Question

Show that $\sum\limits_{n\leq x} \varphi(n)=\frac{1}{2}\sum\limits_{n\leq x}\mu(n)[\frac{x}{n}]^2+\frac{1}{2}$.

score 100 · Answer 1 · 2026-03-27 18:27:45

100

Views

Show that $\sum\limits_{n\leq x} \varphi(n)=\frac{1}{2}\sum\limits_{n\leq x}\mu(n)[\frac{x}{n}]^2+\frac{1}{2}$.

Published on 27 Mar 2026 - 18:27

score 99 · Answer 2 · 2026-03-27 21:20:30

99

Views

Does $\frac{\sigma(q^k)}{n} + \frac{\sigma(n)}{q^k} < 10$ hold in general for an odd perfect number $q^k n^2$ with special prime $q$?

Published on 27 Mar 2026 - 21:20

score 467 · Answer 3 · 2026-03-27 17:57:24

467

Views

Dirichlet series for $\zeta^3(s)/\zeta(2s)$.

Published on 27 Mar 2026 - 17:57

score 125 · Answer 4 · 2022-01-29 13:26:54

125

Views

The map $m \to m\sigma(m)$ is not injective.

Published on 29 Jan 2022 - 13:26

score 40 · Answer 5 · 2026-03-28 10:24:09

40

Views

If $km$ is a Descartes number with quasi-Euler prime $m$, must $\sqrt{k}$ be a squarefree palindrome?

Published on 28 Mar 2026 - 10:24

score 388 · Answer 6 · 2022-02-16 21:02:55

388

Views

On the density of a particular subset of integers

Published on 16 Feb 2022 - 21:02

score 325 · Answer 7 · 2026-03-27 21:19:03

325

Views

What are the remaining cases to consider for this problem, specifically all the possible premises for $i(q)$?

Published on 27 Mar 2026 - 21:19

score 426 · Answer 8 · 2026-03-27 22:57:38

426

Views

Does $\sigma(m^2)/p^k$ divide $m^2 - p^k$, if $p^k m^2$ is an odd perfect number with special prime $p$?

Published on 27 Mar 2026 - 22:57

score 141 · Answer 9 · 2026-03-25 20:12:01

141

Views

Why is this alternating summation involving reciprocals of divisors always positive? A conjecture.

Published on 25 Mar 2026 - 20:12

score 227 · Answer 10 · 2026-03-27 17:51:59

227

Views

Dirichlet Series of Square Full Integers.

Published on 27 Mar 2026 - 17:51

score 182 · Answer 11 · 2022-04-01 15:35:28

182

Views

Is OEIS A046346 sequence a subset $S\subset\mathbb{N}$ s.t. $\sum_S\frac{1}{n-\pi(n)}=1$?

Published on 01 Apr 2022 - 15:35

score 178 · Answer 12 · 2022-04-09 20:02:02

178

Views

Asymptotics of the sum of squares of Von Mangoldt function values.

Published on 09 Apr 2022 - 20:02

score 141 · Answer 13 · 2022-04-10 13:40:16

141

Views

Investigating $\sum \prod_{p\mid n}(1-\frac{1}{p^2})$ as $x\to\infty$

Published on 10 Apr 2022 - 13:40

score 82 · Answer 14 · 2022-04-11 11:15:57

82

Views

The number of solutions $(x,y)$ of the congruence $n \equiv X^2-XY+Y^2$ (mod $p^\alpha$).

Published on 11 Apr 2022 - 11:15

score 68 · Answer 15 · 2022-04-15 20:24:29

68

Views

Different ways to average an arithmetic function

Published on 15 Apr 2022 - 20:24

List Question

Show that $\sum\limits_{n\leq x} \varphi(n)=\frac{1}{2}\sum\limits_{n\leq x}\mu(n)[\frac{x}{n}]^2+\frac{1}{2}$.

Does $\frac{\sigma(q^k)}{n} + \frac{\sigma(n)}{q^k} < 10$ hold in general for an odd perfect number $q^k n^2$ with special prime $q$?

Dirichlet series for $\zeta^3(s)/\zeta(2s)$.

The map $m \to m\sigma(m)$ is not injective.

If $km$ is a Descartes number with quasi-Euler prime $m$, must $\sqrt{k}$ be a squarefree palindrome?

On the density of a particular subset of integers

What are the remaining cases to consider for this problem, specifically all the possible premises for $i(q)$?

Does $\sigma(m^2)/p^k$ divide $m^2 - p^k$, if $p^k m^2$ is an odd perfect number with special prime $p$?

Why is this alternating summation involving reciprocals of divisors always positive? A conjecture.

Dirichlet Series of Square Full Integers.

Is OEIS A046346 sequence a subset $S\subset\mathbb{N}$ s.t. $\sum_S\frac{1}{n-\pi(n)}=1$?

Asymptotics of the sum of squares of Von Mangoldt function values.

Investigating $\sum \prod_{p\mid n}(1-\frac{1}{p^2})$ as $x\to\infty$

The number of solutions $(x,y)$ of the congruence $n \equiv X^2-XY+Y^2$ (mod $p^\alpha$).

Different ways to average an arithmetic function

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