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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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