Questions tagged [arithmetic-functions]

For questions on arithmetic functions, i.e. real or complex valued functions defined on the set of natural numbers.

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What's so special about the number 24 in the definition of the Ramanujan tau function?

I'm learning about the Ramanujan tau function $\tau \colon \mathbb{N} \to \mathbb{Z}$ defined by $$\tau(1)q + \tau(2)q^2 + \tau(3)q^3 + \ldots = q \prod_{n = 1}^{\infty} \left(1 - q^n\right)^{24}.$$ ...
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1 vote
0 answers
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Product of all reduced residues in relation with the function $\mu$: $\prod_{1\leq a\leq n,(a,n)=1}a=n^{\varphi(n)}\prod_{d|n}(d!/d^d)^{\mu(n/d)}$

We know that for any arithmetic function $f$ the Mobius inversion formula gives its inversion. Hence $$ F(n)=\prod_{d|n}f(d)\implies f(n)=\prod_{d|n} F(n/d)^{\mu(d)}.$$ The above statement can be ...
0 votes
0 answers
108 views

If $q^k n^2$ is an odd perfect number with special prime $q$ satisfying $k=1$, then $(q+1)/2$ is squarefree.

The topic of odd perfect numbers likely needs no introduction. Denote the sum of divisors of the positive integer $x$ by $\sigma(x)$, and denote the abundancy index of $x$ by $I(x)=\sigma(x)/x$. Euler ...
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0 answers
13 views

Can this inequality involving the deficiency and sum of aliquot divisors be improved? - Part II

This MSE question (from April 2020) asked whether the inequality $$\frac{D(n^2)}{s(n^2)} < \frac{D(n)}{s(n)}$$ could be improved, where $D(x)=2x-\sigma(x)$ is the deficiency of the positive integer ...
0 votes
1 answer
48 views

Which arithmetic functions satisfy $\sum_{d \mid n} f(\frac{n}{d}) k^d \equiv 0 \pmod{n}$ for all positive integers $n,k$?

There are $$\frac{1}{n} \sum_{d \mid n} \varphi\left(\frac{n}{d}\right) \cdot k^d$$ different $k$-ary necklaces of length $n$ as a result of Pólya's enumeration theorem, where $\varphi$ is Euler's ...
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1 vote
1 answer
109 views

On the equation $s(n^2) = \left(\frac{q-1}{2}\right)\cdot{D(n^2)}$, if $q^k n^2$ is an odd perfect number with special prime $q$

Let $N$ be an odd perfect number given in the so-called Eulerian form $$N = q^k n^2$$ where $q$ is the special prime satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. In what follows, let ...
1 vote
1 answer
43 views

On the second moment of prime divisor function

Let $\omega(n)$ denote the number of distinct prime divisors of $n$. I learned from Cojocaru & Murty that $$ \sum_{n\le x}\omega(n)^2=x(\log\log x)^2+O(x\log\log x).\tag1 $$ I wonder whether it is ...
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0 votes
1 answer
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If $2^r b^2$ is an even almost perfect number that is NOT a power of two, does it follow that $r=1$?

(The following are taken from this preprint by Antalan and Dris.) Antalan and Tagle showed that an even almost perfect number $n \neq 2^t$ must necessarily have the form $2^r b^2$ where $r \geq 1$, $\...
1 vote
1 answer
57 views

Does $\sigma(n^2)/q \mid q^k n^2$ imply $\sigma(n^2)/q \mid n^2$, if $q^k n^2$ is an odd perfect number with special prime $q$? - Part II

(Preamble: This inquiry is an offshoot of this MSE question.) Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$. Denote the abundancy index of $x$ as $I(x)=\...
1 vote
1 answer
78 views

Has this strange real-valued binary function been researched?

Let $X \neq \emptyset$ be an arbitrary set. Let $$ d : X \times X \to \mathbb{R} $$ be a real-valued function with the following properties: (1) $\forall x, \in X: d(x,x) = 0,$ (2) $\forall x,y \in X:...
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0 votes
1 answer
50 views

Are these valid proofs for the equation $\gcd(n,\sigma(n^2))=\gcd(n^2,\sigma(n^2))$, if $q^k n^2$ is an odd perfect number with special prime $q$?

