Infinite Series Concerning Tails of Riemann Zeta Values

Li, Chunli; Chu, Wenchang

doi:10.3390/axioms12080761

Open AccessArticle

Infinite Series Concerning Tails of Riemann Zeta Values

by

Chunli Li

^*

and

Wenchang Chu

^*

School of Mathematics and Statistics, Zhoukou Normal University, Zhoukou 466001, China

^*

Authors to whom correspondence should be addressed.

Axioms 2023, 12(8), 761; https://doi.org/10.3390/axioms12080761

Submission received: 15 June 2023 / Revised: 25 July 2023 / Accepted: 31 July 2023 / Published: 2 August 2023

(This article belongs to the Special Issue Applications of Number Theory to Science and Technology)

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Abstract

Infinite series involving Riemann’s zeta and Dirichlet’s lambda tails, and weighted by three harmonic-like elementary symmetric functions are examined. By means of integral representations of zeta tails together with the telescopic approach, twelve general summation theorems are established that express these series as coefficients of the bivariate beta function

Beta (u, v)

. By further expanding

Beta (u, v)

into Laurent series in u and v, several explicit summation formulae are shown as consequences.

Keywords: Riemann zeta function; Dirichlet lambda function; gamma function; beta function

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MDPI and ACS Style

Li, C.; Chu, W. Infinite Series Concerning Tails of Riemann Zeta Values. Axioms 2023, 12, 761. https://doi.org/10.3390/axioms12080761

AMA Style

Li C, Chu W. Infinite Series Concerning Tails of Riemann Zeta Values. Axioms. 2023; 12(8):761. https://doi.org/10.3390/axioms12080761

Chicago/Turabian Style

Li, Chunli, and Wenchang Chu. 2023. "Infinite Series Concerning Tails of Riemann Zeta Values" Axioms 12, no. 8: 761. https://doi.org/10.3390/axioms12080761

APA Style

Li, C., & Chu, W. (2023). Infinite Series Concerning Tails of Riemann Zeta Values. Axioms, 12(8), 761. https://doi.org/10.3390/axioms12080761

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

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