Question

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

score 4.2k · Answer 1 · 2026-03-27 10:48:01

4.2k

Views

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

Published on 27 Mar 2026 - 10:48

score 128 · Answer 2 · 2013-03-21 05:41:28

128

Views

Prove: $\sum_{k<n, (k,n)=1} k= \frac{1}{2}n \varphi (n)$

Published on 21 Mar 2013 - 5:41

score 3.8k · Answer 3 · 2026-03-26 23:11:19

3.8k

Views

How to prove $ \prod_{d|n} d= n^{\frac{\tau (n)}{2}}$

Published on 26 Mar 2026 - 23:11

score 34 · Answer 4 · 2020-08-01 11:06:26

34

Views

Bounds for $\frac{\sigma(q^k)}{2\sigma(q^{k-1})}$ in terms of $q$ and $k$

Published on 01 Aug 2020 - 11:06

score 522 · Answer 5 · 2020-08-04 07:08:11

522

Views

Does the equation $\sigma(\sigma(x^2))=2x\sigma(x)$ have any odd solutions?

Published on 04 Aug 2020 - 7:08

score 96 · Answer 6 · 2026-03-27 07:20:03

96

Views

Dirichlet series of the $p$-adic valuation

Published on 27 Mar 2026 - 7:20

score 283 · Answer 7 · 2020-09-02 07:53:00

283

Views

How to compute sum of sum of gcd of factor pairs of a number upto a large number efficiently?

Published on 02 Sep 2020 - 7:53

score 120 · Answer 8 · 2026-03-27 08:46:38

120

Views

If $q^k n^2$ is an odd perfect number with special prime $q$, does $q^k < n$ imply that $\sigma(q^k) < n$?

Published on 27 Mar 2026 - 8:46

score 41 · Answer 9 · 2020-09-08 13:41:42

41

Views

Behaviour of a certain arithmetic function

Published on 08 Sep 2020 - 13:41

score 113 · Answer 10 · 2020-09-09 22:52:29

113

Views

An inequality relating the deficiency and sum of aliquot divisors

Published on 09 Sep 2020 - 22:52

score 437 · Answer 11 · 2020-09-15 19:36:16

437

Views

Is the distributive law of multiplication valid when the number of terms are variable?

Published on 15 Sep 2020 - 19:36

score 78 · Answer 12 · 2020-09-17 10:48:07

78

Views

Is it possible to improve the resulting upper bound for $\frac{D(m)}{s(m)}$, given a lower bound for $I(m)$?

Published on 17 Sep 2020 - 10:48

score 48 · Answer 13 · 2020-09-18 05:09:49

48

Views

Is it possible to improve on this lower bound for $I(n)$, if $\sigma(n^2)=q^k n$ and $q^k n^2$ is an odd perfect number with special prime $q$?

Published on 18 Sep 2020 - 5:09

score 216 · Answer 14 · 2020-09-18 06:19:21

216

Views

Is it possible to prove $I(m^2) > \zeta(2) \approx 1.644934$, if $p^k m^2$ is an odd perfect number with special prime $p$?

Published on 18 Sep 2020 - 6:19

score 301 · Answer 15 · 2026-03-27 10:12:55

301

Views

If $N = q^k n^2$ is an odd perfect number with special prime $q$, then must $\sigma(q^k)$ be deficient?

Published on 27 Mar 2026 - 10:12

List Question

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

Prove: $\sum_{k<n, (k,n)=1} k= \frac{1}{2}n \varphi (n)$

How to prove $ \prod_{d|n} d= n^{\frac{\tau (n)}{2}}$

Bounds for $\frac{\sigma(q^k)}{2\sigma(q^{k-1})}$ in terms of $q$ and $k$

Does the equation $\sigma(\sigma(x^2))=2x\sigma(x)$ have any odd solutions?

Dirichlet series of the $p$-adic valuation

How to compute sum of sum of gcd of factor pairs of a number upto a large number efficiently?

If $q^k n^2$ is an odd perfect number with special prime $q$, does $q^k < n$ imply that $\sigma(q^k) < n$?

Behaviour of a certain arithmetic function

An inequality relating the deficiency and sum of aliquot divisors

Is the distributive law of multiplication valid when the number of terms are variable?

Is it possible to improve the resulting upper bound for $\frac{D(m)}{s(m)}$, given a lower bound for $I(m)$?

Is it possible to improve on this lower bound for $I(n)$, if $\sigma(n^2)=q^k n$ and $q^k n^2$ is an odd perfect number with special prime $q$?

Is it possible to prove $I(m^2) > \zeta(2) \approx 1.644934$, if $p^k m^2$ is an odd perfect number with special prime $p$?

If $N = q^k n^2$ is an odd perfect number with special prime $q$, then must $\sigma(q^k)$ be deficient?

Trending Questions

Popular # Hahtags

Popular Questions