On Mobius' Inversion Formula and Closed Sets of Functions

1947 ◽  
Vol 62 (2) ◽  
pp. 213 ◽  
Author(s):  
Otto Szasz
Keyword(s):  
Inversion Formula ◽  
Closed Sets ◽  
Möbius Inversion ◽  
Möbius Inversion Formula

scholarly journals UNCERTAINTY PRINCIPLES CONNECTED WITH THE MÖBIUS INVERSION FORMULA

2013 ◽  
Vol 88 (3) ◽  
pp. 460-472 ◽  
Author(s):  
PAUL POLLACK ◽  
CARLO SANNA
Keyword(s):  
Number Theory ◽  
Uncertainty Principle ◽  
Inversion Formula ◽  
Arithmetic Functions ◽  
Uncertainty Principles ◽  
Elementary Number Theory ◽  
Natural Numbers ◽  
Elementary Number ◽  
Möbius Inversion ◽  
Möbius Inversion Formula

AbstractTwo arithmetic functions $f$ and $g$ form a Möbius pair if $f(n)= {\mathop{\sum }\nolimits}_{d\mid n} g(d)$ for all natural numbers $n$. In that case, $g$ can be expressed in terms of $f$ by the familiar Möbius inversion formula of elementary number theory. In a previous paper, the first-named author showed that if the members $f$ and $g$ of a Möbius pair are both finitely supported, then both functions vanish identically. Here we prove two significantly stronger versions of this uncertainty principle. A corollary is that in a nonzero Möbius pair, one cannot have both ${\mathop{\sum }\nolimits}_{f(n)\not = 0} 1/ n\lt \infty $ and ${\mathop{\sum }\nolimits}_{g(n)\not = 0} 1/ n\lt \infty $.

scholarly journals TWO APPROACHES TO MÖBIUS INVERSION

2011 ◽  
Vol 85 (1) ◽  
pp. 68-78
Author(s):  
I-CHIAU HUANG
Keyword(s):  
Partially Ordered Set ◽  
Inversion Formula ◽  
Ordered Set ◽  
Lagrange Inversion ◽  
Locally Finite ◽  
Partially Ordered ◽  
Möbius Inversion ◽  
Möbius Inversion Formula ◽  
Schauder Bases ◽  
Finite Partially Ordered Set

AbstractThe Möbius inversion formula for a locally finite partially ordered set is realized as a Lagrange inversion formula. Schauder bases are introduced to interpret Möbius inversion.

scholarly journals Möbius inversion formula for the trace group

2004 ◽  
Vol 339 (12) ◽  
pp. 899-904
Author(s):  
Anne Bouillard ◽  
Jean Mairesse
Keyword(s):  
Inversion Formula ◽  
Möbius Inversion ◽  
Möbius Inversion Formula

scholarly journals On Selberg's Lemma For Algebraic Fields

1955 ◽  
Vol 7 ◽  
pp. 138-143 ◽  
Author(s):  
R. G. Ayoub
Keyword(s):  
Prime Ideal ◽  
Simple Proof ◽  
Inversion Formula ◽  
Distribution Of Primes ◽  
Fundamental Lemma ◽  
Möbius Inversion ◽  
Möbius Inversion Formula ◽  
Algebraic Fields ◽  
Prime Ideal Theorem

1. Introduction. Recently two Japanese authors (1) gave a beautifully simple proof of Selberg's fundamental lemma in the theory of distribution of primes. The proof is based on a curious twist in the Möbius inversion formula. The object of this note is to show that their proof may be extended to a proof of the result for algebraic fields corresponding to Selberg's lemma. Shapiro (2) has already derived this result using Selberg's methods and deduced as a consequence the prime ideal theorem.

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