## Modular relation interpretation of the series involving the Riemann zeta values.(English)Zbl 1225.11103

Authors’ summary: “We locate M. Katsurada’s results [Publ. Inst. Math., Nouv. Sér. 62(76), 13–25 (1997; Zbl 0885.11052)], in our framework of modular relations, on two series involving the values of the Riemann zeta-function, which are decisive generalizations of earlier results of S. Chowla and D. Hawkins [J. Indian Math. Soc., New Ser. 26, 115–124 (1962; Zbl 0112.30203)] and of R. G. Buschman and H. M. Srivastava [Monatsh. Math. 115, No. 4, 291–298 (1993; Zbl 0779.11033)] et al. We elucidate these results as an improper or a proper modular relation according as the involved parameter $$\nu$$ exerts effects on the series or not, eventually indicating that they are disguised form of modular relations as given by Theorem 4 in §3.”
Katsurada’s Theorem 1 is a special case of the following more general theorem, which in turn is a simple corollary of Theorem 4.
Theorem 3: $\begin{split} \frac{1}{\Gamma(1-b)}\sum_ {n\geq 0} \frac{(b)_ n}{n!}\zeta(n-\nu)x^ n= x^ {\nu+1}\frac{\Gamma(-\nu-1)}{\Gamma(-\nu-b)}\\ +x^ {\nu+1}\sum_ {k\geq 1}\bigg\{G_ {1,2}^ {2,0}\left(2i\pi kx\left|\frac{-b-\nu}{0,-\nu-1}\right.\right) +G_ {1,2}^ {2,0}\left(-2i\pi kx\left|\frac{-b-\nu}{0,-\nu-1}\right.\right)\bigg\} \end{split}$ where $$G_ {p,q}^ {m,n}$$ stands for the Meijer $$G$$-function.

### MSC:

 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$ 33C60 Hypergeometric integrals and functions defined by them ($$E$$, $$G$$, $$H$$ and $$I$$ functions)

### Citations:

Zbl 0885.11052; Zbl 0112.30203; Zbl 0779.11033
Full Text:

### References:

 [1] R. G. Buschman and H. M. Srivastava, Asymptotic behavior of some power series with $$\zeta$$-functions in the coefficients, Monatsh. Math. 115 (1993), no. 4, 291-298. · Zbl 0779.11033 [2] S. Chowla and D. Hawkins, Asymptotic expansions of some series involving the Riemann zeta function, J. Indian Math. Soc. (N.S.) 26 (1962), 115-124. · Zbl 0112.30203 [3] A. Erdélyi et al., Higher transcendental functions , Vol. I, McGraw-Hill, New York, 1953. · Zbl 0052.29502 [4] S. Kanemitsu, Y. Tanigawa and H. Tsukada, Some number theoretic applications of a general modular relation, Int. J. Number Theory 2 (2006), no. 4, 599-615. · Zbl 1172.11025 [5] M. Katsurada, Power series with the Riemann zeta-function in the coefficients, Proc. Japan Acad. Ser. A Math. Sci. 72 (1996), no. 3, 61-63. · Zbl 0860.11050 [6] M. Katsurada, On Mellin-Barnes type of integrals and sums associated with the Riemann zeta-function, Publ. Inst. Math. (Beograd) (N.S.) 62(76) (1997), 13-25. · Zbl 0885.11052 [7] R. B. Paris and D. Kaminski, Asymptotics and Mellin-Barnes integrals , Cambridge Univ. Press, Cambridge, 2001. · Zbl 0983.41019 [8] A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev, Integrals and series , Supplementary chapters, “Nauka”, Moscow, 1986. (in Russian). · Zbl 0606.33001 [9] H. M. Srivastava, K. C. Gupta and S. P. Goyal, The $$H$$-functions of one and two variables , South Asian Pub., New Delhi, 1982. · Zbl 0506.33007 [10] H. M. Srivastava and J. S. Choi, Series associated with the zeta and related functions , Kluwer Acad. Publ., Dordrecht-Boston-London, 2001. · Zbl 1014.33001 [11] H. Tsukada, A general modular relation in analytic number theory, in Number Theory: Sailing on the Sea of Number Theory , 214-236, World Sci. Publ., Hackensack, NJ, 2007. · Zbl 1170.11031 [12] D. P. Verma, Asymptotic expansion of some series involving the Riemann zeta-function, Indian J. Math. 6 (1964), 121-127. · Zbl 0147.06103
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.