Questions tagged [arithmetic-functions]

Question 1

For integers $n$, the arithmetic derivative $D(n)$ is defined as follows: $D(p) = 1$, for any prime $p$. $D(mn) = D(m)n + mD(n)$, for any $m,n\in\mathbb{N}$ (Leibniz rule). The Leibniz rule implies ...

Question 2

For integers $n$, the arithmetic derivative $D(n)$ is defined as follows: $D(p) = 1$, for any prime $p$. $D(mn) = D(m)n + mD(n)$, for any $m,n\in\mathbb{N}$ (Leibniz rule). The Leibniz rule implies ...

Question 3

Consider Oeis $A065387$: $a(n)=\sigma(n)+\phi(n)$, where $\sigma(n)$ represents the sum of all divisors of $n$ and $\phi(n)$ is Euler's totient function. I ask to prove that the assertion the ...

Question 4

Given some general arithmetic function $A(n)$ with a known asymptotic formula for $\sum_{n \le x}A(n)$, is there any way to figure out the asymptotic formula for $$\sum_{\substack{n \le x \\ \gcd(n,q) ...

Question 5

For integers $n$, the arithmetic derivative $D(n)$ is defined as follows: $D(p) = 1$, for any prime $p$. $D(mn) = D(m)n + mD(n)$, for any $m,n\in\mathbb{N}$ (Leibniz rule). $D(-n) = -D(n)$. The ...

Question 6

For integers $n$, the arithmetic derivative $D(n)$ is defined as follows: $D(p) = 1$, for any prime $p$. $D(mn) = D(m)n + mD(n)$, for any $m,n\in\mathbb{N}$ (Leibniz rule). $D(-n) = -D(n)$. The ...

Question 7

Let $\sigma_{\alpha}(n) = \sum_{d|n}d^{\alpha}$ as usual. Prove that $$\sigma_{\alpha}(m)\sigma_{\alpha}(n) = \sum_{d|\gcd(m, n)}d^{\alpha}\sigma_{\alpha}(\frac{mn}{d^2})$$ I am utterly lost on this. ...

Question 8

When looking into the derivation of Perron's formula, I found that it seems to come from using the inverse Mellin transform of the equation $$\mathcal{M}_x\left[\sum_{n \le x} a_n\right](s) = \frac{1}{...

Question 9

Given two positive integers $n$ and $k$, I want to evaluate the following sum $$ g(n,k) := \sum_{1\le m_1, m_2 \le n \atop \gcd(m_1,m_2,n)=1} \omega^{km_2}, $$ where $\omega = e^{2\pi i/n}$ is a ...

Question 10

Suppose that we have an infinite array $[a_{ij}]_{i,j\in\mathbb{N}}$ consisting entirely of natural numbers. Also, suppose that for each $i,j$ we have $a_{ij}\leq ij$. Prove that for each natural ...

Question 11

At the moment, I'm working on the following problem and I'm stuck... Let $Y = \{(x, y) \in \mathbb{Z}^2 : \gcd(x, y) = 1\}$ and $Y(N) = \{(x, y) \in Y : |x| \leq N, |y| \leq N\}$. Compute the limit $\...

Question 12

I had wondered, if you begin at $0$, and count up until the resultant number is all $9$s ($0$ to $9$, $0$ to $99$, etc.), how many times the number $1$ will appear in each number along the way? In ...

Question 13

This is from Hardy and Wright’s An Introduction to the Theory of Numbers, Section 18.2. "The average order of $d(n)$". Here, $d(n)$ denotes the number of divisors of $n$. The section states ...

Question 14

I want to evaluate asymptotically the following: $$\sum_{ab\le N}\frac{(\log ab)(\log b)\Lambda(a)}{ab}$$ where $\Lambda$ is the von Mangoldt function. I have a few thoughts for how to do this, but so ...

Question 15

I'm trying to find examples of multiplicative functions $f$ that satisfy some interesting convolution identities. Here, $*$ denotes the Dirichlet convolution, $\mu$ is the Möbius function, and $1(n) = ...

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Questions tagged [arithmetic-functions]

Iterating the map $f(n)=1+D(\sigma(n)-1)$

A singular identity involving the arithmetic derivative and the divisor counting function

An equivalent formulation for the twin prime conjecture

Sum of an arithmetic function upto x with gcd(n, q) = 1 for fixed q [closed]

Iterating the arithmetic-derivative map $U(n)=n+D(n)−1$

A sequence based on arithmetic derivative that always converges to prime numbers

Product of sum of divisors function [duplicate]

Proof that $\mathcal{M}_x\left[\displaystyle\sum_{n \le x} a_n\right](s) = \displaystyle\sum_{n=1}^\infty \frac{a_n}{n^s}$?

Sum of $n$-th roots of unity over pairs of integers, whose gcd is coprime to $n$

Prove that a natural number occurs an arbitrary amount of times in this array

Compute the limit $\lim_{N \to \infty} \frac{\#Y(N)}{N^2}$

Using an observation-based formula to determine the total times the number 1 will appear as one counts from 0 to 9,999,999,999.

Ramanujan's theorems about the sums of powers of the number of divisors

Evaluating a number theoretic sum asymptotically

Looking for multiplicative functions satisfying $f * 1 = \mu f $ and $f * 1 = |f|$

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