The multiple Dirichlet product and the multiple Dirichlet series

Preprint English OPEN
Onozuka, Tomokazu (2016)
  • Subject: 11M32, 11A25 | Mathematics - Number Theory
    arxiv: Mathematics::Spectral Theory | Mathematics::Analysis of PDEs
First, we define the multiple Dirichlet product and study the properties of it. From those properties, we obtain a zero-free region of a multiple Dirichlet series and a multiple Dirichlet series expression of the reciprocal of a multiple Dirichlet series.
  • References (11)
    11 references, page 1 of 2

    [1] S. Akiyama, S. Egami and Y. Tanigawa, Analytic continuation of multiple zeta-functions and their values at non-positive integers, Acta Arith, 98 (2001), 107-116.

    [2] S. Akiyama and H. Ishikawa, On analytic continuation of multiple Lfunctions, Analytic Number Theory, C. Jia and K. Matsumoto, eds. (2002), 1-16.

    [3] E. D. Cashwell and C. J. Everett, The ring of number-theoretic functions, Pacific J. Math. 9 (1959), 975-985.

    [4] R. de la Bret´eche, Estimation de sommes multiples de fonctions arith´etiques, Compositio Mathematica 128 (2001), 261-298.

    [5] M. Hoffman, Multiple harmonic series, Pacific J. Math., 152 (1992), 275-290.

    [6] K. Matsumoto, On analytic continuation of various multiple zetafunctions, Number Theory for the Millenium (Urbana, 2000), Vol. II, M. A. Bennett et. al. (eds.), A. K. Peters, Natick, MA, 2002, pp. 417- 440.

    [7] K. Matsumoto, On Mordell-Tornheim and other multiple zeta-functions, In: Proc. Session in Analytic Number Theory and Diophantine Equations, (eds. D. R. Heath-Brown and B. Z. Moroz), Bonner Math. Schriften, 360, Bonn, 2003, n.25, 17pp.

    [8] K. Matsumoto and Y. Tanigawa, The analytic continuation and the order estimate of multiple Dirichlet series, J. Th´eorie des Nombres de Bordeaux, 15 (2003), 267-274.

    [9] R. Spira, Zero-free regions of ζ(k)(s), J. London Math. Soc., 40 (1965), 677-682.

    [10] L. T´oth, Multiplicative arithmetic functions of several variables: a survey, preprint, arXiv:1310.7053.

  • Metrics
    No metrics available
Share - Bookmark