A proof of $\sum{\mu(n)/n}=0$

Question

I am looking for a proof (or references) of the following statement $$\sum_{n=1}^{\infty}{\frac{\mu(n)}{n}}=0$$ where $\mu$ is the Möbius function.

Many thanks !

Answer 1

(I remember being utterly astonished by this result when I first came across it, in Apostol's introductory book on analytic number theory, and I mean to study it properly some time.)

According to Apostol, there's a proof in Ayoub's book (see BibTeX items below) that it is equivalent to the Prime Number Theorem.

From section 5.6 of Edwards' book one learns that even before Landau, de la Vallee Poussin proved it in 1899, obtaining the estimate $|\sum \mu(n)/n| < K/\log x$ for all sufficiently large $x$.

Indeed, the result was already stated by Euler! On reflection, that's not surprising, because he probably merely assumed that you could expand and rearrange the product.

According to Ivic (pp. 50-51), the result was first proved by H. von Mangoldt in 1897. He refers the reader to Titchmarsh for details.

@BOOK{apostol:analytic, AUTHOR = "Tom M. Apostol", TITLE = "Introduction to Analytic Number Theory", PUBLISHER = "Springer", ADDRESS = "New York", YEAR = "1976"}

@book{ayoub:analytic, author = "Raymond G. Ayoub", title = "An Introduction to the Analytic Theory of Numbers", publisher = "American Mathematical Society", year = "1963"}

@book{edwards:riemann, author = "H. M. Edwards", title = "Riemann's Zeta Function", publisher = "Dover", address = "New York", year = "2001", note = "Republication of the edition published by Academic Press in 1974."}

@book{ivic:riemann, author = "Aleksandar Ivi\'c", title = "The Riemann Zeta-Function", publisher = "Dover", address = "New York", year = "2003", note = "Republication of the edition published by Wiley in 1985."}

@article{mangoldt:moebius, author = "H. von Mangoldt", title = "Beweis der {Gleichung} $\sum_{k = 1}^\infty \mu(k)/k = 0$", journal = "Sitz. Preuss. Akad. Wiss., Berlin", year = "1897", pages = "835-852"}

@book{titchmarsh:riemann, author = "E. C. Titchmarsh", title = "The Theory of the Riemann Zeta-function", edition = "Second", publisher = "Oxford University Press", year = "1987", note = "Revised by D. R. Heath-Brown."}