Let $N = q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$. It is known that $$i(q)=\gcd(n^2,\sigma(n^2))=\frac{\sigma(n^2)}{q^k}=\frac{n^2}{\sigma(q^k)/...
2 votes
1 answer
39 views

On a curious PDE involving $I(q^k)+\frac{2}{I(q^k)}$

Let $I(x)=\sigma(x)/x$ be the abundancy index of the positive integer $x$, where $\sigma(x)=\sigma_1(x)$ is the classical sum of divisors of $x$. Consider $$f(q,k)=I(q^k)+\frac{2}{I(q^k)}$$ where $5 \...
1 vote
1 answer
94 views

Are the (Bezout) coefficients for $\gcd(n^2, \sigma(n^2)) = q\sigma(n^2) - 2(q - 1)n^2$ (where $q^k n^2$ is an odd perfect number) unique?

In what follows, denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$, the deficiency of $x$ by $D(x)=2x-\sigma(x)$, and the aliquot sum of $x$ by $s(x)=\sigma(x)...
5 votes
1 answer
334 views

Why counting the number of 1 digits that appear in all integers in 0-9, 0-99, 0-999, 0-9999 follow an arithemtic-geometric sequence?

I noticed that 0-9 = has only 1 '1' 0-99 = has 20 '1's [1,10,11,12,13,14,15,16,17,18,19,21,31,41,51,61,71,81,91] 0-999 = 300 0-9999 = 4000 It follows the formula of n = number of digits in the ...
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-6 votes
1 answer
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If $p$ and $2p + 1$ are odd primes and $n = 4p$, prove that $φ(n + 2) = φ(n) + 2$. [closed]

Prove: if $p$ and $2p + 1$ are odd primes and $n = 4p$, show that $φ(n + 2) = φ(n) + 2$. I'm stuck on this simple question about Euler's theorem. Any help would be welcome. Thanks in advance!
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0 votes
1 answer
83 views

Is the argument used in this proof that $k=1$ logically sound, where $q^k n^2$ is an odd perfect number with special prime $q$?

The topic of odd perfect numbers likely needs no introduction. Euler proved that a hypothetical odd perfect number $N$, if one exists, must have the so-called Eulerian form $N=q^k n^2$, where $q$ is ...
0 votes
2 answers
82 views

$\varphi(x,n)=\sum_{d|n}\mu(d)\left\lfloor\frac xd\right\rfloor$ and $\sum_{d|n}\varphi\left(\frac xd,\frac nd\right)=\lfloor x\rfloor$

If $x$ is real, $x\ge 1$, let $\varphi(x,n)$ denote the number of positive integers less than or equal to $x$ that are relatively prime to $n$. Prove that $$\varphi(x,n)=\sum_{d|n}\mu(d)\left\lfloor\...
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1 vote
1 answer
96 views

Product form of Mobius Inversion formula: $g(n)=\prod_{d|n}f(d)^{a(n/d)}\iff f(n)=\prod_{d|n} g(d)^{b(n/d)}$

Product form of the Möbius inversion formula: If $f(n)>0$ for all $n$ and if $a(n)$ is real, $a(1)\neq 0$, prove that $$g(n)=\prod_{d|n}f(d)^{a(n/d)}\iff f(n)=\prod_{d|n} g(d)^{b(n/d)}$$ where $b=a^...
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4 votes
1 answer
115 views

Prove that $\sum_{d|n}\mu(d)\log^m d=0$

Prove that $$\sum_{d|n}\mu(d)\log^m d=0$$ if $m\ge 1$ and $n$ has more than $m$ distinct prime factors. I tried using Induction and kind of succeeded in the sense that if we write down the case $m=2$ ...
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0 votes
0 answers
32 views

If $q^k n^2$ is an odd perfect number with special prime $q$, is it possible to express $\sigma(n^2)/q^k$ as a function of only $q$ and $k$?

Let $q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)...
0 votes
0 answers
57 views

Relation between Mertens function and Chebyshev function

In Wikipedia article about Mertens function we can read: "A curious relation given by Mertens himself involving the second Chebyshev function is $$\displaystyle \psi (x)=\sum _{n=1}^{\infty } M\...
4 votes
2 answers
120 views

Slight variation of Mertens' third theorem. Do we have an estimate of $\sum_{d \mid n\#}(-1)^{\omega(d)}\dfrac{\sigma_0(d)}{d}$?

