Sum of digamma asymptotic error terms of an arithmetic series

2023/04/03 by Zhiqi Huang, Huang, Zhiqi
Computer Science · Mathematics · #11 #FOS: Mathematics #General Mathematics (math.GM) #Mathematical Analysis and Transform Methods #Mathematical and Theoretical Analysis #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2304.01112

openalex publication_date 2023/04/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We define an S function as the sum of the asymptotic error terms of digamma function of an arithmetic series, S(a) ≡ ∑_n=1^∞ [ln(n)/(a) - (a)/(2n)-ψ((n)/(a))], and show a few properties of it. Using the S function, we construct a real and positive ϕ function. Riemann hypothesis holds if ϕ(k), the complex Fourier transform of ϕ, has only real zeros.

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