Glasnik Matematicki, Vol. 55, No. 2 (2020), 253-265.

DIRICHLET PRODUCT AND THE MULTIPLE DIRICHLET SERIES OVER FUNCTION FIELDS

Yoshinori Hamahata

Department of Applied Mathematics, Okayama University of Science, Ridai-cho 1-1, Okayama, 700-0005, Japan
e-mail: hamahata@xmath.ous.ac.jp

Abstract.   We define the Dirichlet product for multiple arithmetic functions over function fields and consider the ring of the multiple Dirichlet series over function fields. We apply our results to absolutely convergent multiple Dirichlet series and obtain some zero-free regions for them.

2010 Mathematics Subject Classification.   11R58, 11A25, 11M32, 11M41

Key words and phrases.   Arithmetic function, Dirichlet product, Dirichlet series, zeta function, function field

Full text (PDF) (free access)

https://doi.org/10.3336/gm.55.2.06

References:

  1. E. Alkan, A. Zaharescu and M. Zaki, Arithmetical functions in several variables, Int. J. Number Theory 1 (2005), 383-399.
    MathSciNet     CrossRef

  2. E. Alkan, A. Zaharescu and M. Zaki, Multidimensional averages and Dirichlet convolution, Manuscripta Math. 123 (2007), 251-267.
    MathSciNet     CrossRef

  3. T. Apostol, Introduction to analytic number theory, Springer, 1976.
    MathSciNet    

  4. E. Cashwell and C. Everett, The ring of number-theoretic functions, Pacific J. Math. 9 (1959), 975-985.
    MathSciNet     CrossRef

  5. P. Haukkanen, Derivation of arithmetical functions under the Dirichlet convolution, Int. J. Number Theory 14 (2018), 1257-1264.
    MathSciNet     CrossRef

  6. R. Masri, Multiple zeta values over global function fields, Proc. Sympos. Pure Math. 75, 2006, 157-175.
    MathSciNet     CrossRef

  7. K. Matsumoto, On analytic continuation of various multiple zeta-functions, In: Number Theory for the Millennium, Vol. 2, A. K. Peters, 2002, pp. 417-440.
    MathSciNet    

  8. K. Matsumoto, On Mordell-Tornheim and other multiple zeta-functions, In: Proceedings of the Session in Analytic Number Theory and Diophantine Equations, 2003, No. 25, 17pp.
    MathSciNet    

  9. T. Onozuka, The multiple Dirichlet product and the multiple Dirichlet series, Int. J. Number Theory 13 (2017), 2181-2193.
    MathSciNet     CrossRef

  10. M. Rosen, Number theory in function fields, Springer, 2002.
    MathSciNet     CrossRef

  11. R. Sivaramakrishnan, Classical theory of arithmetic functions, Marcel Dekker, Inc., New York, 1989.
    MathSciNet    

  12. D. Thakur, Function field arithmetic, World Scientific, River Edge, 2004.
    MathSciNet     CrossRef

  13. L. Tóth, Multiplicative arithmetic functions of several variables: a survey, In: T. Rassias and P. Pardalos (eds.), Mathematics without boundaries, Springer, 2014, 483-514.
    MathSciNet    
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