Question

Is known the function $\sum_{n=1}^\infty\frac{(-1)^n\mu(n)}{n^s}$, where $s$ is the complex variable and $\mu(n)$ the Möbius function?

score 153 · Answer 1 · 2026-03-27 02:02:48

153

Views

Is known the function $\sum_{n=1}^\infty\frac{(-1)^n\mu(n)}{n^s}$, where $s$ is the complex variable and $\mu(n)$ the Möbius function?

Published on 27 Mar 2026 - 2:02

score 78 · Answer 2 · 2018-01-02 04:00:50

78

Views

Why $\sum_{d|n} \mu(d)q^{n/d}= \sum_{d|n} \mu(n/d)q^d$

Published on 02 Jan 2018 - 4:00

score 101 · Answer 3 · 2018-01-03 12:44:19

101

Views

How to find a chord with length $1/11$ of the function $\sum_{n=1}^\infty\frac{\mu(n)}{n}x^n$, where $x\in[0,1]$ and $\mu(n)$ is the Möbius function?

Published on 03 Jan 2018 - 12:44

score 184 · Answer 4 · 2018-01-04 12:49:29

184

Views

Convergence of $\sum_{n=1}^\infty\mu(n)\int_0^1\cos\left(\sqrt{\pi n z}\right)dz$, where $\mu(n)$ is the Möbius function

Published on 04 Jan 2018 - 12:49

score 46 · Answer 5 · 2018-01-06 18:50:43

46

Views

An inequality involving the Möbius function deduced using the Steffensen's inequality

Published on 06 Jan 2018 - 18:50

score 131 · Answer 6 · 2026-02-23 03:31:10

131

Views

Real analysis and convergence versus $\int_0^1\sum_{n=1}^\infty x^2\mu(n)\frac{1-\cos(nx)}{n(1-\cos(x))}dx$, where $\mu(n)$ is the Möbius function

Published on 23 Feb 2026 - 3:31

score 3.3k · Answer 7 · 2026-03-26 01:03:35

3.3k

Views

Prove that $\sum_{d|n} \mu(d)\sigma(d) = (-1)^{k} \prod_{i=1}^{k} p_i$

Published on 26 Mar 2026 - 1:03

score 6.4k · Answer 8 · 2011-03-07 05:10:32

6.4k

Views

Prove sum of primitive roots congruent to $\mu(p-1) \pmod{p}$

Published on 07 Mar 2011 - 5:10

score 374 · Answer 9 · 2011-07-26 20:27:21

374

Views

How do I show that $\frac{n}{\varphi(n)}=\sum\limits_{d \mid n} \frac{\mu^2(d)}{\varphi(d)}$?

Published on 26 Jul 2011 - 20:27

score 4.3k · Answer 10 · 2026-03-26 01:03:05

4.3k

Views

Is there a "nice" formula for $\sum_{d|n}\mu(d)\phi(d)$?

Published on 26 Mar 2026 - 1:03

score 567 · Answer 11 · 2012-12-31 13:50:37

567

Views

Möbius function from random number sequence

Published on 31 Dec 2012 - 13:50

score 4.2k · Answer 12 · 2013-03-17 07:45:08

4.2k

Views

Prove $\sum_{d \leq x} \mu(d)\left\lfloor \frac xd \right\rfloor = 1 $

Published on 17 Mar 2013 - 7:45

score 244 · Answer 13 · 2020-08-08 17:19:32

244

Views

Problem with the Möbius Inversion Formula

Published on 08 Aug 2020 - 17:19

score 68 · Answer 14 · 2020-08-13 07:00:04

68

Views

estimate for Möbius function of order k

Published on 13 Aug 2020 - 7:00

score 62 · Answer 15 · 2020-09-08 13:01:14

62

Views

Non-trivial bound on $\sum_{k=0}^{N-1} \sum_{n=1}^N \frac{\mu(n+k)}{n}$

Published on 08 Sep 2020 - 13:01

List Question

Is known the function $\sum_{n=1}^\infty\frac{(-1)^n\mu(n)}{n^s}$, where $s$ is the complex variable and $\mu(n)$ the Möbius function?

Why $\sum_{d|n} \mu(d)q^{n/d}= \sum_{d|n} \mu(n/d)q^d$

How to find a chord with length $1/11$ of the function $\sum_{n=1}^\infty\frac{\mu(n)}{n}x^n$, where $x\in[0,1]$ and $\mu(n)$ is the Möbius function?

Convergence of $\sum_{n=1}^\infty\mu(n)\int_0^1\cos\left(\sqrt{\pi n z}\right)dz$, where $\mu(n)$ is the Möbius function

An inequality involving the Möbius function deduced using the Steffensen's inequality

Real analysis and convergence versus $\int_0^1\sum_{n=1}^\infty x^2\mu(n)\frac{1-\cos(nx)}{n(1-\cos(x))}dx$, where $\mu(n)$ is the Möbius function

Prove that $\sum_{d|n} \mu(d)\sigma(d) = (-1)^{k} \prod_{i=1}^{k} p_i$

Prove sum of primitive roots congruent to $\mu(p-1) \pmod{p}$

How do I show that $\frac{n}{\varphi(n)}=\sum\limits_{d \mid n} \frac{\mu^2(d)}{\varphi(d)}$?

Is there a "nice" formula for $\sum_{d|n}\mu(d)\phi(d)$?

Möbius function from random number sequence

Prove $\sum_{d \leq x} \mu(d)\left\lfloor \frac xd \right\rfloor = 1 $

Problem with the Möbius Inversion Formula

estimate for Möbius function of order k

Non-trivial bound on $\sum_{k=0}^{N-1} \sum_{n=1}^N \frac{\mu(n+k)}{n}$

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