Question

Show $\frac{1}{s-1}+\sum_{n=1}^{\infty}\left[\frac{1}{n^s}+\frac{1}{s-1}\left(\frac{1}{(n+1)^{s-1}} - \frac{1}{n^{s-1}}\right)\right] = \zeta(s)$

score 134 · Answer 1 · 2023-12-25 00:38:04

134

Views

Show $\frac{1}{s-1}+\sum_{n=1}^{\infty}\left[\frac{1}{n^s}+\frac{1}{s-1}\left(\frac{1}{(n+1)^{s-1}} - \frac{1}{n^{s-1}}\right)\right] = \zeta(s)$

Published on 25 Dec 2023 - 0:38

score 446 · Answer 2 · 2026-03-25 12:50:46

446

Views

Prove the zeta log gamma integral $\int_0^1{\zeta(1-n,1-t)\ln\Gamma{(t)}}dt = \frac{H_n\zeta(-n)+\zeta’(-n)}{n}$

Published on 25 Mar 2026 - 12:50

score 62 · Answer 3 · 2026-03-26 06:11:00

62

Views

Better bounds in the error term of the summatory function of Von-Mangoldt function and the Riemann Hypothesis

Published on 26 Mar 2026 - 6:11

score 38 · Answer 4 · 2023-12-28 03:00:12

38

Views

On the number of Mobius $\mu =0$ numbers less than a given number.

Published on 28 Dec 2023 - 3:00

score 142 · Answer 5 · 2023-12-29 07:16:43

142

Views

Definite integral of $x\sin^n(x)$, recurrence relations, $\zeta(3)$

Published on 29 Dec 2023 - 7:16

score 89 · Answer 6 · 2024-01-07 20:31:51

89

Views

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score 101 · Answer 7 · 2024-01-09 14:25:33

101

Views

Kloosterman's identity for deriving the functional equation of the zeta function

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score 111 · Answer 8 · 2026-03-26 17:34:28

111

Views

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score 52 · Answer 9 · 2024-01-11 05:45:13

52

Views

Estimate for sum over zeros of Riemann zeta function

Published on 11 Jan 2024 - 5:45

score 263 · Answer 10 · 2024-01-11 09:03:01

263

Views

On the 'reverse theorem' for functional equation of the Riemann zeta function

Published on 11 Jan 2024 - 9:03

score 253 · Answer 11 · 2024-01-11 19:55:26

253

Views

Show that $\dfrac{1}{4} \sum_{n=1}^{\infty} \dfrac{r(n)}{n^w} - \zeta(2w) = \zeta(w) \eta(w) - \zeta(2w)$.

Published on 11 Jan 2024 - 19:55

score 80 · Answer 12 · 2024-01-11 23:32:18

80

Views

On Ingham's proof of $\zeta(1+it) \ne 0$

Published on 11 Jan 2024 - 23:32

score 71 · Answer 13 · 2026-03-31 10:57:22

71

Views

On inequalities for $\dfrac{1}{\zeta(s)}$, $\dfrac{\zeta'(s)}{\zeta(s)}$ and $\log (\zeta(s))$

Published on 31 Mar 2026 - 10:57

score 70 · Answer 14 · 2026-03-26 06:11:02

70

Views

number of n.t. zeros in an interval in the limit

Published on 26 Mar 2026 - 6:11

score 86 · Answer 15 · 2026-03-26 17:33:16

86

Views

Is the hypersurface of revolution associated to the generator $\zeta_d(t)$ related to the primes/Riemann zeta function?

Published on 26 Mar 2026 - 17:33

List Question

Show $\frac{1}{s-1}+\sum_{n=1}^{\infty}\left[\frac{1}{n^s}+\frac{1}{s-1}\left(\frac{1}{(n+1)^{s-1}} - \frac{1}{n^{s-1}}\right)\right] = \zeta(s)$

Prove the zeta log gamma integral $\int_0^1{\zeta(1-n,1-t)\ln\Gamma{(t)}}dt = \frac{H_n\zeta(-n)+\zeta’(-n)}{n}$

Better bounds in the error term of the summatory function of Von-Mangoldt function and the Riemann Hypothesis

On the number of Mobius $\mu =0$ numbers less than a given number.

Definite integral of $x\sin^n(x)$, recurrence relations, $\zeta(3)$

How to understand asymptotic equivalence when graph oscillates?

Kloosterman's identity for deriving the functional equation of the zeta function

Analytical continuation of $\zeta(s)$ at $\operatorname{Re}(s) = 0$

Estimate for sum over zeros of Riemann zeta function

On the 'reverse theorem' for functional equation of the Riemann zeta function

Show that $\dfrac{1}{4} \sum_{n=1}^{\infty} \dfrac{r(n)}{n^w} - \zeta(2w) = \zeta(w) \eta(w) - \zeta(2w)$.

On Ingham's proof of $\zeta(1+it) \ne 0$

On inequalities for $\dfrac{1}{\zeta(s)}$, $\dfrac{\zeta'(s)}{\zeta(s)}$ and $\log (\zeta(s))$

number of n.t. zeros in an interval in the limit

Is the hypersurface of revolution associated to the generator $\zeta_d(t)$ related to the primes/Riemann zeta function?

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