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Publication# The Shifted Convolution of Generalized Divisor Functions

Abstract

We prove an asymptotic formula for the shifted convolution of the divisor functions d(k)(n) and d(n) with k >= 4, which is uniform in the shift parameter and which has a power saving error term, improving results obtained previously by Fouvry and Tenenbaum and, more recently, by Drappeau.

Official source

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Related concepts (6)

Asymptotic analysis

In mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing limiting behavior. As an illustration, suppose that we are interested in the properties of a function f (n) as n becomes very large. If f(n) = n2 + 3n, then as n becomes very large, the term 3n becomes insignificant compared to n2. The function f(n) is said to be "asymptotically equivalent to n2, as n → ∞". This is often written symbolically as f (n) ~ n2, which is read as "f(n) is asymptotic to n2".

Convolution

In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions (f and g) that produces a third function () that expresses how the shape of one is modified by the other. The term convolution refers to both the result function and to the process of computing it. It is defined as the integral of the product of the two functions after one is reflected about the y-axis and shifted. The choice of which function is reflected and shifted before the integral does not change the integral result (see commutativity).

Parameter

A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when identifying the system, or when evaluating its performance, status, condition, etc. Parameter has more specific meanings within various disciplines, including mathematics, computer programming, engineering, statistics, logic, linguistics, and electronic musical composition.