Define $f(n)$ to be: $$ \sum_{d \mid n\#}(-1)^{\omega(d)}\dfrac{\sigma_0(d)}{d} $$ But $\sigma_0(d) = 2^{\omega(d)}$ for any $d \mid n\#$ a primorial, so: $$ f(n) = \prod_{p \text{ prime} \\ p \leq n} ...
1 vote
0 answers
108 views

Does the following lower bound improve on $I(q^k)+I(n^2) > \frac{57}{20}$, where $q^k n^2$ is an odd perfect number?

Preamble: This question is an offshoot of this earlier post. (This inquiry has likewise been cross-posted to MO last June $10, 2022$.) Let $N = q^k n^2$ be an odd perfect number with special prime $q$...
1 vote
0 answers
68 views

Follow-up to MSE question 3738458

This is a follow-up inquiry to this MSE question. Let $\sigma(x)$ denote the sum of divisors of the positive integer $x$. A number $M$ is said to be perfect if $\sigma(M)=2M$. For example, $6$ and $...
0 votes
2 answers
56 views

What are the fixed points of the arithmetic derivative over the non-negative integers?

I just watched When the derivative of a number is not zero -- The arithmetic derivative by Michael Penn. I like to explore things visually and computationally, so I found this recursive implementation ...
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0 votes
0 answers
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Full derivation inside of twin prime statement in terms of multiplicative arithmetic functions. How can the last formula be rearranged?

Let $(\cdot\mid\cdot) : \Bbb{N}\times\Bbb{N} \to \Bbb{Z}_2$ be the divisibility function which takes on the value $(x|y) = 1$ whenever $x$ divides $y$ and the value $(x|y) = 0$ whenever it does not ...
0 votes
0 answers
181 views

Characterization of primes of the form $n^n+1$ by using number-theoretic functions

It is known that there is a unsolved problem related to primes of the form $n^n+1$ as is expained in page 160 of [1] (see also page 156, and the OEIS page related to this integer sequence A121270). In ...
0 votes
1 answer
89 views

What conditions on $X$ will guarantee that $\gcd(\text{square part of } X,\text{squarefree part of } X)=1$, if $X$ is neither a square nor squarefree?

The following query is an offshoot of this answer to a closely related post. Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$. The topic of odd perfect ...
2 votes
0 answers
68 views

Using Mobius Inversion Formula

We are given the following: $f(n)=\prod_{d|n}g(d)$ and asked to show: $g(n)=\prod_{d|n} f(d)^{\mu(\frac{n}{d})}$ The hint given says to use logarithms Here's what I tried doing: $log(f(n))=\prod_{d|n}...
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0 votes
0 answers
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Does $I = \gcd(n,\sigma(n^2)) = (\frac{n}{\sigma(q^k)/2})\cdot\gcd(\sigma(q^k)/2,n)$ imply that $\sigma(q^k)/2 \mid n$ holds?

Let $N = q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. Define the GCDs: $$G = \gcd\bigg(\sigma(q^k),\sigma(n^2)\bigg)$$ $$H = \...
2 votes
0 answers
55 views

Different ways to average an arithmetic function

Consider the following ways to average a multiplicative arithmetic function $f$ over $\mathbb{N}$: Arithmetic: $\displaystyle\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^N f(n)$ Factorized: $\displaystyle\...
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2 votes
1 answer
57 views

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

Is there general formula of the solutions of the congruence? \begin{equation} n\equiv X^2-XY+Y^2 \pmod r, \end{equation} where $n\in\Bbb Z$ and $r\in\Bbb N$. If we define an arithmetic function (two ...
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2 votes
1 answer
110 views

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

I'm investigating the behavior of the following function as $x\to \infty$: $$f(x):=\sum_{1\le n\le x}\frac{J_2(n)}{n^2}$$ where $J_k(n)$ is the Jordan's totient function $$J_k(n):=n^k\prod_{p\mid n}(1-...
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0 votes
1 answer
92 views

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

This question was asked in my quiz of number theory and I was not able to make any progress in it. Question: Show that $$\sum_{ n\leq x}{\Lambda(n)}^2 = x\log x- x+o(x),$$ where $\Lambda(n)$ is the ...
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8 votes
2 answers
178 views

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

OEIS A046346 sequence lists composite numbers divisible by the sum of their prime factors, counted with multiplicity. $$S=\Big\{n\in\mathbb{N}\space not\space prime,\space n=\prod_k p_k^{\alpha_k}\...
0 votes
1 answer
140 views

Dirichlet Series of Square Full Integers.

