## Journal of the Mathematical Society of Japan

### Functional distribution for a collection of Lerch zeta functions

Hidehiko MISHOU

#### Abstract

Let 0 < $\alpha$ < 1 be a transcendental real number and $\lambda_1,\ldots,\lambda_r$ be real numbers with $0 \le \lambda_j$ < 1. It is conjectured that a joint universality theorem for a collection of Lerch zeta functions $\{L(\lambda_j,\alpha,s)\}$ will hold for every numbers $\lambda_j$'s which are different each other. In this paper we will prove that the joint universality theorem for the set $\{L(\lambda_j,\alpha,s)\}$ holds for almost all real numbers $\lambda_j$'s.

#### Article information

Source
J. Math. Soc. Japan, Volume 66, Number 4 (2014), 1105-1126.

Dates
First available in Project Euclid: 23 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1414090236

Digital Object Identifier
doi:10.2969/jmsj/06641105

Mathematical Reviews number (MathSciNet)
MR3272593

Zentralblatt MATH identifier
1317.11089

#### Citation

MISHOU, Hidehiko. Functional distribution for a collection of Lerch zeta functions. J. Math. Soc. Japan 66 (2014), no. 4, 1105--1126. doi:10.2969/jmsj/06641105. https://projecteuclid.org/euclid.jmsj/1414090236

#### References

• B. Bagchi, The statistical behavior and universality properties of the Riemann zeta-function and other allied Dirichlet series, Ph. D. Thesis. Calcutta, Indian Statistical Institute, 1981.
• A. Laurinčikas, Limit Theorems for the Riemann Zeta-Function, Math. Appl., 352, Kluwer Academic Publishers Group, Dordrecht, 1996.
• A. Laurinčikas, The universality of the Lerch zeta function, Lith. Math. J., 37 (1997), 275–280.
• A. Laurinčikas and R. Garunkstis, The Lerch Zeta-Function, Kluwer Academic Publishers, 2003.
• A. Laurinčikas and K. Matsumoto, The joint universality and the functional independence for Lerch zeta-functions, Nagoya Math. J., 157 (2000), 211–227.
• M. Lerch, Note sur la fonction $\mathfrak{K}(w,x,s)=\sum^{\infty}_{k= 0}\exp (2k\pi ix)/(w+k)^{s}$, Acta Math., 11 (1887), 19–24.
• H. Mishou and H. Nagoshi, Functional distribution of $L(s,\chi_d)$ with real characters and denseness of quadratic class numbers, Trans. Amer. Math. Soc., 358 (2006), 4343–4366.
• H. Nagoshi, On the set of Lerch zeta-functions with transcendental $\alpha$, preprint.
• H. Niederreiter, Applications of Diophantine approximations to numerical integration, In: Diopahntine Approximation and Its Applications, (ed. C. F. Osgood), Academic Press, New York, 1973, pp.,129–199.
• W. M. Schmidt, Metrical theorems on fractional parts of sequences, Trans. Amer. Math. Soc., 110 (1964), 493–518.
• J. Steuding, Value-Distribution of $L$-Functions, Lecture Notes in Math., 1877, Springer-Verlag, 2007.
• E. C. Titchmarsh, The Theory of Functions. 2nd ed., Oxford University Press, 1976.
• A. A. Karatsuba and S. M. Voronin, The Riemann zeta-function, de Gruyter Exp. Math., 5, Walter de Gruyter & Co., New York, 1992.