Transcendental Values of Some Dirichlet Series

M. Ram Murty and Purusottam Rath
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M. Ram Murty: Queen’s University, Department of Mathematics and Statistics
Purusottam Rath: Chennai Mathematical Institute

Chapter Chapter 22 in Transcendental Numbers, 2014, pp 123-129 from Springer

Abstract: Abstract There is a large collection of Dirichlet series defined purely arithmetically that have been conjectured to have analytic continuation and functional equations. Deligne [38] has formulated a far-reaching conjecture regarding the special values of these series at special points in the complex plane and one would like to know if these special values are transcendental numbers or not. The most notable example is the L-function attached to an elliptic curve and the Birch and Swinnerton-Dyer conjecture. Birch and Swinnerton-Dyer conjecture In a lecture at the Stony Brook conference on number theory in the summer of 1969, Sarvadaman Chowla Chowla, S. posed the following question. Does there exist a rational-valued arithmetic function f, periodic with prime period p such that ∑ n = 1 ∞ f ( n ) n $$\displaystyle{\sum _{n=1}^{\infty }{f(n) \over n} }$$ converges and equals zero? In 1973, Baker, Birch and Wirsing ([10], see also [29], [31] and [101]) answered this question in the following theorem: Baker, A. Birch, B. Wirsing, E.

Keywords: Dirichlet Series; Transcendent Values; Swinnerton-Dyer Conjecture; Stony Brook Conference; Chowla (search for similar items in EconPapers)
Date: 2014
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DOI: 10.1007/978-1-4939-0832-5_22

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