Correlations of multiplicative functions and applications

Abstract

We give an asymptotic formula for correlations $$\begin{eqnarray}\mathop{\sum }_{n\leqslant x}f_{1}(P_{1}(n))f_{2}(P_{2}(n))\cdots f_{m}(P_{m}(n)),\end{eqnarray}$$ where $f,\ldots ,f_{m}$ are bounded ‘pretentious’ multiplicative functions, under certain natural hypotheses. We then deduce several desirable consequences. First, we characterize all multiplicative functions $f:\mathbb{N}\rightarrow \{-1,+1\}$ with bounded partial sums. This answers a question of Erdős from $1957$ in the form conjectured by Tao. Second, we show that if the average of the first divided difference of the multiplicative function is zero, then either $f(n)=n^{s}$ for $\operatorname{Re}(s)<1$ or $|f(n)|$ is small on average. This settles an old conjecture of Kátai. Third, we apply our theorem to count the number of representations of $n=a+b$, where $a,b$ belong to some multiplicative subsets of $\mathbb{N}$. This gives a new ‘circle method-free’ proof of a result of Brüdern.

Rigidity theorems for multiplicative functions

“…We have a slight edge when considering pretentious functions (in which case the work [15] of the first author is of relevance), though in this case the question is still difficult to answer in full generality.…”

On Binary Correlations of Multiplicative Functions

“…Indeed, allowing the multiplicative functions g j to depend on x is crucial for applications to smooth numbers and to Burgess-type bounds. The asymptotic formula in [20] is a sieve-theoretic product of local mean values, but one cannot express the density of smooth numbers as such a product.…”

The Logarithmically Averaged Chowla and Elliott Conjectures for Two-Point Correlations

“…thus (20) tells us that |X | ε. From complete multiplicativity and the definition of c p we see that…”

“…Some condition of the form (8) must be needed in order to derive the conclusion (9), as one can see by considering examples such as g 1 (n) := χ (n)n it and g 2 (n) := g 1 (n), where χ is a Dirichlet character of bounded conductor, t is a real number of size t = o(x), and w is set equal to (for instance) (x/|t|) 1/2 . More precise asymptotics of sums such as those in (9) in the 'pretentious' case when g 1 and g 2 both behave like twisted Dirichlet characters n → χ (n)n it were computed in the recent preprint of Klurman [20].…”