All Questions
Tagged with dirichlet-series dirichlet-convolution
23
questions
3
votes
1
answer
79
views
Convolution Method for Bound
I am reading A survey of gcd-sum functions where the following result is stated:
Let $P(n)$ be the Pillai's arithmetical function. The Dirichlet series of $P$ is given by:
$$\sum_{n=1}^\infty \frac{P(...
0
votes
1
answer
49
views
What is the Dirichlet serie of The function $A$ (OEIS A001414) which gives the sum of prime factors (with repetition) of a number $n$?
The function $A$ (OEIS A001414) which gives the sum of prime factors (with repetition) of a number $n$ and defined by
$$
A(n)=\sum \limits_{p^{\alpha}\parallel n}\alpha p
$$
is this serie calculated ...
0
votes
0
answers
62
views
Dirichlet's Series - Convergence
Calculate the expression of the following Dirichlet's series:
$$ \dfrac{\zeta(s-1)}{\zeta(s)} = \sum_{n=1}^{\infty} \dfrac{\varphi(n)}{n^s} $$
$$ \dfrac{\zeta(2s)}{\zeta(s)}=\sum_{n=1}^{\infty} \dfrac{...
1
vote
1
answer
40
views
Term-wise product of arithmetic functions and its Dirichlet generating function
If we know Dirichlet generating function F(s) of $f(n)$ and G(s) of g(n) we can express generating function of Dirichlet convolution of $f(n)$ and $g(n)$ as product of the two generating functions $F(...
- 1,206
1
vote
1
answer
67
views
Necessary and sufficient condition to be completely multiplicative
I want to prove that $f*f=f \tau$ iff $f$ is completely multiplicative. The "if" part was relatively easy, using $f(g*h)=(fg)*(fh)$ and plug $g=h=1$ for all $n$.
Juxtaposition is ordinary, ...
- 918
0
votes
2
answers
103
views
How to find the coefficent of a term in a Dirichlet generating function in Mathematica?
For a normal Dirichlet generating function like $Zeta[s]^2$, I can get the coefficient of the n-th term by applying Dirichlet convolution of the two constant functions. But how to find the coefficient ...
- 271
0
votes
0
answers
88
views
Dirichlet Inverses of Binomial Coefficients
Let $\omega$ be a real number between $0$ and $1$, and let:
$$\mathbf{c}\left(n\right)=\binom{\omega+n-1}{n}$$
for all positive integers $n$. Is there a closed form for the Dirichlet inverse $\mathbf{...
- 2,219
2
votes
2
answers
203
views
Is it possible turn the Dirichlet ring into a Banach algebra?
The set of all arithmetic functions $f:\mathbb{Z}^{+}\to\mathbb{C}$, under pointwise addition and Dirichlet convolution, is a commutative ring, not all functions are Dirichlet invertible.
So my ...
user764256
6
votes
1
answer
376
views
Dirichlet series and Dirichlet convolution
Let $f$ and $g$ be an arithmetic functions, and let $f*g$ be the Dirichlet convolution of $f$ and $g$.
As known from fundamental analytic number theory, the Dirichlet series generating function is: $...
- 1,804
3
votes
1
answer
244
views
A certain identity of a Dirichlet series
I have encountered this problem:
I need to prove that
$\sum_{n=1}^{\infty} \frac{d(n^2)}{n^s} = \frac{\zeta^{3}(s)}{\zeta(2s)}$.
Now, I already know that $\frac{\zeta(s)}{\zeta(2s)} = \sum_{n is ...
- 51
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
1
answer
102
views
Question on the coefficient of the Dirichlet series related to $\frac{\zeta(s+2)}{\zeta(s)}$
This question is about the evaluation of $a(n)$ defined in (1) below which is related to the Riemann zeta function $\zeta(s)$ as illustrated in (2) below.
(1) $\quad a(n)=\sum\limits_{d|n}\frac{\mu\...
- 7,631
1
vote
3
answers
187
views
How does one obtain an expression for the Dirichlet series $g(s, \theta) = \sum_{n=1}^{\infty} \frac{\cos(n \theta)}{n^{s}} $?
I would like to obtain an expression for the function $$g(s, \theta) = \sum_{n=1}^{\infty} \frac{\cos(n \theta)}{n^{s}} \qquad (\#).$$
Here is what I've tried so far: we know from the definition of ...
- 7,148
3
votes
3
answers
448
views
Inversion theorem for Dirichlet series
Can someone come up with a proof for this little theorem?
Suppose that $F_a(s)$ is a Dirichlet series and $a(n)$ is its associated arithmetic function, that is:
$$F_a(s)=\sum_{n=1}^{\infty}\frac{a(n)...
user630964
-2
votes
1
answer
153
views
Dirichlet-convolution
Above is the definition i got from my note.
I was trying to do these and i get stuck when i complete setting up the definition. I am trying to break down $c(n) = ((e_1 - 2e_2) * u)n$
= $( u * e_1 - ...
- 173