Question

Prove that: $\zeta(3)=\lim_{N\to \infty}{1\over N}\sum_{k=1}^{N}{1\over k^{\phi^2}\ln{\left(1+{k^{\phi^{-2}}\over N}\right)}}$

score 162 · Answer 1 · 2016-12-27 21:21:56

162

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Prove that: $\zeta(3)=\lim_{N\to \infty}{1\over N}\sum_{k=1}^{N}{1\over k^{\phi^2}\ln{\left(1+{k^{\phi^{-2}}\over N}\right)}}$

Published on 27 Dec 2016 - 21:21

score 235 · Answer 2 · 2016-12-29 11:30:39

235

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Trouble understanding Hadamard's proof that there are no zeros of the zeta function with real part 1

Published on 29 Dec 2016 - 11:30

score 77 · Answer 3 · 2016-12-31 14:53:14

77

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Another method of proving:$\zeta(s)={1\over 1-2^{1-s}}\sum_{n=0}^{\infty}{1\over 2^{n+1}}\sum_{k=0}^{n}(-1)^k{n\choose k}(k+1)^{-s}$

Published on 31 Dec 2016 - 14:53

score 221 · Answer 4 · 2026-03-31 17:49:02

221

Views

Is there a fixed real number for which $\displaystyle\zeta(x)$ is integer?

Published on 31 Mar 2026 - 17:49

score 495 · Answer 5 · 2026-03-26 06:30:57

495

Views

Integral Representation of Gamma & Zeta Functions Inside the Critical Strip

Published on 26 Mar 2026 - 6:30

score 511 · Answer 6 · 2017-01-04 21:36:56

511

Views

Deriving a Series Representation of an Integral (Riemann Zeta Function)

Published on 04 Jan 2017 - 21:36

score 7.8k · Answer 7 · 2026-03-26 06:29:03

7.8k

Views

Intuition behind $\zeta(-1)$ = $\frac{-1}{12}$

Published on 26 Mar 2026 - 6:29

score 89 · Answer 8 · 2017-01-09 01:06:41

89

Views

Zero free region: there exists $c>0$ such that if $\sigma\geq 1-c/\log(|t|+4)$ then $\zeta(s)\neq 0$

Published on 09 Jan 2017 - 1:06

score 267 · Answer 9 · 2026-03-28 15:20:22

267

Views

About the number of zeros of the zeta function?

Published on 28 Mar 2026 - 15:20

score 472 · Answer 10 · 2026-03-26 01:14:30

472

Views

Is this Dirichlet "$\eta$ version" of the inverse relationship between $\zeta(s)$ and the Möbius function correct?

Published on 26 Mar 2026 - 1:14

score 241 · Answer 11 · 2017-01-11 12:28:24

241

Views

Zeta function but with other set of numbers?

Published on 11 Jan 2017 - 12:28

score 137 · Answer 12 · 2026-03-26 06:18:56

137

Views

A telescoping series with the Möbius function. Does a closed form exist?

Published on 26 Mar 2026 - 6:18

score 240 · Answer 13 · 2026-03-28 20:51:58

240

Views

Are non-squares primitive roots?

Published on 28 Mar 2026 - 20:51

score 803 · Answer 14 · 2017-01-14 04:31:46

803

Views

More elegant $\zeta(s)$ zeros counting function than $N(T)$

Published on 14 Jan 2017 - 4:31

score 152 · Answer 15 · 2026-04-01 04:38:47

152

Views

Motivations and applications for the functional equation of $\zeta$

Published on 01 Apr 2026 - 4:38

List Question

Prove that: $\zeta(3)=\lim_{N\to \infty}{1\over N}\sum_{k=1}^{N}{1\over k^{\phi^2}\ln{\left(1+{k^{\phi^{-2}}\over N}\right)}}$

Trouble understanding Hadamard's proof that there are no zeros of the zeta function with real part 1

Another method of proving:$\zeta(s)={1\over 1-2^{1-s}}\sum_{n=0}^{\infty}{1\over 2^{n+1}}\sum_{k=0}^{n}(-1)^k{n\choose k}(k+1)^{-s}$

Is there a fixed real number for which $\displaystyle\zeta(x)$ is integer?

Integral Representation of Gamma & Zeta Functions Inside the Critical Strip

Deriving a Series Representation of an Integral (Riemann Zeta Function)

Intuition behind $\zeta(-1)$ = $\frac{-1}{12}$

Zero free region: there exists $c>0$ such that if $\sigma\geq 1-c/\log(|t|+4)$ then $\zeta(s)\neq 0$

About the number of zeros of the zeta function?

Is this Dirichlet "$\eta$ version" of the inverse relationship between $\zeta(s)$ and the Möbius function correct?

Zeta function but with other set of numbers?

A telescoping series with the Möbius function. Does a closed form exist?

Are non-squares primitive roots?

More elegant $\zeta(s)$ zeros counting function than $N(T)$

Motivations and applications for the functional equation of $\zeta$

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