Multiplicative function mean values: asymptotic estimates

Mean values of multiplicative functions

Let f(n) be a totally multiplicative function such that |f(n)| ≤ 1 for all n, and let F (s) = ∑∞ n=1 f(n)n−s be the associated Dirichlet series. A variant of Halász’s method is developed, by means of which estimates for ∑N n=1 f(n)/n are obtained in terms of the size of |F (s)| for s near 1 with 1. The result obtained has a number of consequences, particularly concerning the zeros of the p...

Decay of Mean-values of Multiplicative Functions

p 1−f(p) p diverges then the limit in (1.1) exists, and equals 0 = Θ(f,∞). Wirsing’s result settled an old conjecture of P. Erdős and Wintner that every multiplicative function f with −1 ≤ f(n) ≤ 1 had a mean-value. The situation for complex valued multiplicative functions is more delicate. For example, the function f(n) = n (0 6= α ∈ R) does not have a mean-value because 1 x ∑ n≤x n iα ∼ x 1+i...

Asymptotic mean values of Gaussian polytopes

We consider geometric functionals of the convex hull of normally distributed random points in Euclidean space R. In particular, we determine the asymptotic behaviour of the expected value of such functionals and of related geometric probabilities, as the number of points increases.

Inequalities and Asymptotic Estimates

Unless specified otherwise, we use µ, σ 2 to denote the mean and variance of the the variable under consideration. This note shall be updated throughout the seminar as I find more useful inequalities. 1 Basic inequalities Theorem 1.1 (Markov's Inequality). If X is a random variable taking only non-negative values, then for any a > 0 Pr[X ≥ a] ≤ E[X] a. (1) Proof. We show this for the discrete c...

Asymptotic Hölder Absolute Values