I. Schur and G. Schur proved that, for all completely multiplicative functions f : N → {−1, 1}, with the exception of two character-like functions, there is always a solution of f (n) = f (n + 1) = f (n + 2) = 1. Hildebrand proved that for the Liouville λ-function each of the eight possible sign combinations (λ(n), λ(n + 1), λ(n + 2)) occurs infinitely often. We prove for completely multiplicative functions f : N → {−1, 1}, satisfying certain necessary conditions, that any sign pattern ( 1, 2, 3, 4), i ∈ {−1, 1}, occurs for infinitely many 4-term arithmetic progressions (f (n), f(n + d), f(n + 2d), f(n + 3d)).The proof introduces graph theory and new combinatorial methods to the subject.

The purpose of this paper is to obtain an effective estimate of the exponential sum n x Λ(n)e a q + β n (where e(α) = e 2πiα , α, β ∈ R, (a, q) = 1 and Λ is the von Mangoldt function) in the range (log x) 1/2+ε q x (log log log x) 1+ε and |β| < 1 q(log log log x) 1+ε . It improves Daboussi's estimate [2, Theorem 1] in the range q (log x) D and x(log x) −D q, D > 0 and is valid in a wider range for β.

We determine asymptotically the maximal order of log d(d(n)), where d(n) is the number of positive divisors of n. This solves a problem first put forth by Ramanujan in 1915.

We present a population genetic algorithm which satisfies detailed balance, and which has a stationary distribution that factorises into an explicit form for arbitrary fitness functions. For a population size of 1, it is the Metropolis algorithm with a 'breeder' proposal distribution; it extends to larger populations in a natural way, and the stationary (that is, the mutation-selection equilibrium) distribution is exactly known in a simple form for any population size. We term this algorithm exchangeable breeding tuple product sampling (EBT).EBT is closely related to some non-parametric Bayesian Markov-chain Monte Carlo sampling algorithms. EBT can also be viewed as a generalisation of the Moran process.

No abstract