## 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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