Open AccessThe Möbius transform and the infinitude of primes
Author(s) -
Paul Pollack
Publication year - 2011
Publication title -
elemente der mathematik
Language(s) - English
Resource type - Journals
eISSN - 1420-8962
pISSN - 0013-6018
DOI - 10.4171/em/178
Subject(s) - mathematics , lemma (botany) , möbius function , finite set , combinatorics , dirichlet distribution , inversion (geology) , arithmetic function , number theory , discrete mathematics , pure mathematics , mathematical analysis , ecology , paleontology , poaceae , structural basin , biology , boundary value problem
The well-known Mobius inversion formula ([2, Theorems 266, 267]) says precisely that the Mobius and Dirichlet transforms are inverses of each other: for any f , we have f qp f pq f . Our proof of the infinitude of primes is based on the following lemma. By the support of f , we mean the set of natural numbers n for which fpnq 0. Lemma (Uncertainty principle for the Mobius transform). If f is an arithmetic function which does not vanish identically, then the support of f and the support of q f cannot both be finite. Proof. Suppose for the sake of contradiction that both f and q f are of finite support. Let F pzq 8
Accelerating Research
Robert Robinson Avenue,
Oxford Science Park, Oxford
OX4 4GP, United Kingdom
Address
John Eccles HouseRobert Robinson Avenue,
Oxford Science Park, Oxford
OX4 4GP, United Kingdom