Multiplicative functions on shifted primes

Abstract

Let f be a positive multiplicative function and let $k\ge 2$ be an integer. We prove that if the prime values $f(p)$ converge to 1 sufficiently slowly as $p\to +\infty$, in the sense that ${\sum }_p|f(p)-1|=\infty$, there exists a real number $c>0$ such that the k-tuples $(f(p+1),\dots ,f(p+k))$ are dense in the hypercube ${[0,c]}^k$ or in ${[c,+\infty )}^k$. In particular, the values $f(p+1),\dots ,f(p+k)$ can be put in any increasing order infinitely often. Our work generalises previous results of De Koninck and Luca.

Introduction

The understanding of the local behavior of arithmetic functions has been the subject of research of many mathematicians. Part of this research involves the study of the values of an arithmetic function on consecutive integers. For example, if we denote the divisor function by τ, in 1952 Erdös and Mirsky [1] asked whether the equation $\tau (n)=\tau (n+1)$ admits infinitely many solutions in the set of natural numbers, a question that can be considered as a close relative of the twin prime conjecture. It remained open for about thirty years until Heath-Brown [3] answered it affirmatively in 1984 by showing that

$$\mathrm{\#}\{n\le x:\tau (n)=\tau (n+1)\}\gg x{(\log x)}^{-7}.$$

The method of Heath-Brown also yielded that there exist infinitely many positive integer solutions to the equation $\mathrm{\Omega }(n)=\mathrm{\Omega }(n+1)$, where Ω counts the number of prime factors of n with multiplicity. His method, however, was not successful in proving that the analogous equation $\omega (n)=\omega (n+1)$ has infinitely many positive integer solutions for the function ω which counts the number of distinct prime factors of n. It was Schlage-Puchta [8] who proved for the first time that even the equality $\omega (n)=\omega (n+1)$ holds for infinitely many natural numbers n.

In 2011, Goldston, Graham, Pintz and Yildirim [2] made a significant breakthrough. They did not just prove that there are infinitely many integer solutions of those equations, but they also showed that the value of the relevant arithmetic function $\tau ,\mathrm{\Omega }$ or ω can be specified. For instance, they proved that there are infinitely many integers n such that $\omega (n)=\omega (n+1)=3$.

Arithmetic functions, such as τ, are very sensitive on the exact number of prime factors of their input. In particular, they are highly sensitive on the large prime factors of their input. On the other hand, if we consider functions that are less sensitive to large prime factors, we can say more. Indeed, recently, De Koninck and Luca [6] proved that for any fixed integer $k\ge 2$ we have that

$$\frac{\max \{\phi (n+1),\dots ,\phi (n+k)\}}{\min \{\phi (n+1),\dots ,\phi (n+k)\}}$$

is arbitrarily close to 1 infinitely often. Here φ stands for the Euler totient function. They also established the same result for the additive functions ω, Ω, the sum of divisors function ${\sigma }_1$ and the kernel function γ, which is the multiplicative function defined by the relation $\gamma (p^m)=p$ for any prime power $p^m$.

They had also proved [5, Problem 8.6] that if $\{i_1,\dots ,i_k\}$ is a permutation of $\{1,\dots ,k\}$, the inequalities

$$\phi (n+i_1)<\dots <\phi (n+i_k)$$

hold for an infinite set of natural numbers n.

The purpose of the present paper is to extend these results. We state below our main result and an immediate consequence of it. Both of them are in the same spirit as the work of De Koninck and Luca.

Theorem 1.1

Let k be a positive integer and let f be a positive multiplicative function such that ${\sum }_p|f(p)-1|=\infty$ and $f(p)\to 1$ when $p\to \infty$. There exists a positive real number c such that the set of tuples $(f(p+1),...,f(p+k))$ is dense either in ${[0,c]}^k$ or in ${[c,+\infty )}^k$.

Corollary 1.2

Let $k\ge 2$ be an integer and let f be a positive multiplicative function such that ${\sum }_p|f(p)-1|=\infty$ and $f(p)\to 1$ when $p\to \infty$. If $\{i_1,\dots ,i_k\}$ is a permutation of $\{1,\dots ,k\}$, the inequalities

$$f(p+i_1)<\dots <f(p+i_k)$$

hold for infinitely many primes p.

Proof

In Theorem 1.1, we may choose sufficiently small or large (this depends on whether we have density in ${[0,c]}^k$ or ${[c,+\infty )}^k$) non-negative numbers $c_1,\dots ,c_k$ such that $c_{i_1}<\dots <c_{i_k}$ and $f(p+i_j)\to c_{i_j}$ for all $j\in \{1,\dots ,k\}$ on a subsequence of the primes. Then, for $ϵ>0$ such that $ϵ<\min \{(c_{i_{j+1}}-c_{i_j})/2,j=1,\dots ,k-1\}$, we have that

$$f(p+i_j)<c_{i_j}+ϵ<c_{i_{j+1}}-ϵ<f(p+i_{j+1})$$

for infinitely many primes p. □

Remark 1.1

In Theorem 1.1 we assumed that the function f is positive. However, we can have an analogous result for f being zero only on a finite set of primes by replacing the shifts $i\in \{1,\dots ,k\}$ by the k first multiples of the product of those primes. For example, if $f(2)=0$, we can modify the proofs below by replacing the shifts $i\in \{1,\dots ,k\}$ by the even shifts 2i. Doing so implies that there exists a constant $c>0$ such that the set of tuples $(f(p+2),\dots ,f(p+2k))$ is dense either in ${[0,c]}^k$ or ${[c,+\infty )}^k$.

Remark 1.2

This theorem extends (1.1), proven by De Koninck and Luca. It is also worth noting that there are additive functions whose values on consecutive integers can be ordered. For example, De Koninck, Friedlander and Luca [4] proved that the inequalities

$$\omega (n+1)<\dots <\omega (n+k)$$

hold infinitely often. They also proved that ω can be replaced by Ω in (1.2). Furthermore, minor changes in the solution of Problem 7.26 in [5] can lead to these results with $\{i_1,\dots ,i_k\}=\{1,\dots ,k\}$.

Theorem 1.1 may be directly deduced from its more technical analogue, which is the following theorem that we prove in Section 2.

Theorem 1.3

Let $k\ge 2$ be an integer and let f be a positive multiplicative function such that $f(p)\to 1$ as $p\to \infty$. Let also $K:=k(k+1){((k-1)!)}^2$.

(a) If

$$\sum _{f(p)>1}(f(p)-1)=\infty ,$$

then the set of tuples $(f(p+1),...,f(p+k))$ is dense in $[f(K),\infty )\times [f(1),\infty )\times \cdots \times [f(k-1),\infty )$.

(b) If

$$\sum _{f(p)<1}(1-f(p))=\infty ,$$

the set of tuples $(f(p+1),...,f(p+k))$ is dense in $[0,f(K)]\times [0,f(1)]\times \cdots \times [0,f(k-1)]$.

Notation and definitions. If n is a positive integer such that $n\ge 2$, then $P^{+}(n)$ denotes the largest prime factor of n and $P^{-}(n)$ denotes the smallest prime factor of n. We also define $P^{+}(1)=1$ and $P^{-}(1)=+\infty$.

Let $y>0$ and let n be a natural number. The y-rough part of n is defined to be equal to ${\prod }_{p^a\Vert n,p>y}{p}^a$ and its y-smooth part is given by ${\prod }_{p^a\Vert n,p\le y}{p}^a$.