Questions tagged [dirichlet-series]

For questions on Dirichlet series.

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34 views

$L(1,\chi)$ exact value where $\chi$ is a non-principal Dirichlet character mod 3

If $\chi$ is a Dirichlet character, one defines its Dirichlet L-series by $L(s,\chi) = \sum_{n=1}^\infty \frac{\chi(n)}{n^s}$. Let $\chi$ be a non-principal Dirichlet character mod 3. What is the ...
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1answer
21 views

Picking the start of convergence $c$ for an integral expression of the Dirichlet $\eta$-function. What happens when $c \in \mathbb{C}$?

Using the following expression for the Dirichlet $\eta(s)$-function: $$\normalsize \eta(s,c) = -\frac{1}{\Gamma(s-c)} \int_0^{\infty} x^{s-c-1}\,\text{Li}_c(-\text{e} ^{-x}) \mathop{dx}$$ one could ...
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3answers
22 views

How does this identity so elegantly combine an infinite sum in $\eta$ and an improper integral in $\Gamma$?

This is all well and good, but where did this come from? In the article on the Gamma function, Wikipedia shows most of its alternate definitions with clear proofs, yet in the article on the Dirichlet ...
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0answers
42 views

Does my odd proof for the Abel sum for $\eta(-2)$ work?

EDIT: The correct answer to the Abel sum of $\eta(-2)$ has been given by the comments under this post. The focus of the question is now whether there is any sense to my method and my "proof" ...
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1answer
36 views

The Dirichlet series for the Liouville function related to the Riemann zeta function

$$\sum_{n=1}^{\infty} \frac{λ(n)}{n^s}=\frac{ζ(2s)}{ζ(s)}$$ Let $λ(n) = (−1)^k$, where $k$ is the number of prime factors of $n$, counting multiplicities. (Liouville function) for $Re(s)>1$, where $...
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0answers
43 views

Is there a name for this generalization of the Riemann Zeta Function?

The usual Riemann Zeta function is $$ \zeta(s) = \sum_n \frac{1}{n^s} $$ Suppose we modify the denominator instead to $$ \zeta_?(s) = \sum_n \frac{1}{(n\sqrt{1+Bn^2})^s} $$ We can do some elementary ...
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0answers
23 views

on the derivatives of the solution of the Dirichlet problem in the unit disk

studying the Dirichlet problem on the unit disk of Rafael Iorio's book "Fourier Analysis and partial Differential Equations" I came across a step that is not entirely clear to me to ...
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2answers
65 views

What is the limit of this Goldbach Conjecture related Dirichlet series?

Background Let us recollect the Goldbach's conjecture. For any $n \geq 2$ there exists primes $p_i$ and $p_j$ such that: $$ 2n = p_i + p_j $$ where $p_k$ is the $k$'th prime. Now we define a function $...
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1answer
25 views

Dirichlet L-series inequality

Let $h$ be a class number of an imaginary quadratic number field of discriminant $d$. It holds that $h = k(d)\cdot L_d(1)$ where $k(d)$ is the Dirichlet structure constant and $L_d$ is the Dirichlet $...
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1answer
37 views

Question Regarding Relationship between the Riemann Zeta Function and the Dirichlet Eta Function

So I was looking at the Riemann hypothesis and I saw the relationship between the Riemann zeta function and the Dirichlet eta function which really confused me because I didn't understand how a ...
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1answer
78 views

Can we find $\lim_{N\to \infty} \frac{1}{\sqrt{N}} \sum_{n \le N} \mu(n)/\sqrt{n}$

Here $\mu(n)$ is möbius function. Without assuming RH can we find if $\lim_{N \to \infty} \frac{1}{\sqrt{N}} \sum_{n=1}^N \frac{\mu(n)}{\sqrt{n}}$ exists, and if yes, what it may be. Calculations ...
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1answer
57 views

Show that $\sum_{n\geq 1} \frac{a_n}{n^z}$ defines a holomorphic function in $\{ \operatorname{Re}z > 1\}$

