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Period functions and cotangent sums
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Period functions and cotangent sums
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We investigate the period function of $\sum_{n=1}^\infty\sigma_a(n)\e{nz}$, showing it can be analytically continued to $|\arg z|<\pi$ and studying its Taylor series. We use these results to give a simple proof of the Voronoi formula and to prove an exact formula for the second moments of the Riemann zeta function. Moreover, we introduce a family of cotangent sums, functions defined over the rationals, that generalize the Dedekind sum and share with it the property of satisfying a reciprocity formula. In particular, we find a reciprocity formula for the Vasyunin sum.
Forward citations
Cited by 2 Pith papers
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A few remarks on the Baez-Duarte Criterion
A canonical third-order truncation of a Vasyunin cotangent sum reduces the open boundedness problem in the Baez-Duarte criterion to an explicit bilinear cancellation estimate.
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A few remarks on the Baez-Duarte Criterion
Several lemmas related to the Báez-Duarte criterion for the Riemann Hypothesis are claimed to be derived.
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