CONVERGENCE OF DIRICHLET SERIES AND EULER PRODUCTS

Doug S. Phillips, Peter Zvengrowski

Abstract


The first part of this paper deals with Dirichlet series, and convergence theorems are proved that strengthen the classical convergence theorem as found e.g. in Serre’s “A Course in Arithmetic.” The second part deals with Euler-type products. A convergence theorem is proved giving sufficient conditions for such products to converge in the half-plane having real part greater than 1/2. Numerical evidence is also presented that suggests that the Euler products corresponding to Dirichlet L-functions L(s, χ), where χ is a primitive Dirichlet character, converge in this half-plane.

Keywords


Dirichlet series, Euler products, L-functions, primitive Dirichlet character

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References


T. M. Apostol, Mathematical Analysis, Second Edition, Addison-Wesley, Menlo Park CA (1974).

D. H. Bailey, “MPFUN2015 : A thread-safe arbitrary precision package” http://www.davidhbailey.com/dhbpapers/mpfun2015.pdf.

R. G. Bartle, The Elements of Real Analysis, John Wiley & Sons, N.Y., London, Sydney (1964).

J. B. Conrey, A. Ghosh, On the Selberg class of Dirichlet series : small degrees, Duke Math. J. 72 (1993), 673-693.

A. Granville, G. Martin, Prime number races, Amer. Math. Monthly 113, No. 1 (2006), 1-33.

G. J. O. Jameson, The Prime Number Theorem. London Math. Soc., Student Texts 53, Cambridge Univ. Press,

Cambridge (2003).

J. Kaczoworski, A. Petrelli, On the structure of the Selberg class, 0 ≤ d ≤ 1, Acta Math. 182 (1999), 207-241.

S. Lang, Complex Analysis, Fourth Edition, Graduate Texts in Mathematics 103, Springer-Verlag, N.Y., Berlin

(1999).

Maple (17), Maplesoft, a division of Waterloo Maple Inc., Waterloo, Ontario.

M. R. Murty, Problems in Analytic Number Theory, Graduate Texts in Mathematics 206, Springer-Verlag,

N.Y., Berlin, Heidelberg (2001).

J. P. Serre, A Course in Arithmetic, Graduate Texts in Mathematics 7, Springer-Verlag, N.Y., Berlin (1973).

Wolfram Research, Inc., Mathematica, Version 9.0, Champaign, IL (2012).




DOI: http://dx.doi.org/10.20903/csnmbs.masa.2017.38.2.111

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