Question

How can I prove the equality of these two infinite sums: $\sum\limits_{n=1}^{\infty}[ 2\, n\,\pi\coth^{-1}(2n)-\pi]= \frac\pi2-\frac{\pi\,\ln(2)}2$

score 75 · Answer 1 · 2026-03-30 03:39:13

75

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How can I prove the equality of these two infinite sums: $\sum\limits_{n=1}^{\infty}[ 2\, n\,\pi\coth^{-1}(2n)-\pi]= \frac\pi2-\frac{\pi\,\ln(2)}2$

Published on 30 Mar 2026 - 3:39

score 141 · Answer 2 · 2026-03-27 14:58:54

141

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Proof that $\zeta(11) > \frac{\pi^3}{31}$.

Published on 27 Mar 2026 - 14:58

score 8.4k · Answer 3 · 2026-03-27 13:27:30

8.4k

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Unexpected appearances of $\pi^2 /~6$.

Published on 27 Mar 2026 - 13:27

score 62 · Answer 4 · 2020-01-19 09:04:40

62

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Riemann $\xi$ function is holomorphic

Published on 19 Jan 2020 - 9:04

score 70 · Answer 5 · 2026-03-25 16:13:59

70

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Questions on explicit formulas related to $\frac{L_{k,j}(s)}{\zeta(s)}$

Published on 25 Mar 2026 - 16:13

score 231 · Answer 6 · 2026-03-30 15:14:58

231

Views

Do we know it is impossible to find a closed form of the Riemann Zeta function?

Published on 30 Mar 2026 - 15:14

score 64 · Answer 7 · 2026-02-23 08:38:42

64

Views

A question regarding Euler's number, Riemann's zeta function and the prime-number theorem

Published on 23 Feb 2026 - 8:38

score 936 · Answer 8 · 2026-03-26 01:14:36

936

Views

What is the value of ζ(1)?

Published on 26 Mar 2026 - 1:14

score 78 · Answer 9 · 2026-03-25 14:21:44

78

Views

How to analytically continue the ‘two-dimension’ zeta function?

Published on 25 Mar 2026 - 14:21

score 213 · Answer 10 · 2026-03-26 02:57:39

213

Views

Prove $\int^\infty_0 \frac{e^{-t}}{t}\left[\frac1{t^2}-\frac1{(1-e^{-t})^2}+\frac1{1-e^{-t}}-\frac1{12}\right]dt=\frac34-\zeta'(-1)+\zeta'(0)$

Published on 26 Mar 2026 - 2:57

score 69 · Answer 11 · 2026-03-30 05:13:11

69

Views

Summation of gamma function in terms of reimann zeta function

Published on 30 Mar 2026 - 5:13

score 177 · Answer 12 · 2026-02-24 07:18:32

177

Views

Uniform convergence of $f: \mathbb{R} \rightarrow \mathbb{R} $ Where $f(\sigma) = \sum_{n=1}^{\infty}\mu(n)/n^\sigma$

Published on 24 Feb 2026 - 7:18

score 43 · Answer 13 · 2026-03-30 10:11:45

43

Views

Significance of two trigonometric function in theory of numbers:

Published on 30 Mar 2026 - 10:11

score 89 · Answer 14 · 2020-02-08 00:17:17

89

Views

How to compute an upper bound of $1 + \frac{1}{2^a} + \frac{1}{3^a} + \cdots$?

Published on 08 Feb 2020 - 0:17

score 125 · Answer 15 · 2026-03-25 14:22:59

125

Views

The Riemann zeta function and analytic continuation, how many solutions are possible to his analytic continuation?

Published on 25 Mar 2026 - 14:22

List Question

How can I prove the equality of these two infinite sums: $\sum\limits_{n=1}^{\infty}[ 2\, n\,\pi\coth^{-1}(2n)-\pi]= \frac\pi2-\frac{\pi\,\ln(2)}2$

Proof that $\zeta(11) > \frac{\pi^3}{31}$.

Unexpected appearances of $\pi^2 /~6$.

Riemann $\xi$ function is holomorphic

Questions on explicit formulas related to $\frac{L_{k,j}(s)}{\zeta(s)}$

Do we know it is impossible to find a closed form of the Riemann Zeta function?

A question regarding Euler's number, Riemann's zeta function and the prime-number theorem

What is the value of ζ(1)?

How to analytically continue the ‘two-dimension’ zeta function?

Prove $\int^\infty_0 \frac{e^{-t}}{t}\left[\frac1{t^2}-\frac1{(1-e^{-t})^2}+\frac1{1-e^{-t}}-\frac1{12}\right]dt=\frac34-\zeta'(-1)+\zeta'(0)$

Summation of gamma function in terms of reimann zeta function

Uniform convergence of $f: \mathbb{R} \rightarrow \mathbb{R} $ Where $f(\sigma) = \sum_{n=1}^{\infty}\mu(n)/n^\sigma$

Significance of two trigonometric function in theory of numbers:

How to compute an upper bound of $1 + \frac{1}{2^a} + \frac{1}{3^a} + \cdots$?

The Riemann zeta function and analytic continuation, how many solutions are possible to his analytic continuation?

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