Correlations and distribution of essentially bounded functions in short intervals

Abstract

We say that an arithmetic function f is essentially bounded if $$f(n)\ll _{\varepsilon } n^{\varepsilon }$$ for all $$\varepsilon >0$$ . A particular case of essentially bounded function is the convolution product of the constantly 1 function and an essentially bounded g such that $$g(q)=0$$ for $$q>Q$$ . In this case, we say that $$f=g*{\mathbf{1}}$$ is a sieve function of range Q. Here we survey some of our recent results concerning the theory of the correlations and the distribution of essentially bounded functions in short intervals, with a special attention to the case of the sieve functions. Further, we take this opportunity to refine some proofs and generalize some of the achievements by giving their weighted version for pairs of arithmetic functions.