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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Contiguous generalized hypergeometric functions modifying only denominator variables

The standard contiguous relations for the Gaussian Hypergeometric Functions can be stacked/repeated to relate $_2F_1(a,b;c;z)$ in many ways to sums of the formats $_2F_1(a\pm,b\pm ; c\pm ;z)$ , see e....
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1 vote
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How fast does the proportion guaranteed by dirichlet converge?

I'm working on a counting problem and I'm using Dirichlets theorem (weak form) at some point in the counting. The problem is I don't know how fast something converges and I'm not very knowledgeable in ...
1 vote
36 views

Connection between multiplication table $n * k$ and partial sums of the partial sums of the Dirichlet inverse of the Euler totient function.

I am watching this video: L-functions and the Langlands program (RH Saga S1E2) This reminds me of a recurrence: Let $c=1$ and a recurrence be: ...
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1 vote
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Verify if exists $n$ such that $\phi(n)$ equals $\frac{n}{d}$, for a constant $d$

Taking $n$ for $n=p_1^{a_1}...p_k^{a_k}$ (being $p_i$ prime numbers)and applying it to Euler's totient function, we would get $\phi(n) = p_1^{a_1-1}...p_k^{a_k-1}(p_1-1)...(p_k-1)$ If we take $d=2$, ...
1 vote
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How to get the Euler product for $\sum_{n = 1}^{\infty} \mu(n)/\phi(n^k)$.

Let $\mu$ be the Mobius function, and $\phi$ Euler's totient function. I am reading a proof found in this paper (Theorem 2 on page 17), and I can't quite figure why I'm getting something different ...
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If an arithmetic function is multiplicative, non-zero at a prime, and "prime-linear", is it the identity?

Let $f:\mathbb{N}\to\mathbb{N}\cup\{0\}$ be a function. Let $f(1)=1,$ and $f(ab)=f(a)f(b)$ whenever $\gcd(a,b)=1.$ Note that I am assuming that $f$ is multiplicative but not completely multiplicative. ...
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1 vote
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A problem on von Mangoldt function.

Suppose $n$ is odd natural number. Define $r(n)=\sum_{n_1+n_2+n_3=n} \Lambda(n_1)\Lambda(n_2)\Lambda(n_3)$ and $r'(n)=\sum_{p_1+p_2+p_3=n}(\log p_1)(\log p_2)(\log p_3)$ where $p_1,p_2,p_3$ are ...
182 views

Does an odd perfect number have a divisor (other than $1$) which must necessarily be almost perfect?

Let $\sigma(x)=\sigma_1(x)$ be the classical sum of divisors of the positive integer $x$. Denote the aliquot sum of $x$ by $s(x)=\sigma(x)-x$ and the deficiency of $x$ by $d(x)=2x-\sigma(x)$. ...
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Almost exact formula for sum of Euler Phi function $\sum_{k=1}^n \varphi(k)$

Solving a problem, I needed a formula for $\Phi(n)=\sum_{k=1}^n \varphi(k)$, where $\varphi(k)$ is Euler's notorious totient function, that counts the number of numbers between $1$ and $k$ that are ...
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1 vote
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Pointwise Extension of Multiplicativity

A standard exercise in elementary number theory is to prove that if two multiplicative functions agree on the set of prime powers, then they are identical arithmetic functions. Consider the following ...
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1 vote
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Is there an algorithm to solve number manipulation problems?

There are this kind of problems where, given a certain amount of the same numbers, it's needed to manipulate them with functions or operators in a way to get a certain result. It's allowed to glue ...
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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 $$\tau(1)q + \tau(2)q^2 + \tau(3)q^3 + \ldots = q \prod_{n = 1}^{\infty} \left(1 - q^n\right)^{24}.$$ ...
• 387
1 vote
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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 ...
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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 ...
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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 PΓ³lya's enumeration theorem, where $\varphi$ is Euler's ...
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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
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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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1 vote
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1 vote