## About Summit

# On the correlation of completely multiplicative functions

This dissertation focuses on a conjecture of S. Chowla which asserts the equidistribution of the parity of the number of primes dividing the integers represented by a polynomial of degree d. This involves the behaviour of the classical Liouville function λ(n) which captures the parity of the total number of prime factors of an integer n. For any non-square polynomial f(n) with integral coefficients of degree d we consider the distribution of the sequence {λ(f(n))}. Chowla conjectured that the partial sum average of this sequence goes to zero. In the first two chapters we study a weaker form of this conjecture for polynomials of degree 2 with integer and rational coefficients and prove that this sequence takes the values −1 and +1 infinitely often. In the final chapter we show that this partial sum average goes to zero when f (n) = n(n + 1)(n + 2) and λ(n) is replaced by the truncated Liouville function λ_y(n).