2011arXiv (Cornell University)Open access

List of conjectural series for powers of $\pi$ and other constants

Zhi‐Wei Sun

Open full text 46 citations

Abstract

The author gives the full list of his conjectures on series for powers of $\pi$ and other important constants scattered in some of his public papers or his private diaries. The list contains 234 reasonable conjectural series. On the list there are 178 reasonable series for $\pi^{-1}$, four series for $\pi^2$, two series for $\pi^{-2}$, four series for $\pi^4$, two series for $\pi^5$, three series for $\pi^6$, seven series for $\zeta(3)$, one series for $\pi\zeta(3)$, two series for $\pi^2\zeta(3)$, one series for $\zeta(3)^2$, three series involving both $\zeta(3)^2$ and $\pi^6$, one series for $\zeta(5)$, three series involving both $\zeta(5)$ and $\zeta(2)\zeta(3)$, two series involving both $\pi\zeta(5)$ and $\pi^3\zeta(3)$, three series involving $\zeta(7)$, three series for $K=L(2,(\frac{\cdot}{3}))$, one series for the Catalan constant $G$, two series for $\pi G$, one series involving both $\pi^3G$ and $\pi^2\zeta(3)$, two series for $\pi K$, two series involving $L=L(4,(\frac{\cdot}3))$, three series involving $\beta(4)=L(4,(\frac{-4}{\cdot}))$, and four series for $\pi^2\log a$ with $a=2,3,(\sqrt5+1)/2$. The code of a conjectural series is underlined if and only if a complete proof of the identity is available.

Open-access reader

About this research paper

What this paper is about

The author gives the full list of his conjectures on series for powers of $\pi$ and other important constants scattered in some of his public papers or his private diaries. The list contains 234 reasonable conjectural series. On the list there are 178 reasonable series for $\pi^{-1}$, four series for $\pi^2$, two series for $\pi^{-2}$, four series for $\pi^4$, two series for $\pi^5$, three series for $\pi^6$, seven series for $\zeta(3)$, one series for $\pi\zeta(3)$, two series for $\pi^2\zeta(3)$, one series for $\zeta(3)^2$, three series involving both $\zeta(3)^2$ and $\pi^6$, one series for $\zeta(5)$, three series involving both $\zeta(5)$ and $\zeta(2)\zeta(3)$, two series involving both $\pi\zeta(5)$ and $\pi^3\zeta(3)$, three series involving $\zeta(7)$, three series for $K=L(2,(\frac{\cdot}{3}))$, one series for the Catalan constant $G$, two series for $\pi G$, one series involving both $\pi^3G$ and $\pi^2\zeta(3)$, two series for $\pi K$, two series involving $L=L(4,(\frac{\cdot}3))$, three series involving $\beta(4)=L(4,(\frac{-4}{\cdot}))$, and four series for $\pi^2\log a$ with $a=2,3,(\sqrt5+1)/2$. The code of a conjectural series is underlined if and only if a complete proof of the identity is available.

Why it matters

OpenAlex reports 46 citations for this work. Citation counts describe recorded attention and do not establish research quality.

Key contribution

A contribution statement is not available in the OpenAlex record.

Method / approach

Method details are not available in the OpenAlex metadata.

Main findings

Findings are not separately available in the OpenAlex metadata.

Limitations

Limitations are not available in the OpenAlex metadata.

Applications

Application details are not available in the OpenAlex metadata.

Available abstract

The author gives the full list of his conjectures on series for powers of $\pi$ and other important constants scattered in some of his public papers or his private diaries. The list contains 234 reasonable conjectural series. On the list there are 178 reasonable series for $\pi^{-1}$, four series for $\pi^2$, two series for $\pi^{-2}$, four series for $\pi^4$, two series for $\pi^5$, three series for $\pi^6$, seven series for $\zeta(3)$, one series for $\pi\zeta(3)$, two series for $\pi^2\zeta(3)$, one series for $\zeta(3)^2$, three series involving both $\zeta(3)^2$ and $\pi^6$, one series for $\zeta(5)$, three series involving both $\zeta(5)$ and $\zeta(2)\zeta(3)$, two series involving both $\pi\zeta(5)$ and $\pi^3\zeta(3)$, three series involving $\zeta(7)$, three series for $K=L(2,(\frac{\cdot}{3}))$, one series for the Catalan constant $G$, two series for $\pi G$, one series involving both $\pi^3G$ and $\pi^2\zeta(3)$, two series for $\pi K$, two series involving $L=L(4,(\frac{\cdot}3))$, three series involving $\beta(4)=L(4,(\frac{-4}{\cdot}))$, and four series for $\pi^2\log a$ with $a=2,3,(\sqrt5+1)/2$. The code of a conjectural series is underlined if and only if a complete proof of the identity is available.

Key concepts: Series (stratigraphy), Alternating series, Function series, Neumann series, Geometric series, Mathematics, General Dirichlet series, Power series

Related papers

Back to paper searchBrowse research topicsOriginal source
List of conjectural series for powers of $\pi$ and other constants — Research Paper | ScholarLens