We study how dense a multiplicative basis of order $h$ for $\mathbb Z^+$ can be, improving on earlier results. Upon introducing the notion multiplicative complement, we present some tight density bounds.

A famous conjecture of Littlewood (c. 1930) concerns approximating two real numbers by rationals the same denominator, multiplying errors. In a lesser-known paper, Wang and Yu (Chin Ann Math 2:1–12, 1981) established an asymptotic formula for number such approximations, valid almost always. Using quantitative Koukoulopoulos–Maynard theorem Aistleitner–Borda–Hauke, together with bounds arising f...

We develop the theory of multiplicative Ehresmann connections for Lie groupoid submersions covering identity, as well their infinitesimal counterparts. construct obstructions to existence such connections, and we prove several interesting classes groupoids algebroids, including all proper groupoids. show that many notions from principal bundle have analogues in this general setup, 1-forms, curv...

A multiplicative function f is said to be specially multiplicative if there is a completelymultiplicative function fA such that f(m)f(n)= ∑ d|(m,n) f (mn/d)fA(d) for allm andn. For example, the divisor functions and Ramanujan’s τ-function are specially multiplicative functions. Some characterizations of specially multiplicative functions are given in the literature. In this paper, we provide so...