Related papers: The maximal order of the shifted-prime divisor fun…

Let $\omega^*(n)$ denote the number of divisors of $n$ that are shifted primes, that is, the number of divisors of $n$ of the form $p-1$, with $p$ prime. Studied by Prachar in an influential paper from 70 years ago, the higher moments of…

Let $\omega^*(n) = \{d|n: d=p-1, \mbox{$p$ is a prime}\}$. We show that, for each integer $k\geq2$, $$ \sum_{n\leq x}\omega^*(n)^k \asymp x(\log x)^{2^k-k-1}, $$ where the implied constant may depend on $k$ only. This confirms a recent…

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…

Let $a,b\in\mathbb{Z}\setminus\{0\}$. For every $n\in\mathbb{N}$, denote by $\omega_a^*(n)$ the number of shifted-prime divisors $p-a$ of $n$, where $p>a$ is prime. In this paper, we study the moments of $\omega_a^*$ over shifted primes…

Let $\sigma(n)$ denotes the sum of divisors function of a positive integer $n$. Robin proved that the Riemann hypothesis is true if and only if the inequality $\sigma(n) < e^{\gamma}n \log \log n$ holds for every positive integer $n \geq…

The well-known Hardy--Ramanujan inequality states that if $\omega(n)$ denotes the number of distinct prime factors of a positive integer $n$, then there is an absolute constant $C>0$ such that uniformly for $x\ge2$ and $k\in\mathbb{N}$,…

Let $ x\geq 1 $ be a large number, let $ [x]=x-\{x\} $ be the largest integer function, and let $ \sigma(n)$ be the sum of divisors function. This note presents the first proof of the asymptotic formula for the average order $ \sum_{p\leq…

Let $\omega^*(n)$ be the number of primes $p$ such that $p-1$ divides $n$. Assuming the Elliott--Halberstam Conjecture, we prove a conjecture posted by M. R. Murty and V. K. Murty in 2021 which states that $$\sum_{n\leqslant…

Following Wigert, various authors, including Ramanujan, Gronwall, Erd\H{o}s, Ivi\'{c}, Schwarz, Wirsing, and Shiu, determined the maximal order of several multiplicative functions, generalizing Wigert's result $$\max_{n\leq x} \log d(n) =…

Define $s (n) := n^{- 1} \sigma (n)$ ($\sigma (n):=\sum_{d|n}d )$ and $\omega(n)$ is the number of prime divisors of $n$. One of the properties of $s$ plays a central role: $s (p^a) > s (q^b)$ if $p < q$ are prime numbers, with no special…

The divisor function $\sigma(n)$ denotes the sum of the divisors of the positive integer $n$. For a prime $p$ and $m \in \mathbb{N}$, the $p$-adic valuation of $m$ is the highest power of $p$ which divides $m$. Formulas for…

We denote by $P^+(n)$ the largest prime factor of the integer $n$. In 1935, Erd\H os studied the quantity $T_c(x)$ defined by $$ T_c(x)=\big|\big\{p\le x: P^+(p-1)\ge p^c\big\}\big|, $$ and he proved $$ \limsup_{x\rightarrow…

Let $(u_n)_{n \ge 0}$ be a nondegenerate Lucas sequence and $g_u(n)$ be the arithmetic function defined by $\gcd(n, u_n).$ Recent studies have investigated the distributional characteristics of $g_u$. Numerous results have been proven based…

We determine asymptotically the maximal order of log d(d(n)), where d(n) is the number of positive divisors of n. This solves a problem first put forth by Ramanujan in 1915.

We prove the infinitude of shifted primes $p-1$ without prime factors above $p^{0.2844}$. This refines $p^{0.2961}$ from Baker and Harman in 1998. Consequently, we obtain an improved lower bound on the the distribution of Carmichael…

It is known that there are infinitely-many prime numbers which take the form of a polynomial of degree one with integer coefficients, this is Dirichlet's theorem. We use an elementary sieving argument together with bounds on the prime…

We show that there are infinitely many primes $p$ such that $p-1$ is divisible by a square $d^2 \geq p^\theta$ for $\theta=1/2+1/2000.$ This improves the work of Matom\"aki (2009) who obtained the result for $\theta=1/2-\varepsilon$ (with…

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…

A modified Dirichlet character $f$ is a completely multiplicative function such that for some Dirichlet character $\chi$, $f(p)=\chi(p)$ for all but a finite number of primes $p\in S$, and for those exceptional primes $p\in S$, $|f(p)|\leq…

In 1876, Edouard Lucas showed that if an integer $b$ exists such that $b^{n-1} \equiv 1 (\mathrm{mod} \ n)$ and $b^{(n-1)/p} \not\equiv 1( \mathrm{mod} \ n)$ for all prime divisors $p$ of $n-1$ , then $n$ is prime, a result known as Lucas's…