Sum of digamma asymptotic error terms of an arithmetic series

Zhiqi Huang

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arXiv · 2304.01112

Sum of digamma asymptotic error terms of an arithmetic series

Zhiqi Huang

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Abstract

We define an S function as the sum of the asymptotic error terms of digamma function of an arithmetic series, $S(a) \equiv \sum_{n=1}^\infty \left[\ln\frac{n}{a} - \frac{a}{2n}-\psi\left(\frac{n}{a}\right)\right]$, and show a few properties of it. Using the S function, we construct a real and positive $\phi$ function. Riemann hypothesis holds if $\tilde{\phi}(k)$, the complex Fourier transform of $\phi$, has only real zeros.

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Zhiqi Huang. 2023-04-03. Sum of digamma asymptotic error terms of an arithmetic series. https://arxiv.org/abs/2304.01112

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