I am trying to make the following demonstration: let $(a_n)_n\subset \mathbb{C}$ be a sequence fulfilling that for all $\delta>0$, $$\sup_n \dfrac{|a_n|}{n^\delta}<+\infty. $$ Show that the ...
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1answer
29 views

Proving $ B(x) = 2 \sqrt{x} L(1, \chi) + O(1) $ where $ B(x) = \sum_{n \le x} \frac{1}{\sqrt{n}} \sum_{d \mid n} \chi(d) $

Theorem 6.20 of Introduction to Analytic Number Theory (Tom M. Apostol) states: Theorem 6.20 For any real-valued nonprincipal character $\chi \bmod k$, let $$ A(n) = \sum_{d\mid n} \chi(d) \hspace{...
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0answers
65 views

Proof of Vaughan's identity, based on comparing the Dirichlet coefficients

I am trying to figure out how to prove $- \frac{\zeta'}{\zeta}(s) = F(s) -G(s)\zeta'(s) -F(s)G(s)\zeta(s) -(\zeta(s)G(s) -1) \Big{(} - \frac{\zeta'}{\zeta}(s) - F(s) \Big{)}$ to prove $\Lambda(n) = ...
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1answer
33 views

$L$-function of elliptic curves expansion into Dirichlet series

Let $E/\mathbb{Q}$ be an elliptic curve. The $L$-function of $E$ is defined to be the Euler product $$ L_E(s) = \prod_{\text{ bad }p} (1 - a_p p^{-s})^{-1} \prod_{\text{ good }p} (1 - a_p p^{-s} + p^{...
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1answer
82 views

Dirichlet L functions [closed]

I would like to know more special Dirichlet L functions (like Zeta function for instace). Despite Zeta, Beta, Eta, Lambda and Hurwitz zeta are there more special Dirichlet L functions? I went to the ...
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32 views

Solve Laplace equation for annular region

I have to solve the following boundary value problem by using a series expansion. $\Delta u(x,y) = 0 \ \ \ in \ \ \ \Omega = \left\{(x, y) \in R^2 :\ 0 \lt R_1^2 \lt x^2 + y^2 \lt R_2^2 \lt \infty\...
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49 views

Where does $s-1$ come from in this $\zeta(s)$ equation?

I have been working my way through this Arxiv paper concerning the analytic continuation of the zeta function. I don't understand the first equality in equation (19), page 6. In equation (11), the ...
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1answer
71 views

Does the Dirichlet series for $\frac{\zeta(s+1)}{\zeta(s)}$ converge for $s>0$ as well as $\Re(s)>\sigma_c>0$?

This question pertains to the Fundamental Theorem of Dirichlet series which is stated on Wikipedia as follows (where $s=\sigma+i\,t$): There are now three possibilities regarding the convergence of ...
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3answers
67 views

$\int_{0}^{\pi} D_{n}(y)dy=\frac{1}{2}$ Dirichlet

I need to calculate that $\int_{0}^{\pi} D_{n}(y)dy=\frac{1}{2}$ with $D_{n}(y)= \frac{1}{2\pi}\frac{\sin((n+\frac{1}{2})y)}{\sin(\frac{y}{2})}$ from Dirichlet. Now I tried to do this with the known ...
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42 views

Analytical continuation for polylogarithm

It is known that the series presentation $$L_s(z)=\sum_{n=1}^{\infty}\frac{z^n}{n^s}$$ for the polylogarithm is valid only in the open disk $|z|<1$. Outside this region, the polylogarithm is ...
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0answers
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How to prove that width of general dirichlet series is given by this

My instructor gave me this question for the assignment and I am not able to solve this. Prove that $ 0\leq \sigma_{a} -\sigma_{c} \leq L$, where $\displaystyle L=\limsup_{ n\to \infty} \frac{\log n} ...
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1answer
24 views

Not able to find abcissa of absolute convergence of this general dirichlet series

I am reading about general dirichlet series from my class notes and I also referred to wikipedia here: https://en.m.wikipedia.org/wiki/General_Dirichlet_series In the 2nd section : Abcissa of ...
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1answer
31 views

