All Questions
Tagged with dirichlet-series mobius-function
17
questions
3
votes
1
answer
238
views
Alternating Dirichlet series involving the Möbius function.
It is well known that:
$$\sum_{n=1}^\infty \frac{\mu(n)}{n^s} = \frac{1}{\zeta(s)} \qquad \Re(s) > 1$$
with $\mu(n)$ the Möbius function and $\zeta(s)$ the Riemann Zeta function.
Numerical ...
- 3,191
0
votes
1
answer
49
views
What is the Dirichlet serie of The function $A$ (OEIS A001414) which gives the sum of prime factors (with repetition) of a number $n$?
The function $A$ (OEIS A001414) which gives the sum of prime factors (with repetition) of a number $n$ and defined by
$$
A(n)=\sum \limits_{p^{\alpha}\parallel n}\alpha p
$$
is this serie calculated ...
0
votes
0
answers
117
views
Perron's formula application - zeros of $L(s,\chi )$
After applying Perron's formula I have a complex integral involving something like
\[ \sum _{n=1}^\infty \frac {\mu (n)}{n^s}e\left (\frac {an}{q}\right ).\]
As usual ("usual" meaning e.g. ...
- 1,662
0
votes
1
answer
96
views
Need help with finding Dirichlet generating function
I am unable to find the following generating function:
$$\tag{1}{\sum _{n=1}^{\infty } \frac{2^{-k_n} \mu \big(\frac{n}{2^{k_n}}\big)}{n^s}}$$
$\mu$ is Möbius function, $k_n$ is the highest integer ...
- 1,206
1
vote
2
answers
235
views
Prove $\sum_{d | n} \mu(d) (\log(d))^2=0$ [duplicate]
If $n$ is a positive integer with more than 2 distinct prime factors, how to prove that $\sum_{d | n} \mu(d) (\log(d))^2=0$?
I struggle on how to continue from this.
Suppose $n=p_1 p_2 ... p_r$, ...
- 21
1
vote
0
answers
52
views
Variant of Möbius inversion: $b(n) = \sum_{d^2 \mid n} a(n/d^2) d^\alpha$
I'm trying to understand a step in a classic paper of Rankin. In Rankin's paper Contributions to the theory of Ramanujan's function $\tau(n)$ and similar arithmetical functions, he defines
$$ b(n) := \...
- 11
0
votes
3
answers
109
views
Proving identity using Dirichlet L functions
I'm trying to prove the following identity using Dirichlet L functions :
${\displaystyle \sum _{d\mid n}\varphi (d)=n}$
I have shown proved that the Dirichlet Series of $\varphi (n)$ equals to
${\...
- 105
2
votes
0
answers
113
views
Question about Dirichlet Series Related to Formula for $\frac{1}{e}$
This question is related to the three functions defined in (1) to (3) below where $\coth(z)$ gives the hyperbolic cotangent of $z$.
(1) $\quad M(x)=\sum\limits_{n=1}^x\mu(n)\quad\text{(Mertens ...
- 7,631
0
votes
1
answer
84
views
Questions on $f(x)=\sum\limits_{n=1}^{x}a(n)$ with an infinite number of positive integer zeros
This question is related to a class of functions that meet the following conditions.
(1) $\quad f(x)=\sum\limits_{n=1}^{x}a(n)$
(2) $\quad f(x)=0$ for an infinite number of values of $x\in\mathbb{Z}^...
- 7,631
0
votes
1
answer
378
views
Dirichlet series for 1/ζ(s)
Prove that for Re(s)>1
$$\frac{1}{\zeta(s)}=\sum_{n=1}^{\infty}\frac{\mu(n)}{n^s}$$
Where $\mu(n)$ is the Möbius function defined by:
$\mu(n)=1, \mbox{if }n=1$
$\mu(n)=(-1)^k, \mbox{if }n=p_1,p_2,....
1
vote
1
answer
1k
views
Dirichlet Series of Absolute value of Mobius Function equals Ratio of Riemann Zeta
I would like to prove this using Euler products:
$$\frac {\zeta(s)}{\zeta(2s)} = \sum_{n=1}^{\infty}\frac {\lvert \mu(n) \rvert}{n^s}$$
I have gotten here, but don't know if this is a correct ...
- 395
3
votes
1
answer
152
views
Is known the function $\sum_{n=1}^\infty\frac{(-1)^n\mu(n)}{n^s}$, where $s$ is the complex variable and $\mu(n)$ the Möbius function?
Let $\mu(n)$ the Möbius function, see its definition for example from this MathWorld, and we denote with $s$ the complex variable.
I'm curious to know if some case of the series $$\sum_{n=1}^\infty\...
user243301
1
vote
1
answer
64
views
On $\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\eta'(s)\frac{x^s}{s\zeta(s)}ds$, for $c>1$, where $\eta(s)$ is the Dirichlet Eta function
When I was combining the identities from this article from Wikipedia for the Mertens function, I've asked my an open question, if you can solve it from a standard viewpoint it is appreciated, and ...
user243301
6
votes
1
answer
503
views
Is this Dirichlet "$\eta$ version" of the inverse relationship between $\zeta(s)$ and the Möbius function correct?
With $\mu(n)$ the Möbius function and $\zeta(s)$ the Riemann zeta function, it is well-known that:
$$\frac 1{\zeta(s)}=\sum_{n=1}^\infty \frac{\mu(n)}{n^s}$$
that is proven to converge for $\Re(s)&...
- 3,191
6
votes
2
answers
732
views
Dirichlet series associated to squared Möbius
I would like to estimate the Dirichlet series of a multiplicative function. Consider the following:
$$\sum_{m \leqslant X} \frac{\mu^2(m)}{m^s}$$
When does it converges when $X$ grows? What is an ...
- 1,509