For questions on Dirichlet series.
3
votes
1answer
52 views
A numeral system built around Dirichlet series, by analogy of how positional numeral systems are built around power series?
For any natural number and chosen base p, the number admits a unique expression of the form $a_np^n + ... + a_2p^2 + a_1p^1 + a_0$, where $a_k < p$ for all k. This property is effectively what ...
5
votes
1answer
105 views
Interesting phenomenon with the $\zeta(3)$ series
I noticed that if one takes certain partial sums of the series for $\zeta(3)$:
$$\zeta(3) = \sum_{n=1}^{\infty} \frac{1}{n^3} \approx \sum_{n=1}^{N} \frac{1}{n^3}$$
an interesting phenomenon occurs ...
0
votes
0answers
36 views
Convergence of $\sin(x)/x$ using Dirichlets Test [duplicate]
How do you go about proving convergence of $$\sum_{k=1}^{\infty} \dfrac{\sin(k)}k$$ using Dirichlet's Test?
3
votes
0answers
37 views
Dirichlet series' help
If $f(n)$ is an arithmetic function with $|f(n)|=1$, and $$\lim_{s\to+1} (s-1)\sum_{n=1}^\infty\frac{f(n)}{n^s}=0$$
Can I deduce that $$\lim_{s\to +1}(s-1)^2\sum_{n=1}^\infty\frac{f(n)\ln(n)}{n^s}=0$$
...
0
votes
0answers
47 views
Limits as a representation of the Dirichlet function
I read that the Dirichlet function (1 if Rational, 0 else) can be written as:
What is the proof of that? Are those limits commutative? Is there any other closed formula for Dirichlet function? (With ...
7
votes
2answers
146 views
How can I prove my conjecture for the coefficients in $t(x)=\log(1+\exp(x)) $?
I'm considering the transfer-function
$$ t(x) = \log(1 + \exp(x)) $$
and find the beginning of the power series (simply using Pari/GP) as
$$ t(x) = \log(2) + 1/2 x + 1/8 x^2 – 1/192 x^4 + 1/2880 x^6 - ...
0
votes
1answer
77 views
Sines and cosines of angles in arithmetic progression
Prove that if $\phi$ is not equal to $2k\pi$ for any integer $k$, then
$$\sum_{t=0}^{n} \sin{(\theta + t \phi)}=\frac{\sin({\frac{(n+1)\phi}2})\sin{(\theta+\frac{n \phi}2)}}{\sin{(\frac{\phi}2)}}$$
...
1
vote
0answers
54 views
Can we express $x$ as a Dirichlet series?
Can we express $x$ as a Dirichlet series ?
Thus $x = a_0 + a_1 2^{-x} + a_2 3^{-x} + a_3 4^{-x} + ...$ where the $a_i$ are real numbers ? I do not know how to get a nondivergent solution.
3
votes
0answers
46 views
Can that two double series representations of the $\eta$/$\zeta$ function be converted into each other?
By an analysis of the matrix of Eulerian numbers(see pg 8) I came across the representation for the alternating Dirichlet series $\eta$:
$$ \eta(s) = 2^{s-1} \sum_{c=0}^\infty \left( ...
2
votes
1answer
85 views
Absolute Convergence of Dirichelet Series
The exact series I must show converges absolutely is:
$$\sum_{n=1}^{\infty}{\frac{d(n)^r}{n^s}}$$
for $s > 1$, $r\in \mathbb{N}$ and where $d(n)=\#\text{ of divisors of } n$. I've been able to ...
1
vote
1answer
48 views
Multiplicative Dirichlet Series
How do I go about proving that where $f(k)$ is multiplicative that: $$\sum_{k\geq 1}{\frac{f({k})}{k^{s}}}=\prod_{p}\sum_{k\geq 0}{\frac{f(p^{k})}{p^{ks}}}$$
I first tried to use the fundamental ...
1
vote
0answers
52 views
How write Dirichlet character sums for the terms of the von Mangoldt function?
The way to separately write the terms of the von Mangoldt function $\Lambda$ as Dirichlet character sums seems to be:
$$\Lambda (1) = \sum\limits_{n=1}^{\infty } \frac{(e^{\Lambda (1)} \chi ...
1
vote
1answer
87 views
Euler product of Dirichlet Series
For $n$ a positive integer, let $f(n)$ be the squarefree part of $n$.
Find the Euler product for $\mathfrak D_{f}(s)$ where $\mathfrak D_{f}(s)$ is the Dirichlet Series of $f$.
2
votes
2answers
155 views
Does this Dirichlet series converge to zero?
Consider the periodic Dirichlet series that has this iterative definition:
...
9
votes
2answers
402 views
$\# \{\text{primes}\ 4n+3 \le x\}$ in terms of $\text{Li}(x)$ and roots of Dirichlet $L$-functions
In a paper about Prime Number Races, I found the following (page 14 and 19):
This formula, while
widely believed to be correct, has not yet been proved.
$$
\frac{\int\limits_2^x{\frac{dt}{\ln ...
0
votes
0answers
86 views
$L(1+it,\chi)\neq 0 $ whenever $t \neq 0 \in \mathbb{R}$
I understand that the proof of the assertion in the title uses the same method which proves that zeta function satisfies $\zeta(1+it)\neq 0$, where the above $L$ is Dirichlet L-function.
I.e, you ...
5
votes
1answer
125 views
Exponentiation of a Dirichlet series
I'm trying to understand a proof in Chandrasekharan's Introduction to Analytic Number Theory. Specifically, the proof of the lemma on p.118 before Dirichlet's theorem on primes in arithmetic ...