Dirichlet series $\sum a_n n^{-x}$ where $(a_1+\dots+a_n)/n \rightarrow A$

I am working on some Dirichlet series and I am familiar with the proofs of convergence of the functions $\zeta$ and $\eta$. And I was wondering if $g(x)=\sum a_n n^{-x}$ was normally convergent on $[a,...
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3answers
201 views

Dirichlet series for square root of Riemann Zeta function

Can we obtain Dirichlet series for the function $\sqrt{\zeta(s)}$? Is it possible via Euler product for $\zeta(s)$?
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1answer
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Property on average orders of multiplicative functions

I read the statement that, for a multiplicative function $h(n)$ we have $$\sum_{n<x} h(n)n^a \sim R \frac{x^{a+1}}{a+1}$$ where $$R := \prod_p \left(1 - \frac 1p \right) \left( 1 + \frac{h(p)}{p} + ...
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1answer
63 views

How to determine the residue of an (arithmetic) Dirichlet series

Consider a multiplicative function $f(n)$ that we write in the form $f(n) = h(n)n^a$ for a certain $a>0$ and $h(n)$ a multiplicative function such that $h(n) \asymp 1$ (more precisely, we can take $...
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2answers
56 views

Want to show $\sum_{n>N}n^{-1/2} \ll N^{-1/2}$

I feel like I can use the result $$\sum_{n=x}^N \frac{a_n}{n^s} = A(N)N^{-s} + s \int_x^N A(t)t^{-s-1}dt$$ where $s=1/2$ to verify the $\ll$ approximation. If I pick $A(n)=\sum_{x\leq t\leq n}\chi(t)$,...
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0answers
50 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{...
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1answer
34 views

Analytic continuation of a Dirichlet series

Suppose we have a Dirichlet series $$ D(s) = \sum_{n=1}^\infty \frac{a(n)}{n^s} $$ which we know is absolutely convergent for $Re(s)>1$. Suppose that we prove that $\lim_{s\to 1^+}D(s) < \infty$....
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2answers
66 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 ...
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1answer
69 views

Proof: If convergence of Dirichlet series in one point, then uniform convergent in a sector

I'm currently reading the proof the theorem: if a Dirichlet serie converges at some point, $s_0$, then the serie is uniformly convergent in a sector around that point. (Montgomery and Vaughan: ...
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26 views

Regularity of Epstein zeta Z|h,0|(A; s) in h

My question considers the regularity of the Epstein zeta function in one of it modules. Let $h\in \mathbb R^d$ and $A\in \mathbb R^{d\times d}$ positive definite (we can also assume $A=I$ for ...
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0answers
18 views

Generalization of convolution operation? [duplicate]

In integral transforms, convolution is defined by $$ (f*g)(t)=\int_{-\infty}^\infty f(\tau)g(t-\tau)\mathrm d\tau $$ satisfying the commutative, associative property, and $$ \mathcal F\{f*g\}=\mathcal ...
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50 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: $...
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20 views

Generalized Dirichlet Series

I am looking into Dirichlet series book by Mandelbrojt but unfortunately I don't find results regarding boundary behavior of the type of series $\sum a_ne^{-\lambda_ns}$ in which the density ($\limsup ...
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0answers
23 views

Property of Dirichlet series which have $\sigma_{a}=1$

If the Dirichlet series $$ \sum_{n=1}^{\infty}\frac{f(n)}{n^s} $$ converges absolutely for $\Re(s)>1$, does it follow that the Dirichlet series $$ \sum_{n=1}^{\infty}\frac{f(n^2)}{n^s} $$ also ...
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25 views

Function $\epsilon(s, \chi) $ related to Dirichlet L -function

I am trying some exercises in number theory from Tom M Apostol as my instructor doesn't gives any assignment. I am struck on this particular problem. ( Problem 12.9 on page 274). Can you please tell ...
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136 views

Binomial transform of Dirichlet series

First off, i appologise for the long question, but it seems this is the only way i can convey my thoughts. Referring to this unanswered question on MO, i have thought for some time about it, and came ...
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1answer
46 views

An $L-$function and a $J-$function. Related?

