg, D. Bump, D. Goldfeld and J. Hoffstein), Proc. Sympos. Pure Math., 75, Amer. Math. Soc., Providence, RI, 2006, pp.,91–114.
• V. Bykovskiĭ, Functional equations for Hecke-Maass series, Funct. Anal. and Appl., 34 (2000), 98–105.
• N. Diamantis and D. Goldfeld, A converse theorem for double Dirichlet series, Amer. J. Math., 133 (2011), 913–938.
• D. Goldfeld and J. Hoffstein, Eisenstein series of $1/2$-integral weight and the mean value of real Dirichlet $L$-series, Invent. Math., 80 (1985), 185–208.
• H. Maass, Lectures on Modular Functions of One Complex Variable, 2nd ed., Tata Inst. Fund. Res. Lectures on Math. and Phys., 29, Tata Institute of Fundamental Research, Bombay, 1983.
• F. Olver, D. Lozier, R. Boisvert and C. Clark, NIST Handbook of Mathematical Functions, U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge, 2010.
• W. Roelcke, Das Eigenwertproblem der automorphen Formen in der hyperbolischen Ebene. I, II, Math. Ann., 167 (1966), 292–337; ibid., 168 (1966), 261–324.
• M. J. Razar, Modular forms for $\Gamma_0(N)$ and Dirichlet series, Trans. Amer. Math. Soc., 231 (1977), 489–495.
• H. Saito, On $L$-functions associated with the vector space of binary quadratic forms, Nagoya Math. J., 130 (1993), 149–176.
• F. Sato, Zeta functions in several variables associated with prehomogeneous vector spaces I: Functional equations, Tôhoku Math. J. (2), 34 (1982), 437–483.
• C. L. Siegel, Die Funktionalgleichungen einiger Dirichletscher Reihen, Math. Z., 63 (1956), 363–373.
• G. Shimura, On the holomorphy of certain Dirichlet series, Proc. London Math. Soc. (3), 31 (1975), 79–98.
• T. Shintani, On zeta-functions associated with the vector space of quadratic forms, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 22 (1975), 25–65.
• E. Whittaker and G. Watson, A Course of Modern Analysis, An introduction to the general theory of infinite processes and of analytic functions: with an account of the principal transcendental functions. 4th ed., Reprinted Cambridge University Press, New York, 1962.