Questions tagged [dirichlet-series]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
0
votes
0answers
65 views

Abscissa of convergence of transformed Dirichlet series

Let $$F(s)=\sum_{k=1}^\infty \frac{f(k)}{k^s} \mbox{ and }F^*(s)=\sum_{k=2}^\infty \frac{f(k)g(k)}{k^s},$$ where the infinite sum $\sum f(k)$ diverges, $f(k)$ and $g(k)$ are real numbers, $s$ a ...
7
votes
0answers
86 views

Are there natural Dirichlet series whose completions have poles in the region of absolute convergence?

The Selberg class of $L$-functions are Dirichlet series $$ L(s, f) = \sum_{n \geq 1} \frac{a(n)}{n^s}, $$ satisfying certain properties that can be abbreviated as analyticity, a Ramanujan conjecture, ...
1
vote
0answers
168 views

Analytic continuation of Euler product $\phi(s)=\prod_p(1+p^{-s})^{-1}$

I am actually interested in the analytic continuation of $\phi_w(s)=\prod_p(1+w\cdot p^{-s})^{-1}$. Here $w$ is rational, or the imaginary unit multiplied by a rational. Consider for now that $w=1$. ...
6
votes
4answers
300 views

Reference for behavior of Artin $L$-functions at $\Re(s) = 1$

Would anyone know a reference that proves the basic facts about Artin $L$-functions at $\Re(s) = 1$? Namely, the non-vanishing and holomorphicity for non-trivial characters. I assume this was done in ...
2
votes
0answers
66 views

An analytic function, asymptotically expandable in a Dirichlet series, is the sum of this series

Let there be a function $F(s)$ that is analytic in some half-plane $\sigma>\sigma_0$ (where $s=\sigma + it $). Let the function $F(s)$ have an asymptotic expansion of the form $F(s)\sim\sum\limits_{...
1
vote
1answer
147 views

Continuing an analytic continuation of the Dirichlet $\eta$-function?

The Dirichlet $\eta$-function is defined as: $$\eta(s) = \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n^s} \qquad \Re(s) > 0$$ and has the full analytical continuation: $$\eta(s) = \sum_{n=1}^N \frac{(-1)^{...
2
votes
1answer
71 views

Are there theorems dealing with the “amount of oscillatory divergence” of series?

Are there a set of theorems dealing with "amount of divergence" series? Let me explain by example. The Dirchlet $\eta$ series $\sum_n (-1)^{n-1} n^{-x}$ converges when $x > 0$. We may say ...
3
votes
0answers
59 views

Continuation of $\sum \sigma_\nu(n) a(n) n^{-s}$ for $a(\cdot)$ coming from a half-integral weight form

In some of my work, I've run into a wall trying to understand whether a Dirichlet series has a meromorphic continuation or not. Let $f(z) = \sum_{n \geq 1} a(n) e(nz)$ be a half-integral weight ...
0
votes
0answers
104 views

Some properties of special Dirichlet series, connection to Riemann Hypothesis

The series in question is $$\phi(\sigma,\mu,t; \alpha,\beta,\gamma) = \sum_{n=2}^\infty (-1)^{n+1} \cdot \frac{W\big(\gamma +\alpha t+\beta t\lambda(n)\big)}{n^\sigma (\log n)^\mu}$$ The wave $W$ is a ...
2
votes
0answers
126 views

On a generalization of the Möbius function from number theory

Let $\omega$ be a positive real number, and define: $$\mathbf{1}_{\omega}\left(n\right)\overset{\textrm{def}}{=}\left(-1\right)^{n}\binom{-\omega}{n}=\binom{\omega-1+n}{n}$$ for all positive integers $...
12
votes
1answer
294 views

Convergence of the series involving Mobius functions $\sum_{k,d} \mu(d) x_{kd}$

(I originally asked this question here, but the problem appears much more difficult than I think after a moment of thought, so I think it might be more suitable to post it here. Please tell me if this ...
2
votes
0answers
110 views

