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Two problems related to the non-vanishing of \(L(1,\chi)\). (English) Zbl 0929.11003

For a real number \(u\) let \(((u)):= u-[u]-1/2\) denote the centered fractional part of \(u\). The authors consider two different problems both leading to the study of the ranks of matrices with entries zero or \(((xy/q))\), where \(0\leq x\), \(y<q\). They observe also that these ranks are closely connected with the nonvanishing of the Dirichlet functions \(L(s,\chi)\) at \(s=1\).
The first problem has its origin in the diophantine geometry and asks for the number of linearly independent functions of the form \[ f_{a,q}(n) =n/q+ \bigl((an/q) \bigr)- \biggl(\bigl(a+1) n/q\bigr) \biggr)=n/q+ g_{a, q}(n),\quad a,q\in\mathbb{Z},\;q>0. \] It is proved (Theorem 1) that the vector space generated by the functions \(g_{a,q}\) \((f_{a,q}\text{ resp.)}\) is equal to \([(q-3)/2]+d(q)\) \(([(q-1)/2]+d(q)\) resp.).
The second problem arises from Fourier analysis and concerns functions of the form \[ g(x)=\sum^\infty_{n=1}{f(n)\over n}\cos 2 \pi nx, \] where \(f\) is complex-valued and periodic modulo \(q\) for a positive integer \(q\). It is proved that if all sums \(\delta_h(d):={1\over d}\sum^d_{a=1} h(a/d)\) vanish for \(d\geq 1\) then \(f\) is identically zero. Hence the sums \(\delta_h (d)\) uniquely determine \(h\) (Theorem 2).

MSC:

11A25 Arithmetic functions; related numbers; inversion formulas
15A03 Vector spaces, linear dependence, rank, lineability
11M20 Real zeros of \(L(s, \chi)\); results on \(L(1, \chi)\)
42A16 Fourier coefficients, Fourier series of functions with special properties, special Fourier series
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References:

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