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A question about Mertens function $M(n)=\sum_{k=1}^n\mu(n)$

score 139 · Answer 1 · 2026-03-22 11:12:00

139

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A question about Mertens function $M(n)=\sum_{k=1}^n\mu(n)$

Published on 22 Mar 2026 - 11:12

score 164 · Answer 2 · 2026-03-20 03:17:57

164

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A good estimation of the inverse $f^{-1}(x)$ of $-\sum_{n=2}^\infty\frac{\mu(n)}{n}x^n$, over the unit inverval if it has mathematical meaning

Published on 20 Mar 2026 - 3:17

score 49 · Answer 3 · 2026-03-22 17:38:03

49

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Find a continuous function, over the unit interval, satisfying $\int_0^x f(u)du\geq \sum_{n=1}^\infty\frac{\mu(n)}{n^2}x^{n-1}$ for each $ x\in[0,1]$

Published on 22 Mar 2026 - 17:38

score 113 · Answer 4 · 2026-03-23 03:09:54

113

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A rigorous proof of $\Re\int_0^{2\pi}\left(\sum_{n=1}^{\infty}\frac{\mu(n)}{n}\log\left(\frac{1}{1-ne^{i\theta}}\right)\right)d\theta=2\pi$

Published on 23 Mar 2026 - 3:09

score 87 · Answer 5 · 2026-03-22 11:43:08

87

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An inequality deduced for $-\sum_{n=1}^\infty\frac{\mu(n)}{n}x^{n-1}$ on assumption of convexity, invoking a theorem due to Dragomir

Published on 22 Mar 2026 - 11:43

score 169 · Answer 6 · 2026-03-22 20:55:56

169

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Why $\sum\frac{\mu(h)\mu(k)}{hk}\gcd(h,k)=\prod\limits_{p\le x}\left(1-\frac1p\right)$, where the sum enumerates the pairs $(h,k)$ of primes below $x$

Published on 22 Mar 2026 - 20:55

score 40 · Answer 7 · 2026-03-22 20:59:40

40

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Two questions concerning series involving the Möbius function and trigonometric functions

Published on 22 Mar 2026 - 20:59

score 117 · Answer 8 · 2026-03-22 14:34:02

117

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Where is positive or negative the function $\sum_{n=1}^\infty\frac{\mu(n)}{n}\left(\frac{\cos(nx)}{n}\right)^2$ over the set $[0,2\pi]$?

Published on 22 Mar 2026 - 14:34

score 1.2k · Answer 9 · 2026-03-23 03:04:36

1.2k

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Prove $ \left\vert \sum_{n=1}^N \frac{\mu(n)}{n} \right\vert \leqslant 1,$ where $\mu(n)$ is the Mobius function.

Published on 23 Mar 2026 - 3:04

score 88 · Answer 10 · 2026-03-24 17:10:13

88

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Justify an approximation of $-\sum_{n=2}^\infty H_n\left(\frac{1}{\zeta(n)}-1\right)$, where $H_n$ denotes the $n$th harmonic number

Published on 24 Mar 2026 - 17:10

score 44 · Answer 11 · 2026-03-22 19:19:07

44

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polynomial annihilators of finite rings - proof generalization sought

Published on 22 Mar 2026 - 19:19

score 352 · Answer 12 · 2026-03-25 03:04:17

352

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What is the inverse of $\left[ \sum_{k=1}^{j} \left\lfloor \frac{i}{k} \right\rfloor \right]_{n \times n}$?

Published on 25 Mar 2026 - 3:04

score 113 · Answer 13 · 2026-03-20 03:12:22

113

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Divergence of $\sum_{n=1}^\infty\frac{\mu(n)}{\sqrt{n}}\cos\left(n^2 \pi \gamma\right)$, where $\gamma$ is the Euler-Mascheroni constant

Published on 20 Mar 2026 - 3:12

score 271 · Answer 14 · 2026-03-24 05:00:21

271

Views

The meaning and definition of $\psi^{(-2)}(x)$, and the convergence of some related series involving the Möbius function

Published on 24 Mar 2026 - 5:00

score 314 · Answer 15 · 2026-03-25 04:34:51

314

Views

Proof with Möbius function, divisor function and little omega function

Published on 25 Mar 2026 - 4:34

List Question

A question about Mertens function $M(n)=\sum_{k=1}^n\mu(n)$

A good estimation of the inverse $f^{-1}(x)$ of $-\sum_{n=2}^\infty\frac{\mu(n)}{n}x^n$, over the unit inverval if it has mathematical meaning

Find a continuous function, over the unit interval, satisfying $\int_0^x f(u)du\geq \sum_{n=1}^\infty\frac{\mu(n)}{n^2}x^{n-1}$ for each $ x\in[0,1]$

A rigorous proof of $\Re\int_0^{2\pi}\left(\sum_{n=1}^{\infty}\frac{\mu(n)}{n}\log\left(\frac{1}{1-ne^{i\theta}}\right)\right)d\theta=2\pi$

An inequality deduced for $-\sum_{n=1}^\infty\frac{\mu(n)}{n}x^{n-1}$ on assumption of convexity, invoking a theorem due to Dragomir

Why $\sum\frac{\mu(h)\mu(k)}{hk}\gcd(h,k)=\prod\limits_{p\le x}\left(1-\frac1p\right)$, where the sum enumerates the pairs $(h,k)$ of primes below $x$

Two questions concerning series involving the Möbius function and trigonometric functions

Where is positive or negative the function $\sum_{n=1}^\infty\frac{\mu(n)}{n}\left(\frac{\cos(nx)}{n}\right)^2$ over the set $[0,2\pi]$?

Prove $ \left\vert \sum_{n=1}^N \frac{\mu(n)}{n} \right\vert \leqslant 1,$ where $\mu(n)$ is the Mobius function.

Justify an approximation of $-\sum_{n=2}^\infty H_n\left(\frac{1}{\zeta(n)}-1\right)$, where $H_n$ denotes the $n$th harmonic number

polynomial annihilators of finite rings - proof generalization sought

What is the inverse of $\left[ \sum_{k=1}^{j} \left\lfloor \frac{i}{k} \right\rfloor \right]_{n \times n}$?

Divergence of $\sum_{n=1}^\infty\frac{\mu(n)}{\sqrt{n}}\cos\left(n^2 \pi \gamma\right)$, where $\gamma$ is the Euler-Mascheroni constant

The meaning and definition of $\psi^{(-2)}(x)$, and the convergence of some related series involving the Möbius function

Proof with Möbius function, divisor function and little omega function

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