All Questions
Tagged with dirichlet-series arithmetic-functions
24
questions
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How fast does the proportion guaranteed by dirichlet converge?
I'm working on a counting problem and I'm using Dirichlets theorem (weak form) at some point in the counting. The problem is I don't know how fast something converges and I'm not very knowledgeable in ...
- 899
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1
answer
78
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How to get the Euler product for $\sum_{n = 1}^{\infty} \mu(n)/\phi(n^k)$.
Let $\mu$ be the Mobius function, and $\phi$ Euler's totient function. I am reading a proof found in this paper (Theorem 2 on page 17), and I can't quite figure why I'm getting something different ...
- 937
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1
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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 ...
1
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1
answer
135
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Show the function for which the Dirichlet generating series is $\zeta(2s)$ using only $\tau,\varphi,\sigma\text{ and }\mu$ or some explicit formula.
I'm trying to find the function with Dirichlet generating series $\zeta(2s)$, I know that this relates somehow to the Liouville function but I am trying to express it in terms of only the standard ...
- 299
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1
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246
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Dirichlet Series of Square Full Integers.
As in the title, I want to find the Dirichlet series $F$ of the indicator function for cube full integers $f(n)=1 \iff p^3|n, \forall p|n$ and $f(n)=0$ otherwise. Since $f$ is clearly multiplicative, $...
- 193
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1
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530
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Dirichlet series for $\zeta^3(s)/\zeta(2s)$.
I am currently studying number theory and our instructor refers to Apostol's book on Analytic number theory for the chapter Dirichlet series.In that book,there is an exercise which is as follows:
Let $...
- 3,619
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2
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213
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Show that $\sum\limits_{n\in \mathbb N} \frac{2^{\omega(n)}}{n^s}=\frac{\zeta^2(s)}{\zeta(2s)}$.
I am a graduate student of Mathematics. I have started reading number theory. I encountered a problem of analytic number theory.
Show that $\sum\limits_{n\in \mathbb N} \frac{2^{\omega(n)}}{n^s}=\frac{...
- 3,619
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2
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235
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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
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206
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Convergence of a Dirichlet series
For a fixed positive integer $j$, consider the arithmetical function :
$$\vartheta _{j}(k+1)=\left\{\begin{matrix}
1 \;\;, & k+1=j^{l}\;\;(l=1,2,3...)\\
0 \;\;, & \text{otherwise}
\end{...
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Question on the distribution of the values of $f(x)=\sum\limits_{n=1}^x a(n)$ where $a(n)=\sum\limits_{d|n}\mu(d)\ \mu\left(\frac{n}{d}\right)$
Consider the function $a(n)$ defined in formula (1) below and it's summatory function $f(x)$ defined in formula (2) below where $f(x)$ is related to the Riemann zeta function $\zeta(s)$ as illustrated ...
- 7,631
2
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2
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203
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Is it possible turn the Dirichlet ring into a Banach algebra?
The set of all arithmetic functions $f:\mathbb{Z}^{+}\to\mathbb{C}$, under pointwise addition and Dirichlet convolution, is a commutative ring, not all functions are Dirichlet invertible.
So my ...
user764256
1
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1
answer
102
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Dirichlet series of the $p$-adic valuation
Recall that given some prime $q$, the $q$-adic valuation of an integer $n\geqslant1$ is defined as : $$\nu_q(n)=\max\{\nu\geqslant0\;/\;q^\nu|n\}.$$
From the prime factorization, it follows that $\...
- 1,270
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75
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What is meant by $\sum_{d \le x}f(d)$ in this equation?
Wikipedia's page (here) on the average order of arithmetic functions gives the following as a means of finding such an order using Dirichlet Series:
Define $f$ as an arithmetic function on $n$, and ...
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3
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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
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1
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Dirichlet Series in analytic number theory
I have a question about Abscissa of Convergence of Dirichlet series. The question is ;
"Let $\sigma_{1}$ and $\sigma_{2}$ be real numbers with $\sigma_{1} \leq \sigma_{2} \leq \sigma_{1}+1 .$ ...
user759578