Sacha Mangerel

Analogues of the binary Goldbach problem for the Liouville function

Motivated by the heuristic relationship between the value distribution of the Liouville function \lambda(n) and the distribution of primes, we will discuss two analogues of the binary Goldbach problem, concerning the occurrence of sign patterns in the sequence of pairs (\lambda(n),\lambda(N-n)), for 1 \leq n < N. These problems are not tractable using current techniques towards correlations of multiplicative functions. I will outline two instances in which rigid hypotheses on the non-existence or paucity of patterns like \lambda(n) = \lambda(N-n) = -1 imply, via a study of exponential sums, ``character-like'' behaviour for \lambda. This leads, in one instance, to the solution of a conjecture of Shusterman, conditional on GRH. Surprisingly, an understanding of the Pierce expansion of rational numbers plays a significant role.