As in the title, I want to find the Dirichlet series $F$ of the indicator function for cube full integers $f(n)=1 \iff p^3|n, \forall p|n$ and $f(n)=0$ otherwise. Since $f$ is clearly multiplicative, $...
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1 vote
0 answers
135 views

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

Conjecture. For all $n,m \in \Bbb{N}$, $$ f(n, m) := \sum_{c\mid d\mid n\# \\ \gcd(c, 2m) = 1}\dfrac{(-1)^{\omega(d)}}{d} $$ is greater than $0$. Example verification code: ...
0 votes
2 answers
415 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 $p$?

The following query is an offshoot of this post 1 and this post 2. Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$. The topic of odd perfect numbers likely ...
0 votes
2 answers
282 views

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

Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$. (Note that the divisor sum $\sigma$ is a multiplicative function.) A number $P$ is said to be perfect if $\...
13 votes
1 answer
368 views

On the density of a particular subset of integers

Given a positive integer $n$ in the standard form $$n=\prod_k p_k^{\alpha_k}$$ and the arithmetic function $$f(n)=\sum_k \alpha_k p_k$$ let's define the subset $F$ of positive integers $$F=\Big\{n\in ...
0 votes
0 answers
34 views

(Applied to OPN) Let $c$ and $d$ be integers, not both equal to zero. If $q$ and $r$ are integers such that $c = dq + r$, then $\gcd(c,d)=\gcd(d,r)$.

I was reading the following web resource, and found the following Lemma 8.1, which I think would be immensely useful for a computational task on odd perfect numbers (hereinafter abbreviated as OPN): ...
1 vote
0 answers
36 views

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

Let $$\sigma(x) = \sum_{d \mid x}{d}$$ denote the sum of divisors of $x \in \mathbb{N}$, where $\mathbb{N}$ is the set of natural numbers or positive integers. Recall that a Descartes number is an odd ...
3 votes
2 answers
114 views

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

Let $$\tau(n) = \sum_{d \mid n}{1}$$ be the divisor function, $$\omega(n) = \sum_{p \mid n}{1}$$ be the prime divisor function, $$\varphi(n) = \#\{1 \leqslant k \leqslant n : \gcd(k,n) = 1\}$$ be ...
0 votes
1 answer
126 views

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

I am currently studying number theory and our instructor refers to Apostol's book on Analytic number theory for the chapter Dirichlet series.In that book,there is an exercise which is as follows: Let $...
0 votes
4 answers
95 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 $q$?

This question is an offshoot of this MSE answer. Let $\sigma(x)$ be the sum of the divisors of the positive integer $x$. (Denote the abundancy index of $x$ by $I(x)=\sigma(x)/x$.) If $\sigma(M) = 2M$...
3 votes
1 answer
63 views

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

I am a graduate student of Mathematics.I am now studying analytic number theory from Apostol's book.In the exercise $3$ there is a question which is as follows: If $x\geq1$,Show that $\sum\limits_{n\...
1 vote
0 answers
71 views

On the reciprocal of $I(n^2) - \frac{2(q - 1)}{q}$, if $q^k n^2$ is a(n) (odd) perfect number with special prime $q$

Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$. Denote the abundancy index of $x$ by $I(x)=\sigma(x)/x$. A number $N$ is said to be perfect if $\sigma(N)=...
1 vote
3 answers
114 views

Does the inequality $I(n^2) \leq 2 - \frac{5}{3q}$ improve $I(q^k) + I(n^2) < 3$, if $q^k n^2$ is an odd perfect number with special prime $q$?

Let $N = q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. Denote the abundancy index of the positive integer $x$ by $I(x)=\sigma(x)/...
1 vote
1 answer
100 views

Bounds for the abundancy index of divisors of odd perfect numbers in terms of the deficiency function - Part II

This post is an offshoot of this MSE question. Motivation Let $x, y$ and $z$ be positive integers. Denote the sum of divisors of $x$ by $\sigma(x)$. Also, denote the deficiency of $y$ by $D(y)=2y-\...
3 votes
1 answer
121 views

If $\gcd(n^2, \sigma(n^2)) = q\sigma(n^2) - 2(q - 1) n^2$, does it follow that $q n^2$ is perfect?

Let $q$ be an (odd) prime, and let $\gcd(q,n)=1$. Denote the classical sum of divisors of the positive integer $x$ by $\sigma(x)=\sigma_1(x)$. A number $N$ is said to be perfect if $\sigma(N)=2N$. ...

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