Articles of arithmetic functions

LCM of binomial coefficients and related functions

I know about the following identity: $$\displaystyle \text{lcm} \left( {n \choose 0}, {n \choose 1}, … {n \choose n} \right) = \frac{\text{lcm}(1, 2, … n+1)}{n+1}$$ 1) Is there any method to find $$\displaystyle \text{lcm} \left( {n \choose 0}, {n \choose 1}, … {n \choose k} \right)$$ for any $k \le n$ without evaluating all the […]

Is there a complex variant of Möbius' function?

When you’re dealing with arithmetic functions, you might have come across the classical Möbius’ function $$ \mu(n)=\begin{cases} (-1)^{\omega(n)}=(-1)^{\Omega(n)} &\mbox{if }\; \omega(n) = \Omega(n)\\ 0&\mbox{if }\;\omega(n) < \Omega(n).\end{cases}, $$ where $ω(n)$ is the number of distinct primes dividing the number $n$ and $Ω(n)$ is the number of prime factors of $n$, counted with multiplicities. Is there […]

If $\sigma(n) \equiv 2 \pmod 4$ and $n$ is odd, does this imply that $n = p^k m^2$?

Let $\sigma(n)$ denote the sum of divisors of $n$. Here is my question: If $\sigma(n) \equiv 2 \pmod 4$ and $n$ is odd, does this imply that $n = p^k m^2$ where $p \equiv k \equiv 1 \pmod 4$ and $\gcd(p,m)=1$? I think the proof must closely parallel the proof for the form of odd […]

The arithmetic function $\lambda(n)=(-1)^{a_1+\cdots +a_k}$

Define $\lambda(1)=1$, and if $n=p_1^{a_1}\cdots p_k^{a_k}$, define $$\lambda(n)=(-1)^{a_1+\cdots +a_k}$$ How can I see that $$\sum_{d\mid n}\lambda(d)=\begin{cases} 1 \,\,\text{ if $n$ is a square}\\ 0 \,\,\text{ otherwise} \end{cases} $$?

Farey Sequence Vector Orthogonality Relation Question

Take the Farey sequence $\mathcal{F}_n$ for $n=39$ with values $a_m\in \mathcal{F}_n$ and put them into a vector $$ \vec v_k=\biggr(\exp(2\pi i k a_m)\biggr)_m $$ Since Merten’s function for $n=39$ is zero $$ M(39)= \sum_{a\in \mathcal{F}_{39}} e^{2\pi i a} =0 , $$ our vector $\vec v_1$ is orthogonal to the vector containing only $1$’s, i.e. $\vec […]

a formula involving order of Dirichlet characters, $\mu(n)$ and $\varphi(n)$

Let $p$ a prime number, ${q_{_1}}$,…, ${q_{_r}}$ are the distinct primes dividing $p-1$, ${\mu}$ is the Möbius function, ${\varphi}$ is Euler’s phi function, ${\chi}$ is Dirichlet character $\bmod{p}$ and ${o(\chi)}$ is the order of ${\chi}$. How can I show that: $$\sum\limits_{d|p – 1} {\frac{{\mu (d)}}{{\varphi (d)}}} \sum\limits_{o(\chi ) = d} {\chi (n)} = \prod\limits_{j = […]

Does $k=9018009$ have a friend?

(Note: This question has been cross-posted to MO.) Let $\sigma(x)$ be the sum of the divisors of the positive integer $x$. For example, $\sigma(6)=1+2+3+6=12$ and $\sigma(28)=1+2+4+7+14+28=56$. Denote the abundancy index $I$ of $x$ by $$I(x)=\frac{\sigma(x)}{x}.$$ If a positive integer $y$ is one of at least two solutions of $$I(y)=\frac{a}{b}$$ for a given rational number $a/b$, […]

Limiting value of $\lim \frac{1}{k}\sum_{n=1}^k \frac{p(n+1)-p(n)}{\log p(n)}$

Empirically it seems $$\lim_{k\to \infty} \frac{1}{k}\sum_{n=1}^k \frac{g(n)}{\log p(n)} = 1\tag{1} $$ in which p(n) is the nth prime and g(n) is the prime gap $p(n+1)-p(n).$ Cramer conjectured that $$g(n) = O\left(\log^2 p(n)\right). $$ My somewhat open-ended question is: Formally how does (1) relate to Cramer’s estimate (if at all)…and can we prove (1)? For instance, […]

How often does $D(n^2) = m^2$ happen, where $D(x)$ is the deficiency of $x$?

Let $\sigma=\sigma_{1}$ denote the classical sum of divisors. For instance, $\sigma(12) = 1 + 2 + 3 + 4 + 6 + 12 = 28$. Let $x \in \mathbb{N}$ ($\mathbb{N}$ is the set of natural numbers/positive integers). Denote the number $2x – \sigma(x)$ by $D(x)$. We call $D(x)$ the deficiency of $x$. Now, let $m, […]

Elements of finite order in the group of arithmetic functions under Dirichlet convolution.

Let $(G, ∗)$ be the group of arithmetic functions $f : N \to C$ that satisfy $f (1)\neq 0$, with group operation given by the Dirichlet product $∗$. The identity function $I$ is the identity element of $G$ defined by $I : N \to Z$ where $I=1$ if $n=1$, $0$ if $n>1$. An element $f […]