# 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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### Follow-up to question 3121498, asked in February 2019

Let $n = p^k m^2$ be an odd perfect number given in Eulerian form (i.e. $p$ is the special/Euler prime satisfying $p \equiv k \equiv 1 \pmod 4$ and $\gcd(p,m)=1$). Denote the classical sum of ...
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### Zero divisors in the Dirichlet ring

I'm trying to determine if the ring $(\mathbb{A}, +, *)$ is an integral domain, where $\mathbb{A}$ is the set of arithmetic functions and $*$ is the Dirichlet convolution. To do so, I'm trying to ...
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### On the quantity $I(q^k) + I(n^2)$ considered as functions of $q$ and $k$, when $q^k n^2$ is an odd perfect number with special prime $q$

Hereinafter, let $N = q^k n^2$ be an odd perfect number with special/Euler prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. Denote the abundancy index of the positive integer $x$ ...
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### Sum of an arithmetic function over the divisors of an integer is multiplicative

I am trying to prove the following claim: Let $f$ be an arithmetic function and, for $n \in \mathbb{Z}$ with $n > 0$, let $$F(n) = \sum_{d\ \vert\ n,\ d > 0}f(d)$$ If $F$ is multiplicative, ...
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### On a conjectured upper bound for $k=\nu_q(N)$, if $N=q^k n^2$ 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 $x$ as $I(x)=\sigma(x)/x$ ...
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### What is known about partial sums involving squares of the totient function?

The behaviour of the sum: $$\displaystyle\sum_{n \leq x} \frac{\varphi(n)}{n}$$ is well-studied. However, I am interested in the behaviour of the following sum, and haven't been able to find any ...
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### Check if an arithmetic function is (completely) multiplicative

In Paul J. McCarthy's book about arithmetic functions, he constructs a completely multiplicative function $g$ by setting $g(p)$ equal to a root of the equation $$X^2+f^{-1}(p)X+f^{-1}(p^2)=0,$$ where ...
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### Inclusion-exclusion of values of $\Omega$ (the arithmetic function).

For any number $n \in \Bbb{Z}$, define $$f(n) = \Omega(n) - \sum_{q \mid n} \Omega(n/q) + \sum_{q,q' \mid n}\Omega(n/(qq')) - \dots \pm \Omega(1),$$ where each sum is taken over only prime divisors ...
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### Asymptotic formula for Mean of Sum of Power of Divisors $\frac{ \sum_{i=1}^{n}\sigma_v(i)}{n}\sim\frac{n^v\zeta(v+1)}{v+1}$

Question: Define the sum of $v$-powers of divisor $\sigma_v(n)=\sum_{d|n}d^v$ for $v \in \mathbb{R}$. Prove that for all $v>0$, $$\frac{ \sum_{i=1}^{n}\sigma_v(i)}{n}\sim\frac{n^v\zeta(v+1)}{v+1}$$ ...
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### Is it possible to improve on the bound $D(q^k) < \varphi(q^k)$ if $k > 1$?

The problem is as is in the title: Is it possible to improve on the bound $$D(q^k) < \varphi(q^k)$$ if $k > 1$? Here, $q$ is a prime number and $k$ is a positive integer. The (deficiency) ...
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### Proving a relation between sum of reciprocal of divisors and $\sigma(n)$

Prove that $\sum_{d\mid n}\dfrac{1}{d}=\dfrac{\sigma(n)}{n}\ \forall\ n\geq 1, n\in \mathbb Z$ My question and my approach is a lot similar to this question and a bit different from this question ...
### Is this expression bounded when $p \equiv 1 \pmod 4$ is prime and $t \equiv 1 \pmod 4$?
Let $\sigma(x)$ denote the sum of divisors of the positive integer $x$. The problem under consideration is as in the title: Is the following expression bounded when $p \equiv 1 \pmod 4$ is prime and \$...