Note on the Theory of Correlation Functions

The purpose of this note is to improve the current theoretical results for the correlation functions of the Mobius sequence $\{\mu(n): n\geq 1 \}$ and the Liouville sequence $\{\lambda(n): n\geq 1 \}$.Comment: Sixty Six Pages. Keywords: Autocorrelation function, Correlation function, Multiplicative function, Liouville function, Mobius function, von Mangoldt function, Exponential Su

Bombieri-Vinogradov for multiplicative functions, and beyond the x1/2x^{1/2}-barrier

Part-and-parcel of the study of "multiplicative number theory" is the study of the distribution of multiplicative functions in arithmetic progressions. Although appropriate analogies to the Bombieri-Vingradov Theorem have been proved for particular examples of multiplicative functions, there has not previously been headway on a general theory; seemingly none of the different proofs of the Bombieri-Vingradov Theorem for primes adapt well to this situation. In this article we find out why such a result has been so elusive, and discover what can be proved along these lines and develop some limitations. For a fixed residue class $a$ we extend such averages out to moduli $\leq x^{\frac {20}{39}-\delta}$.Comment: 54 page

On binary correlations of multiplicative functions

We study logarithmically averaged binary correlations of bounded multiplicative functions $g_1$ and $g_2$. A breakthrough on these correlations was made by Tao, who showed that the correlation average is negligibly small whenever $g_1$ or $g_2$ does not pretend to be any twisted Dirichlet character, in the sense of the pretentious distance for multiplicative functions. We consider a wider class of real-valued multiplicative functions $g_j$, namely those that are uniformly distributed in arithmetic progressions to fixed moduli. Under this assumption, we obtain a discorrelation estimate, showing that the correlation of $g_1$ and $g_2$ is asymptotic to the product of their mean values. We derive several applications, first showing that the number of large prime factors of $n$ and $n+1$ are independent of each other with respect to the logarithmic density. Secondly, we prove a logarithmic version of the conjecture of Erd\H{o}s and Pomerance on two consecutive smooth numbers. Thirdly, we show that if $Q$ is cube-free and belongs to the Burgess regime $Q\leq x^{4-\varepsilon}$, the logarithmic average around $x$ of the real character $\chi \pmod{Q}$ over the values of a reducible quadratic polynomial is small.Comment: 33 pages; Referee comments incorporated; To appear in Forum Math. Sigm