A conjectural asymptotic formula for multiplicative chaos in number theory

An asymptotic formula in additive number theory

1 . Introduction . In his paper [1], Erdös introduced the sequences of positive integers b 1 < b, < . . ., with (b ;, bj ) = 1, for i ~A j, and 'bi 1 < oo . With any such arbitrary sequence of integers, he associated the sequence {di} of all positive integers not divisible by any bj , and he showed that if b1 > 2, there exists a 0 < 1 (independent of the sequence {b i }) such that d i 1 di < d°...

Renormalization of Critical Gaussian Multiplicative Chaos and KPZ formula

Gaussian Multiplicative Chaos is a way to produce a measure on Rd (or subdomain of Rd) of the form eγX(x)dx, where X is a log-correlated Gaussian field and γ ∈ [0, √ 2d) is a fixed constant. A renormalization procedure is needed to make this precise, since X oscillates between −∞ and ∞ and is not a function in the usual sense. This procedure yields the zero measure when γ = √ 2d. Two methods ha...

The Maximum Number of Edges in a Strongly Multiplicative Graph

Multiplicative Problems in Combinatorial Number Theory

In this thesis we look at several problems that lie in the intersection between combinatorial and multiplicative number theory. A common theme of many of these problems are estimates for and properties of the smooth numbers, those integers not divisible by any large prime factors. The first is the Ramsey-theoretic problem to determine the maximal size of a subset of the integers containing no 3...

A conjectural Lefschetz formula for locally symmetric spaces