Fourier series and the special values of \(L\)-functions. (English) Zbl 0649.12009

Let \(k\) be a global function field over a finite field, and let \(A\) be the ring of functions which are regular outside \(\infty\), where \(\infty\) is a place of \(k\). The \(L\)-series \(L(s,\chi)=\sum \chi (B)N(B^{-s})\) is extended over all non-trivial ideals \(B\) of \(A\) being prime to the conductor \(F\) of the idele class character \(\chi\). In former papers S. Galovich and M. Rosen [J. Number Theory 13, 363–375 (1981; Zbl 0473.12014)] and D. Hayes [Invent. Math. 65, 49–69 (1981; Zbl 0491.12014)] proved formulas for the values of \(L(1,\chi)\), \(\chi\neq 1\).
In this paper all these formulas are reproved by applying Fourier analysis on the locally compact field \(k_{\infty}\). Also a functional equation for \(L(s,\chi)\) is obtained.


11R42 Zeta functions and \(L\)-functions of number fields
43A32 Other transforms and operators of Fourier type
11R58 Arithmetic theory of algebraic function fields
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