On some series connected with Riemann zeta function

2017/07/13 by V. E. Shestopal, Shestopal, V. E.
Mathematics · #Advanced Mathematical Theories #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1707.04190

openalex publication_date 2017/07/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Using properties of the Riemann zeta-function we propose two new large classes of evaluated series. Incidentally the first class represents integrals as generalized average on very nonuniform sequences. The second class contains inter alia a lot of new series with the Jacoby theta-functions and rationals of the exponential function. Moreover we propose many functions that can replace the Riemann zeta-function in similar constructions. Two examples: 1) if f(x) has period 1 and is in some Lipschitz class, we have for any natural M>1 ln M⋅∫₀¹f(x)dx = ∑_{n≥ 1} ∑_k=1^M-1[(1)/(Mn-k)f((ln (Mn-k))/(ln M))-(1)/(Mn) f((ln (Mn))/(ln M))], 2) if φ_J,M,N(w) = (-1)^J((d^J)/(dw^J))(\fracNe^Nw-1-\fracM(e^Mw-1), where J,M,N are integer, M>N>1, J≥ 0 and for all n∈ℤ, (e^M(M/N)^n+w - 1) (e^N(M/N)^n+w -1 ) ≠ 0, we have ∑_n∈ℤ(M/N)^(J+1)(n+w) φ_J,M,N((M/N)^n+w)=J!.

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