Related papers: On a family of arithmetic series related to the M\…

Let $\mu(n)$ be the M\"{o}bius function. Let $P^-(n)$ denote the smallest prime factor of an integer $n$. In 1977, Alladi established the following formula related to the prime number theorem for arithmetic progressions \[…

In 1977, the first author observed a duality between the largest and smallest prime factors of integers, and established as a consequence some new results on the M\"obius function $\mu(n)$ using the Prime Number Theorem for Arithmetic…

In the first paper under this title (1977), the first author utilized a duality identity between the largest and smallest prime factors involving the Moebius function, to establish the following result as a consequence of the Prime Number…

If $\gcd(r,t)=1$, then a theorem of Alladi offers the M\"obius sum identity $$-\sum_{\substack{ n \geq 2 \\ p_{\rm{min}}(n) \equiv r \pmod{t}}} \mu(n)n^{-1}= \frac{1}{\varphi(t)}. $$ Here $p_{\rm{min}}(n)$ is the smallest prime divisor of…

Let $\mu(n)$ denote the M\"obius function, define $M(x)= \sum_{n\leq x}^{}\mu (n)$. The main result of this paper is to prove that \begin{equation*} \displaystyle\lim_{x \to +\infty}\frac{M(x)}{x}=0 \end{equation*} which is equivalent to…

In this note, we mainly show the analogue of one of Alladi's formulas over $\mathbb{Q}$ with respect to the Dirichlet convolutions involving the M\"{o}bius function $\mu(n)$, which is related to the natural densities of sets of primes by…

An elementary recursive relation for M$\ddot{\mathrm{o}}$bius function $\mu (n)$ is introduced by two simple ways. With this recursive relation, $\mu (n)$ can be calculated without directly knowing the factorization of the $n$. $\mu (1)…

The main object of this paper is to find closed form expressions for finite and infinite sums that are weighted by $\omega(n)$, where $\omega(n)$ is the number of distinct prime factors of $n$. We then derive general convergence criteria…

We show that, for the M\"obius function $\mu(n)$, we have $$ \sum_{x < n\leq x+x^{\theta}}\mu(n)=o(x^{\theta}) $$ for any $\theta>0.55$. This improves on a result of Ramachandra from 1976, which is valid for $\theta>7/12$. Ramachandra's…

In this paper, we build some ergodic theorems involving function $\Omega$, where $\Omega(n)$ denotes the number of prime factors of a natural number $n$ counted with multiplicities. As a combinatorial application, it is shown that for any…

Let $\omega(n)$ (resp. $\Omega(n)$) denote the number of prime divisors (resp. with multiplicity) of a natural number $n$. In 1917, Hardy and Ramanujan proved that the normal order of $\omega(n)$ is $\log\log n$, and the same is true of…

The Mertens function, $M(x) := \sum_{n \leq x} \mu(n)$, is defined as the summatory function of the classical M\"obius function. The Dirichlet inverse function $g(n) := (\omega+1)^{-1}(n)$ is defined in terms of the shifted strongly…

Given an integer $k\ge2$, let $\omega_k(n)$ denote the number of primes that divide $n$ with multiplicity exactly $k$. We compute the density $e_{k,m}$ of those integers $n$ for which $\omega_k(n)=m$ for every integer $m\ge0$. We also show…

Let $<\P > \subset \N$ be a multiplicative subsemigroup of the natural numbers $\N = \{1,2,3,...\}$ generated by an arbitrary set $\P$ of primes (finite or infinite). We given an elementary proof that the partial sums $\sum_{n \in < \P >: n…

A primitive root modulo an integer $n$ is the generator of the multiplicative group of integers modulo $n$. Gauss proved that for any prime number $p$ greater than $3$, the sum of its primitive roots is congruent to $1$ modulo $p$ while its…

For n=1,2,3,... define S(n) as the smallest integer m>1 such that those 2k(k-1) mod m for k=1,...,n are pairwise distinct; we show that S(n) is the least prime greater than 2n-2 and hence the value set of the function S(n) is exactly the…

Let $\{p_j(n)\}_{j=1}^{\omega(n)}$ denote the increasing sequence of distinct prime factors of an integer $n$. We provide details for the proof of a statement of Erd\H{o}s implying that, for any function $\xi(n)$ tending to infinity with…

Let $\Omega(n)$ denote the number of prime factors of $n$. We show that for any bounded $f\colon\mathbb{N}\to\mathbb{C}$ one has \[ \frac{1}{N}\sum_{n=1}^N\, f(\Omega(n)+1)=\frac{1}{N}\sum_{n=1}^N\, f(\Omega(n))+\mathrm{o}_{N\to\infty}(1).…

Let $g(n)$ be the largest positive integer $k$ such that there are distinct primes $p_i$ for $1\leq i\leq k$ so that $p_i |n+i$. This function is related to a celebrated conjecture of C.A. Grimm. We establish upper and lower bounds for…

We study the values of the M\"obius function $\mu$ of intervals in the containment poset of permutations. We construct a sequence of permutations $\pi_n$ of size $2n-2$ for which $\mu(1,\pi_n)$ is given by a polynomial in $n$ of degree 7.…