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Lecture

Asymptotic Estimations

Description

This lecture covers the concept of completely multiplicative functions, inversion with respect to multiplicative functions, Mobius functions, asymptotic estimation for monotonically increasing or decreasing functions, and the Euler-Maclaurin summation formula. It also delves into rearranging functions, convolution method, and Dirichlet-hyperbola method.

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Related concepts (29)

Möbius function

The Möbius function μ(n) is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated Moebius) in 1832. It is ubiquitous in elementary and analytic number theory and most often appears as part of its namesake the Möbius inversion formula. Following work of Gian-Carlo Rota in the 1960s, generalizations of the Möbius function were introduced into combinatorics, and are similarly denoted μ(x). For any positive integer n, define μ(n) as the sum of the primitive nth roots of unity.

Multiplicative function

In number theory, a multiplicative function is an arithmetic function f(n) of a positive integer n with the property that f(1) = 1 and whenever a and b are coprime. An arithmetic function f(n) is said to be completely multiplicative (or totally multiplicative) if f(1) = 1 and f(ab) = f(a)f(b) holds for all positive integers a and b, even when they are not coprime.

Arithmetic function

In number theory, an arithmetic, arithmetical, or number-theoretic function is for most authors any function f(n) whose domain is the positive integers and whose range is a subset of the complex numbers. Hardy & Wright include in their definition the requirement that an arithmetical function "expresses some arithmetical property of n". An example of an arithmetic function is the divisor function whose value at a positive integer n is equal to the number of divisors of n.

Completely multiplicative function

In number theory, functions of positive integers which respect products are important and are called completely multiplicative functions or totally multiplicative functions. A weaker condition is also important, respecting only products of coprime numbers, and such functions are called multiplicative functions. Outside of number theory, the term "multiplicative function" is often taken to be synonymous with "completely multiplicative function" as defined in this article.

Möbius inversion formula

In mathematics, the classic Möbius inversion formula is a relation between pairs of arithmetic functions, each defined from the other by sums over divisors. It was introduced into number theory in 1832 by August Ferdinand Möbius. A large generalization of this formula applies to summation over an arbitrary locally finite partially ordered set, with Möbius' classical formula applying to the set of the natural numbers ordered by divisibility: see incidence algebra.

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Related concepts (29)

Möbius function

Multiplicative function

Arithmetic function

Completely multiplicative function

Möbius inversion formula