Abstract
In this paper we shall discuss the moment integrals of 1/sint as a generalization of one of the integral representations of the Catalan constant. As an application of our formulas, we shall derive several relations among the Riemann zeta-, the Dirichlet L-, and multiple L-values mod 4.
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References
Akiyama, S., Egami, S., Tanigawa, Y.: Analytic continuation of multiple zeta-functions and their values at non-positive integers. Acta Arith. 98, 107–116 (2001)
Apostol, T.M.: Introduction to Analytic Number Theory. Springer, New York (1976)
Arakawa, T., Kaneko, M.: Multiple zeta values, poly-Bernoulli numbers, and related zeta functions. Nagoya Math. J. 153, 189–209 (1999)
Arakawa, T., Kaneko, M.: On multiple L-values. J. Math. Soc. Jpn. 56, 967–991 (2004)
Arakawa, T., Kaneko, M.: Introduction to Multiple Zeta Values. MI Lecture Note Series, vol. 23. Kyushu University, Kyushu (2010). Tajyu Zeta-chi Nyumon (in Japanese)
Berndt, B.C.: Ramanujan’s Notebooks Part I. Springer, New York (1985)
Bradley, D.: Representations of Catalan’s constant (1998). http://www.math.umaine.edu/~bradley/papers/pub.html
Espinosa, O., Moll, V.: On some integrals involving the Hurwitz zeta function: Part 1. Ramanujan J. 6, 159–188 (2002)
Espinosa, O., Moll, V.: On some integrals involving the Hurwitz zeta function: Part 2. Ramanujan J. 6, 449–468 (2002)
Gradshteyn, I.S., Ryzhik, I.M.: In: Jeffrey, A., Zwillinger, D. (eds.) Table of Integrals, Series and Products, 7th edn. Academic Press, San Diego (2007)
Hardy, G.H.: Divergent Series, 2nd edn. Chelsea, New York (1991)
Kanemitsu, S., Tanigawa, Y., Yoshimoto, M.: Convolution of Riemann zeta-values. J. Math. Soc. Jpn. 57, 1167–1177 (2005)
Ohno, Y.: A generalization of the duality and sum formulas on the multiple zeta values. J. Number Theory 74, 39–43 (1999)
Ramanujan, S.: On the integral \(\int_{0}^{x} \frac{\tan^{-1}t}{t}\,dt\). J. Indian Math. Soc. 7, 93–96 (1915). (Colletcted papers, pp. 40–43. Chelsea (1962))
Srivastava, H.M., Choi, J.: Series Associated with the Zeta and Related Functions. Kluwer Academic, Dordrecht (2001)
Tsumura, H.: On the parity conjecture for multiple L-values of conductor four. Tokyo J. Math. 30, 21–40 (2007)
Zagier, D.: Values of zeta-functions and their applications. In: First European Congress of Mathematics. Vol. II, pp. 497–512. Birkhäuser, Basel (1994)
Acknowledgements
The authors are grateful to the referee for careful readings and kindly comments, and to Professor Heng Huat Chan for his valuable advice and comments.
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M. Minamide is supported by Grant-in-Aid for Young Scientists (B) No.23740009.
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Furuya, J., Minamide, M. & Tanigawa, Y. Moment integrals of 1/sint and related zeta-values. Ramanujan J 33, 423–445 (2014). https://doi.org/10.1007/s11139-013-9467-1
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DOI: https://doi.org/10.1007/s11139-013-9467-1