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An exponentially-averaged Vasyunin formula. (English) Zbl 07352296
Let $$f_\alpha(t)=\frac{1}{\alpha t}-\frac{1}{e^{\alpha t}-1}$$. In the paper under review the authors compute the integral $$\int_0^\infty f_m(t)f_n(t)dt$$ for coprime integers $$m,n\geq 1$$, where the answer may read as a reciprocity formula for cotangent sums. The work is motivated by studying a probabilistic version of the Nyman-Beurling criterion for the Riemann hypothesis.

MSC:
 11L03 Trigonometric and exponential sums (general theory) 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$ 11M26 Nonreal zeros of $$\zeta (s)$$ and $$L(s, \chi)$$; Riemann and other hypotheses 11F20 Dedekind eta function, Dedekind sums
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References:
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