2
votes
0answers
85 views
Dirichlet series 'shifted' by a polynomial
Let $F(x) \in \mathbb{Z}[x]$ and
$$
\xi(s) = \sum^\infty_{n=1}g(n)n^{-s}
$$
be the Dirichlet series associated an arithmetic function $g(n)$. Define a new Dirichlet series
$$
\xi_F(s) = ...
2
votes
2answers
198 views
Convergence of the series $\sum \frac{\sqrt{n+1}-\sqrt{n}}{n^x}$
Could you help me to understand for which $x$ this series converge $\displaystyle\sum \frac{\sqrt{n+1}-\sqrt{n}}{n^x}$?
11
votes
2answers
334 views
regularity of root spacing of $G(z)=\sum\limits_{n=1}^{\infty} \frac{e^{-n^{2}}}{n^{z}}$
Define, on $\mathbb{C}$:
$$G(z)=\sum_{n=1}^{\infty} \frac{e^{-n^{2}}}{n^{z}}$$
A domain colored portrait of $G(z)$ (boxes are supposed to be negative signs):
suggests that the roots of $G(z)$ are ...
6
votes
2answers
224 views
Divergence of Dirichlet series
Suppose $s$ is a complex number with $\Re(s) \in (0,1]$ and $\{a_n\}$ is a complex sequence converging to $a \neq 0$. Must the Dirichlet series $$\sum_{n=1}^\infty\frac{a_n}{n^s}$$ diverge?
2
votes
1answer
149 views
Dirichlet character over Riemann zeta function
Let $\chi$ be a Dirichlet character mod q and let $$L(s,\chi)=\sum_{n\leq x} \frac{\chi(n)}{n^s}.$$ What is the value of $\displaystyle\lim_{s \rightarrow 1} \frac{L(s,\chi)}{\zeta(s)}$ for principal ...
4
votes
1answer
101 views
Relation between zeta function and Dirichlet L-function
Let $$H(s)=\frac{\zeta(s)}{\phi(q)} \sum_{\chi \mod{q}} L(s,\chi)=\sum_{n=1}^{\infty} \frac{h(n)}{n^s}$$ What is the smallest n (as a function of q) such that $h(n)\neq 1$?
3
votes
0answers
116 views
Dirichlet series represents an analytic function
Let $$T(x)=\sum_{n \leq x} t_n$$ and $T(X)=O(x^a)$ for $a \geq 0$. Now let $$F(s)=\sum_{n=1}^{\infty} \frac{t_n}{n^s}$$ What needs to be checked to prove that this Dirichlet series represents an ...
3
votes
2answers
162 views
Approximate Riemann zeta function
Given the function $Z(s,N)= \sum \limits_{n=1}^{N}n^{-s}$.
In the limit $N \to \infty$ the function $Z(s,N) \to \zeta (s)$ Riemann Zeta function.
My question is: Is there a Functional equation for ...
2
votes
2answers
86 views
Convergence of sum in proof that $\Phi(s) - \frac{1}{s-1}$ extends to $\Re(s) > \frac{1}{2}$
Definitions: $\Phi(s) = \displaystyle\sum_{p} \frac{\log p}{p^s}$ where $p$ denotes a prime number.
$\zeta(s) = \displaystyle\sum_{n=1}^{\infty} \frac{1}{n^s}$ denotes the Riemann zeta function.
...
5
votes
0answers
168 views
How to simplify $\newcommand{\bigk}{\mathop{\vcenter{\hbox{K}}}}\prod_{p\in\mathbb{P}}\left(1+\bigk_{k=1}^{\infty }\frac{f_k(s)}{g_k(s)}\right)^{-1}$
I'd like to simplify
$$\newcommand{\bigk}{\mathop{\huge\vcenter{\hbox{K}}}}B(s)=\prod_{p\in\mathbb{P}}\left(1+\bigk_{k=1}^{\infty }\frac{f_{k}(s)}{f_{k}(s)}\right)^{-1}$$ to something of the form ...
5
votes
2answers
375 views
An identity involving the Möbius function
$$\sum_{n=1}^{\infty}\frac{1}{n^s}\sum_{n=1}^{\infty}\frac{\mu(n)}{n^s}=1$$
for $s>1$.
How do I prove this identity?
7
votes
1answer
188 views
Reference request: $L$-series and $\zeta$-functions
Does anyone know a good book, lecture note, article etc. on $L$-series (Dirichlet, Hecke, Artin) and $\zeta$-functions in number theory? I'm especially interested in material explaining the following:
...
2
votes
2answers
103 views
A question about an identity involving Dirichlet characters
Let $\chi$ be a Dirichlet character $\bmod q$. We have
$$\sum_{n=0}^{\infty} (-1)^{n-1} \chi(n) n^{-s}=\prod_p ...
10
votes
2answers
308 views
Two Dirichlet's series related to the Divisor Summatory Function and to the Riemann's zeta-function, $\zeta(s)$
Considering the $\textit{Divisor Summatory Function}$, $D(n)$, defined as
$$
D(n) = \sum_{k=1}^{n}d(k) ,
$$
where
$$
d(n) = \sum_{k|n}^{n}1.
$$
One can observe the following pattern in the values of ...
16
votes
3answers
665 views
On Dirichlet series and critical strips
(I'll keep this one short)
Given a Dirichlet series
$$g(s)=\sum_{k=1}^\infty\frac{c_k}{k^s}$$
where $c_k\in\mathbb R$ and $c_k \neq 0$ (i.e., the coefficients are a sequence of arbitrary nonzero ...
