Question

Show that $\zeta(s) = - \dfrac{\sin \pi s}{\pi} \int_0^{\infty} \bigg( \dfrac{\Gamma'(1+x)}{\Gamma(1+x)} - \log x \bigg) x^{-s} dx$.

score 443 · Answer 1 · 2024-01-18 18:42:39

443

Views

Show that $\zeta(s) = - \dfrac{\sin \pi s}{\pi} \int_0^{\infty} \bigg( \dfrac{\Gamma'(1+x)}{\Gamma(1+x)} - \log x \bigg) x^{-s} dx$.

Published on 18 Jan 2024 - 18:42

score 150 · Answer 2 · 2026-03-31 13:02:10

150

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Theorem 4.15. of Titchmarsh's Book of the Zeta Function

Published on 31 Mar 2026 - 13:02

score 59 · Answer 3 · 2026-03-29 12:05:13

59

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What are these points of lower magnitude in the Riemann Zeta?

Published on 29 Mar 2026 - 12:05

score 126 · Answer 4 · 2026-03-26 06:08:10

126

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Winding of $\frac{\zeta(\sigma+it)}{\zeta^{\prime}(\sigma+it)}$ for $0 \lt \sigma \le \frac{1}{2}$

Published on 26 Mar 2026 - 6:08

score 145 · Answer 5 · 2026-03-29 13:47:25

145

Views

What are some unique integral representations of Apery's constant - $\zeta(3)$?

Published on 29 Mar 2026 - 13:47

score 122 · Answer 6 · 2026-03-28 03:27:40

122

Views

Could this closed form for negative integer values of a series involving $\zeta(s)$ be proven?

Published on 28 Mar 2026 - 3:27

score 63 · Answer 7 · 2024-01-25 18:01:21

63

Views

Nature of $\zeta(s)-\zeta(s-1)$

Published on 25 Jan 2024 - 18:01

score 100 · Answer 8 · 2026-03-26 07:46:30

100

Views

Riemann Hypothesis follows from the statement $M\left(x\right)=o_x(x^{\frac{1}{2}+\varepsilon})$

Published on 26 Mar 2026 - 7:46

score 75 · Answer 9 · 2024-01-28 19:09:54

75

Views

Do the zeros of this series all reside on the critical line $\Re(s) = \frac12$?

Published on 28 Jan 2024 - 19:09

score 67 · Answer 10 · 2024-02-05 00:06:16

67

Views

How $\Xi(t)$ function in the theory of Riemann zeta function is real for real $t$?

Published on 05 Feb 2024 - 0:06

score 57 · Answer 11 · 2026-04-03 12:14:37

57

Views

Proofs for the properties of $\mu(\sigma)$, the order of $\zeta(s)$

Published on 03 Apr 2026 - 12:14

score 77 · Answer 12 · 2026-04-03 08:03:36

77

Views

Probability that two random integers have at most $n$ prime factors in common

Published on 03 Apr 2026 - 8:03

score 58 · Answer 13 · 2026-03-31 12:56:31

58

Views

Understanding Theorem 5.5. of Titchmarsh's Book

Published on 31 Mar 2026 - 12:56

score 90 · Answer 14 · 2026-03-26 07:44:17

90

Views

Using Model-Theoretic Proof of Ax-Grothendieck for the Riemann Hypothesis

Published on 26 Mar 2026 - 7:44

score 167 · Answer 15 · 2026-03-27 21:35:47

167

Views

Integrals of the form $\int_0^1\text{arctanh}^a(y) y^{2b} dy$

Published on 27 Mar 2026 - 21:35

List Question

Show that $\zeta(s) = - \dfrac{\sin \pi s}{\pi} \int_0^{\infty} \bigg( \dfrac{\Gamma'(1+x)}{\Gamma(1+x)} - \log x \bigg) x^{-s} dx$.

Theorem 4.15. of Titchmarsh's Book of the Zeta Function

What are these points of lower magnitude in the Riemann Zeta?

Winding of $\frac{\zeta(\sigma+it)}{\zeta^{\prime}(\sigma+it)}$ for $0 \lt \sigma \le \frac{1}{2}$

What are some unique integral representations of Apery's constant - $\zeta(3)$?

Could this closed form for negative integer values of a series involving $\zeta(s)$ be proven?

Nature of $\zeta(s)-\zeta(s-1)$

Riemann Hypothesis follows from the statement $M\left(x\right)=o_x(x^{\frac{1}{2}+\varepsilon})$

Do the zeros of this series all reside on the critical line $\Re(s) = \frac12$?

How $\Xi(t)$ function in the theory of Riemann zeta function is real for real $t$?

Proofs for the properties of $\mu(\sigma)$, the order of $\zeta(s)$

Probability that two random integers have at most $n$ prime factors in common

Understanding Theorem 5.5. of Titchmarsh's Book

Using Model-Theoretic Proof of Ax-Grothendieck for the Riemann Hypothesis

Integrals of the form $\int_0^1\text{arctanh}^a(y) y^{2b} dy$

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