On equal consecutive values of multiplicative functions

On equal consecutive values of multiplicative functions, Discrete Analysis 2024:12, 20 pp.

It is widely expected that the prime factorizations of a pair of large consecutive natural numbers $n, n+1$ (or of a pair with fixed spacing, such as $n,n+2$), should be statistically independent of each other, taking into account obvious correlations such as the ones arising from the parity of $n$. Confirming this belief in general is quite difficult; in particular, establishing statistical independence of the primality of $n$ and $n+2$ would imply the notorious twin prime conjecture. However, a more feasible task is to study the values $f(n), f(n+1)$ of a given multiplicative function $f$ at consecutive values and show some lack of correlation between them. For instance, in 1987, Erdős, Pomerance, and Sárközy showed that the number of divisors $d(n), d(n+1)$ of two consecutive numbers were unequal for all but a zero density set of numbers $n$. More recently, the author and Klurman showed a similar statement regarding multiplicative functions arising from modular forms.

On the other hand, for multiplicative functions taking values on the unit circle, such as the Liouville function $\lambda$, one would expect consecutive values of this function to agree quite often. Indeed, the prime number theorem tells us that $\lambda(n)$ is equal to $+1$ half the time and $-1$ half the time, so we should have $\lambda(n) = \lambda(n+1)$ for about half of the natural numbers $n$. This is a special case of Chowla’s conjecture, and is known by a result of Tao from 2016, albeit under the caveat that “about half” must be interpreted using the notion of “logarithmic density”. (This result is also related to the solution of the Erdős discrepancy problem, which was previously published in this journal.)

Here, the author shows that this sort of collision is possible only for multiplicative functions $f$ that are frequently on the unit circle. More precisely, it is shown that if $f$ is a multiplicative function for which $f(p)$ does not lie on the unit circle for a “large” set of primes $p$ (where “large” means that the sum of reciprocals is divergent), then $f(n)$ and $f(n+1)$ are unequal for all $n$ outside of a set of (logarithmic) density zero. This recovers versions of the previous results of Erdős-Pomerance-Sárközy and Mangerel-Klurman, while avoiding the use of analytic number theory machinery that is specific to particular multiplicative functions, such as the coefficients of modular forms.

The main strategy of proof is to apply Tao’s results (which apply for all 1-bounded functions) to the multiplicative functions $|f|^{it}$ for various $t$, and then use tools from additive combinatorics to combine the information gleaned from the different choices of $t$.