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On Kronecker’s limit formula for Dirichlet series with periodic coefficients. (English) Zbl 0654.10039
The author derives the so-called Kronecker’s limit formula of Dirichlet series with periodic coefficients. This formula yields some known and some not-so-known representations of the value of a Dirichlet L-function at \(s=1\) for any Dirichlet character not necessarily primitive. Using these representations, he determines the linear independence of certain trigonometric values over the rational numbers. The subject was motivated by S. Chowla [J. Number Theory 2, 120-123 (1970; Zbl 0211.070)]. Assuming that all Dirichlet L-functions for odd characters modulo N do not vanish at \(s=1\), the author proves that the numbers cot(r\(\pi\) /N) (resp. tan(r\(\pi\) /N), sec(r\(\pi\) /N)) with \(r=1,...,[N/2]\) and \((r,N)=1\) are always linearly independent over the rational numbers and that so are the numbers cosec(r\(\pi\) /N) if and only if (i) \(N\equiv 0 mod 4\), (ii) \(N\equiv 2 mod 4\) and the multiplicative order of 2 mod N/2 is even, or (iii) \(N\equiv 1 mod 2\) and that of 2 mod N is even. The above converse is true. This paper also contains some other equivalent statements and some related topics.
Reviewer: T.Funakura

MSC:
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
11L10 Jacobsthal and Brewer sums; other complete character sums
11J81 Transcendence (general theory)
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