Question

Show $\sum_{d|n} d\mu(d) = ((-1)^{ω(n)}ϕ(n)s(n))/n$

score 168 · Answer 1 · 2017-05-01 01:33:05

168

Views

Show $\sum_{d|n} d\mu(d) = ((-1)^{ω(n)}ϕ(n)s(n))/n$

Published on 01 May 2017 - 1:33

score 154 · Answer 2 · 2026-03-25 21:00:13

154

Views

Möbius inversion exercise

Published on 25 Mar 2026 - 21:00

score 120 · Answer 3 · 2017-05-06 11:55:11

120

Views

Maximum of $\sum_{n=1}^\infty\frac{\mu(n)}{n}x^n$, $0<x<1$, where $\mu(n)$ is the Möbius function

Published on 06 May 2017 - 11:55

score 136 · Answer 4 · 2017-05-06 15:34:35

136

Views

On the convergence of $-\int_0^1\left(\int_1^\infty \left(\frac{1}{\zeta(x/y)}-1\right)dx\right)dy$

Published on 06 May 2017 - 15:34

score 50 · Answer 5 · 2017-05-06 22:02:05

50

Views

What about of the convergence of the sequence $\sum_{k=1}^{n-1}\mu(k)\left(\frac{n}{k^2(n-k)}\right)^{2}$, being $\mu(n)$ the Möbius function?

Published on 06 May 2017 - 22:02

score 187 · Answer 6 · 2026-03-25 22:30:42

187

Views

Convergence of the limit of a series involving Möbius and Floor functions

Published on 25 Mar 2026 - 22:30

score 64 · Answer 7 · 2017-05-17 19:28:10

64

Views

About the differentiability of $\sum_{n=1}^\infty\frac{\mu(n)\sin(\sqrt{2n^3}x)}{\sqrt{2n^3}}e^{-i\sqrt{n^3}x}$

Published on 17 May 2017 - 19:28

score 57 · Answer 8 · 2017-05-19 07:51:48

57

Views

On a limit involving $\sum_{n=1}^\infty\frac{\sin(\sqrt{n^3+1}x)}{\sqrt{n^3+1}}$

Published on 19 May 2017 - 7:51

score 42 · Answer 9 · 2017-05-27 13:59:37

42

Views

Prove that 2 is a nonresidue of any integer = ± 3 (mod 8), without using Gauss’s lemma.

Published on 27 May 2017 - 13:59

score 57 · Answer 10 · 2017-06-16 20:38:33

57

Views

On $\sum_{n=1}^\infty\eta\left(\frac{n}{N}\right)\frac{\mu(n)}{n^2}$, where $\eta$ is a cutoff function, in the spirit of Tao's blog

Published on 16 Jun 2017 - 20:38

score 84 · Answer 11 · 2017-06-18 11:13:12

84

Views

Expressing $\sum_{k=1}^\infty\frac{\mu(k)}{k}\cos\left(2\pi\frac{n}{k}\right)$ in terms of $n$

Published on 18 Jun 2017 - 11:13

score 118 · Answer 12 · 2026-03-26 16:56:46

118

Views

On the Möbius function and the difference $(p_{n+1})^{1/2}-(p_{n})^{1/2}$, being $p_n$ the $nth$ prime number: a conjecture

Published on 26 Mar 2026 - 16:56

score 71 · Answer 13 · 2017-07-01 12:37:38

71

Views

squarefree and square full functions

Published on 01 Jul 2017 - 12:37

score 102 · Answer 14 · 2017-07-01 23:18:49

102

Views

Prove that $\sum_{n=1}^\infty\frac{\mu(n)}{n}H_n\sum_{k=n+1}^\infty\frac{\mu(k)}{k^2}$ is convergent

Published on 01 Jul 2017 - 23:18

score 772 · Answer 15 · 2017-07-03 21:49:57

772

Views

Does the Mertens function have an infinite number of zeros?

Published on 03 Jul 2017 - 21:49

List Question

Show $\sum_{d|n} d\mu(d) = ((-1)^{ω(n)}ϕ(n)s(n))/n$

Möbius inversion exercise

Maximum of $\sum_{n=1}^\infty\frac{\mu(n)}{n}x^n$, $0<x<1$, where $\mu(n)$ is the Möbius function

On the convergence of $-\int_0^1\left(\int_1^\infty \left(\frac{1}{\zeta(x/y)}-1\right)dx\right)dy$

What about of the convergence of the sequence $\sum_{k=1}^{n-1}\mu(k)\left(\frac{n}{k^2(n-k)}\right)^{2}$, being $\mu(n)$ the Möbius function?

Convergence of the limit of a series involving Möbius and Floor functions

About the differentiability of $\sum_{n=1}^\infty\frac{\mu(n)\sin(\sqrt{2n^3}x)}{\sqrt{2n^3}}e^{-i\sqrt{n^3}x}$

On a limit involving $\sum_{n=1}^\infty\frac{\sin(\sqrt{n^3+1}x)}{\sqrt{n^3+1}}$

Prove that 2 is a nonresidue of any integer = ± 3 (mod 8), without using Gauss’s lemma.

On $\sum_{n=1}^\infty\eta\left(\frac{n}{N}\right)\frac{\mu(n)}{n^2}$, where $\eta$ is a cutoff function, in the spirit of Tao's blog

Expressing $\sum_{k=1}^\infty\frac{\mu(k)}{k}\cos\left(2\pi\frac{n}{k}\right)$ in terms of $n$

On the Möbius function and the difference $(p_{n+1})^{1/2}-(p_{n})^{1/2}$, being $p_n$ the $nth$ prime number: a conjecture

squarefree and square full functions

Prove that $\sum_{n=1}^\infty\frac{\mu(n)}{n}H_n\sum_{k=n+1}^\infty\frac{\mu(k)}{k^2}$ is convergent

Does the Mertens function have an infinite number of zeros?

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