Questions tagged [mobius-function]
Questions on the Möbius function μ(n), an arithmetic function used in number theory.
431 questions
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Closed form for Dirichlet series whose coefficients are the Möbius function times a geometric series
By definition,
$$
\sum_{n=1}^{+\infty}\frac{1}{n^s} = \zeta(s)
\tag{*}
$$
when the real part of $s$ is large enough ($>1$). I am also aware that
$$
\sum_{n=1}^{+\infty}\frac{\mu(n)}{n^s} = \frac{1}...
- 6,568
2
votes
1
answer
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views
Generalizing Mobius Inversion to Abelian Groups
I am reading through Ireland & Rosen's section on the Mobius function and Dirichlet products (Ch.2). Here is how I understand it:
We consider complex valued functions on the natural numbers. We ...
- 750
1
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2
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Looking for multiplicative functions satisfying $f * 1 = \mu f $ and $f * 1 = |f|$
I'm trying to find examples of multiplicative functions $f$ that satisfy some interesting convolution identities.
Here, $*$ denotes the Dirichlet convolution, $\mu$ is the Möbius function, and $1(n) = ...
- 11
1
vote
1
answer
145
views
Why does $\mu$ arise in $\frac{1}{\zeta(s)} = \sum_{n=1}^{\infty}{\frac{\mu(n)}{n^s}}=1 - \frac{1}{2^s}-\frac{1}{3^s}-\frac{1}{5^s}+ \dots$? [closed]
Recently, I've become interested in the Zeta Function, and as a result I started reading this paper by W. Dittrich. On page 9 of the pdf the following equation, referred to as the Möbius function, is ...
0
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0
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65
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Question about the step in the proof of the Weyl's inquality along primes from the book of Vaughan.
I have been reading Vaughan's book on the Hardy–Littlewood method and in Theorem 3.1 in Chapter III he proves the following estimate for
\begin{equation*}
f(\alpha):=\sum_{p \leqslant n}(\log p) e(\...
- 839
-1
votes
1
answer
89
views
The Dirichlet product $\mu \ast 2^{\omega (n)}$
Let $\omega(n)$ be the function that counts the number of distinct prime factors of n, and let $\mu$ be the Möbius function.
I have seen many questions on the Dirichlet product $\mu \ast \omega$ and i ...
- 181
2
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0
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Sum of weighted Möbius function
It is well known that the Riemann hypothesis is equivalent to the following statement about the growth rate of the Mertens function (see Titschmarsh’s text Section 14.25):
For all $\epsilon>0$
$$M(...
- 477
2
votes
1
answer
84
views
Merten function limit
Considering difference:
$$\sum _{h=1}^n \mu (h)-n \sum _{h=1}^n \frac{\mu (h)}{h}$$
This difference seems to diverge.
If we consider graph $n={1,300000}$:
Looks like that the average is below $0$. ...
- 2,066
0
votes
1
answer
130
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Reference for a proof of the identity $\sum_{k=1}^\infty\frac{\mu(k)\ln^2k}k=-2 \gamma$? [closed]
Can you please provide me a reference for a proof of the identity
$$
\sum_{k = 1}^{\infty}\frac{\mu\left(k\right)\ln^{2}\left(k\right)}k
= -2\gamma\ ?.
$$
This is an identity involving the Mobius ...
2
votes
1
answer
143
views
Proof of $|\sum_{n \leq x} \frac{\mu (n)}{n}| \leq 1$.
I'm reading a proof of $$\left|\sum_{n \leq x} \frac{\mu (n)}{n}\right| \leq 1$$ for all $x \geq 1$. If $x < 2$, there is only one term in the sum, $\mu(1) = 1$. Now assume that $x \geq 2$. For ...
- 1,191
-2
votes
1
answer
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Can we find an exact formula for the number of square-free integers between $1$ and $n$ without summation or product?
The problem of counting the number of square-free integers between $1$ and $n$ is well studied in number theory. The exact solution is traditionally given by a formula involving the Möbius function $\...
user1497226
3
votes
1
answer
173
views
Finding a "Closed Form" Expression for $\sum_{d|n}\mu(n/d)\tau(d)$
I recently got this question on a homework, and after some initial experiments got the hypothesis that $\sum_{d|n}\mu(n/d)\tau(d)=1$. I did prove this, however along the way I also encountered a proof ...
- 69
2
votes
1
answer
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Minimal number of generators of the monoid generated by roots of the unity.
This question is linked to Unicity of decomposition for the monoid generated by roots of the unity.
We set for $n \in \mathbb{N}^*$ $M_n$ the monoid generated by $n$-th roots of the unity and $M_n^\...
- 5,691
2
votes
2
answers
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Unicity of decomposition for the monoid generated by roots of the unity.
This question is linked to Minimal number of generators of the monoid generated by roots of the unity.
The inspiration of this question comes from a proof I made in which I had to decompose relative ...
- 5,691
0
votes
0
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about an estimate concerning the Mobius function
I am studying about the zero density estimates of Dirichlet $L$-series, and in a related paper by O. Ramare, he proves the following explicit estimate: for $X \geq 10^{9}$, we have
\begin{equation}
\...
- 1