Consider a Dirichlet series for a non real character of modulus $q$ $$ L(s,\chi)=\sum_{n=1}^\infty \frac{\chi(n)}{n^s} $$ and $s\in\Bbb C$ with real part greater than one. Consider a $J$-series $$ J(s,...
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1answer
48 views

Why does this equality Dirichlet series hold?

Following on from my question here, I have hit a second roadblock. I am working (very slowly!) through a paper here that demonstrates Riemann's analytic continuation of the zeta function $\zeta(s)=\...
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1answer
55 views

Why this group is cyclic?

Let $m$ be a positive integer, let $\chi$ be the Dirichlet character on $\mathbb{Z}/m\mathbb{Z}$ which means $\chi$ is a group homomorphism from $(\mathbb{Z}/m\mathbb{Z})^{*}$ to $\mathbb{C}^{*}$. We ...
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1answer
56 views

How to show the abscissa of convergence for a Dirichlet series given by a mod 3 function

The first thing I need to do is to write the function $$f(n) = \begin{cases} 1, & \mbox{if } n \ne 0 \mbox{ mod 3} \\ -2, & \mbox{if } n\equiv 0 \mbox{ mod 3} \end{cases}$$ in the form of a ...
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30 views

The formula some simple zeros of the Dirichlet eta function

Let $s$ be a complex variable with $\Re(s)>0$. The Dirichlet eta function $\eta(s)$ is defined by $$\eta(s)=(1-2^{1-s})\zeta(s)$$ where $\zeta(s)$, of course, is the Riemann zeta function. We know ...
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4answers
117 views

Errors are decreasing in series $\sum_{n=1}^\infty(-1)^n/n^4$?

Let $v=\sum_{n=1}^\infty(-1)^n/n^4$ ($v$ for "value"), let $S=(\sum_{n=1}^m(-1)^n/n^4)_{m\in\mathbb Z_{\ge1}}$ be the partial sums, and let $e=(|S_n-v|)_{n\in\mathbb Z_{\ge1}}$ be the errors....
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0answers
43 views

Understanding the functional equation $\left( \frac{2}{\pi} \right)^s \sin \left( \frac{\pi s}{2} \right) \Gamma(s)L(s)= L(1-s).$

In the functional equation $L(s)$ is the Dirichlet Beta function which is defined as $L(s)= \sum_{n=0}^{\infty}\frac{\chi(n)}{n^s}$ where $\chi$ is a Dirichlet character of period 4. Now I know that ...
2
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1answer
100 views

Proving the “identity” $\frac{\zeta^{2}(s)}{\zeta(2 s) J(s)}=\frac{\zeta^{2}(-s)}{\zeta(-2 s) J(-s)}$

Consider $J(s)$ a Dirichlet series defined by its Euler product as follows \begin{align*} J(s)=\prod_{p \in \mathbb{P}}\left(1+\sum_{k=1}^{\infty} \frac{2}{p^{k^{2} s}}\right) \end{align*} After some ...
6
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1answer
57 views

Abelian group zeta function

Let $s \in \mathbb{C}$. What's known about $$\zeta_{\mathrm{ab}}(s) := \sum_G \frac{1}{o(G)^s} \tag{1}$$ where the sum is over all finite abelian groups $G$ up to isomorphism? By the primary ...
5
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1answer
104 views

Asymptotic Expression for $ f(z) = z+ z^\frac{1}{2}+ z^\frac{1}{3}+ z^\frac{1}{4} +\dots + z^\frac{1}{N}$ with complex $z$?

Question (corrected) I managed to prove: $$ f(z) \sim \left\{ \begin{array}{ll} - \ln |z| \int_0^{\frac{-N}{\ln|z|}} e^{-\frac{1}{|y|}} dy & |z|<< 1 \\ ? & |z| \approx 1 \\ ...
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1answer
34 views

What is the limit of this $a_r = e^{i \theta/r}$ Dirichlet Series?

Consider the following Dirichlet Series: $$ D(s) = e^{i \theta} + \frac{e^{i \theta/2}}{2^s} + \frac{e^{i \theta / 3}}{3^s} + \dots$$ Is there a nice limit for the below? $$ \lim_{s \to 1} \frac{D(s)}{...

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