Talk:Möbius function

message: Silly me, the exponential generating function 
:f(x)=\sum_{n=1}^\infty \mu(n) e^{-nx} 
seems to be non-trivial? linas 03:28, 2 July 2006 (UTC)
thread_title:  Exponential generating function? 

message: I understand, of course, that the sentence claiming that μ(0) is generally left undefined is an undisputable fact, so instead I wish to argue the reasoning behind that belief. It would seems to me it would be obvious that the Maple system's definition of -1 is correct. 0 is clearly not square-free, since its divisible by all spmplex numbers. That leaves only 1 and -1. Since 0 is divisible by all complex numbers, and inherently all primes, it has an infinte number of primes divisors. Since infinity is not an integer, and it is also half of itself, it can be shown not to be even. Thus, 0 is non-square-free integer with and odd number of distinct prime divisors, which gives us μ(0) = -1. I may have overlooked something, but if not I simply wonder why it is generally left undefined. -- He Who Is[ Talk ] 22:58, 5 July 2006 (UTC)
thread_title:  Moebius of 0 

message: the connection w/physics and pauli exclusion principle sounds neat but it's possibly a clever physicist messing with our heads.It was put in by 70.16.52.150 on 10/26/05. At about same time that editor stated similar stuff on Mobius inversion formula, later removed by Linas, I think.Rich 10:34, 25 October 2006 (UTC)
replies: text: I think the current version is correct, and that the primon gas article provides the citations, as well as enough details to verify it yourself.  linas (talk) 13:13, 6 May 2009 (UTC)
thread_title: Physics connection needs citation

message: It seems like this phrase near the beginning of the article overspecifies the numbers we're talking about.  If we want to avoid the 0-v-no-0 business mentioned at the natural number page, wouldn't it make more sense to say "positive integers," and just avoid the poorly-defined label "natural" in the first place?  I didn't want to change it myself, because there may be a reason I'm not aware of for keeping it. 129.2.56.220 (talk) 14:09, 15 February 2008 (UTC)
replies: text: Thanks.  I agree, and I changed it.  You may want to read WP:BOLD.  -- Dominus (talk) 14:36, 15 February 2008 (UTC)
thread_title:  Strictly positive natural numbers 

message: What is the overall distribution of the values of the mobius function? Is it -1 1/3 of the time 0 1/3 of the time and 1 1/3 of the time? Woscafrench (talk) 22:28, 29 April 2010 (UTC)
replies: text: The article Square-free integer has a short proof that the density of square-free integers is asymptotically \frac{6}{\pi^2}. These numbers have non-zero values of μ. I do not know if they are equally divided between plus and minus one. Thus 1-\frac{6}{\pi^2} of the integers have μ equal to zero. Virginia-American (talk) 00:43, 4 January 2011 (UTC)
thread_title: Distribution at <math>\infty</math>

message: What makes 0 even?  Is it because 0/2 = 0? 0 is the absence of a number, so can you give to 0 properties of a real number? My 2 Cents' Worth (talk) 12:16, 25 May 2010 (UTC)
replies: text: 0 is commonly considered as an integer, and hence a real number. 0 is not the absence of a number, it is the number that represents the absence of something. If you have 1 apple, and you eat it, you can still ask what number of apples you have, the answer being that you have zero apples. As for the evenness of zero, I'll refer you to the wiki article Parity of zero for a long discussion (though I'll also add that zero is divisible by two (since 2×0=0), so it satisfies one of the defining properties of even numbers). RobHar (talk) 18:00, 25 May 2010 (UTC)
thread_title:  0 is even? 

message: Toward the end of the article this passage occurs:
"The Möbius function also arises in the primon gas or free Riemann gas model of supersymmetry. In this theory, the fundamental particles or "primons" have energies log p. Under second-quantization, multiparticle excitations are considered; these are given by log n for any natural number n. This follows from the fact that the factorization of the natural numbers into primes is unique.
In the free Riemann gas, any natural number can occur, if the primons are taken as bosons."
Aha!  So any natural number can occur.  As what?????  Readers trying to understand the last sentence of this passage will not be able to get even a clue as to its meaning if an essential piece of information is omitted. It makes as much sense as if someone had written just: "In mathematics, any natural number can occur."Daqu (talk) 04:46, 10 February 2013 (UTC)
thread_title:  Incoherent sentence 

message: "The Dirichlet series that generates the Möbius function is the (multiplicative) inverse of the Riemann zeta function
\sum_{n=1}^\infty \frac{\mu(n)}{n^s}=\frac{1}{\zeta(s)}.
This is easy to see from its Euler product"
Why is it clear? Am i being stupid here?  — Preceding unsigned comment added by 109.153.163.53 (talk) 23:23, 29 September 2013 (UTC)
thread_title:  Product formula relation 

message: I have removed material sourced to a blog post on math.stackexchange.com.  Material added must be verified by independent reliable sources.  This does not qualify.  Deltahedron (talk) 15:46, 12 April 2014 (UTC)
thread_title:  Unsourced material 

message: I'm a little confused as to why the function returns positive 1 for the input 1. Wouldn't it be considered to have repeating prime factors (1x1x1x...) and output to 0? Or simply an odd number of factors, a single factor, which is "1". I'm not understanding how 1 has an even number of prime factors. 67.186.128.135 (talk) 06:07, 28 January 2020 (UTC)
replies: text: 1 is not prime...--Jasper Deng (talk) 06:14, 28 January 2020 (UTC)
thread_title:  Möbius function of 1 

message: Regarding this revert...
To answer the question in your edit summary, "more consistent with what?", the consistency is between the inline and standalone formulas. For example, the μ in these formulas:
:
:\sum_{d\mid n} \mu(d) = \begin{cases}1 & \text{if } n=1, \\ 0&\text{if } n>1.\end{cases}
They look slightly different. It's possible some question is left in the mind of the reader as to whether or not they represent the same thing. Sometimes typography does make a difference in math formulas. For example, consider the potential differences and non-differences among R, R, ℝ, r, and r. It would be better if math symbols that mean the same thing retain the same appearance, so the reader has fewer questions to puzzle through. -- Beland (talk) 14:19, 21 September 2020 (UTC)
thread_title:  Typographical consistency