Question regarding proof of Mertens' estimates in Montgomery-Vaughan's “Multiplicative number theory”

I have been trying to read Theorem 2.7 of Montgomery-Vaughan's "Multiplicative number theory" volume 1, and there is an issue I have run into: in the proof of subpart (e), when they write $$\...
2
votes
0answers
113 views

Multiplicity of zeros of partial sums of the Dirichlet Eta function

I am studying ways to approach the problem of the multiplicity of zeros of the partial sums of the Dirichlet Eta functions: $$ \sum_{n=1}^{K}\frac{(-1)^{n-1}}{n^{s_o}} = 0 $$ more in particular, ...
1
vote
2answers
143 views

What is the approximation of $\log(|\zeta'(\frac{1}{2}+it)|)$ in Dirichlet polynomial if it is exists?

I have done some search many times on web to find any approximation of $\log|(\zeta'(s))|$ in Dirichlet polynomial but I didn't got it, Probably that $\log(|\zeta'(s)|$ dosn't have a Dirichlet ...
2
votes
2answers
259 views

Convergence of Euler product and Dirichlet series in the same half-plane?

I'm crossposting this from math.stackexchange because I think it might be inappropriately research-level for the community over there. Suppose we have an Euler product over the primes $$F(s) = \prod_{...
2
votes
1answer
203 views

Binomial transform of Dirichlet series

Let $\Theta(s)$ be a Dirichlet series , and let $\beta$ be its abscissa of convergence: $$\Theta(s)=\sum_{n=1}^{\infty}\frac{\theta(n)}{n^{s}}\;\;\;\;\;\;\Re(s)>\beta$$ And let $\left\{a_{n}\right\}...
7
votes
2answers
328 views

Conditions under which $\lim_{s\to1^+}\sum_{n=1}^{\infty}\frac{a_n}{n^s}=\sum_{n=1}^{\infty}\frac{a_n}{n}$

I was working with some Dirichlet series and I realized that I have never seen any general conditions under which \begin{equation} \sum_{n=1}^{\infty}\frac{a_n}{n}=\lim_{s\to1^+}\sum_{n=1}^{\infty}\...
1
vote
1answer
123 views

Moments of Dirichlet $L$-functions on the critical line

I'm looking for a reference for some questions related to the moments of Dirichlet characters on the critical line, $$ M_k(T;\chi) = \int_T^{2T} |L(1/2+it,\chi)|^{2k}\,dt, $$ where $\chi$ is a ...
1
vote
0answers
121 views

Euler product over subsets of primes

It is well known that $$\prod_p\,(1-p^{-1})=\frac 1 {\zeta(1)}=0$$ Given an arbitrary prime $\,q\,$ is it true that $$\prod_{q\,|\,p+1}\,(1-p^{-1})=0\;\;\;?$$ Thanks.
2
votes
1answer
107 views

Existence of analytic continuation of Dirichlet series corresponding to the indicator sequence of a complement of a special multiplicative set

Let $K/ \mathbb Q $ be a finite Galois extension and let $X$ be a proper non-empty subset of the Galois group $G=Gal(K/ \mathbb Q)$ that is closed under conjugation. Consider a set of integer primes $...
0
votes
1answer
199 views

Properties of Dirichlet series

I have a question about convergence and properties of Dirichlet series. it seems a bit interesting and different about the convergences of Dirichlet Series to me. With $c\in [0,1]$, $$f(n) = \pm 1,...
0
votes
2answers
143 views

What is the dirichlet series of $f(n)=\sum_{d | n}(\log d) / d$ function? [closed]

My opinion is ; We may use id(d)=d arithmetic function and log*id dirichlet convolution in the question. i thought that ; when we multiply and divide n with $(\log d) / d$ we obtain $F(S)=\sum_{n=...
15
votes
2answers
591 views

If $\zeta(s)=0$ with $\Re(s)=\frac{1}{2}$, is then $|\hat{\zeta}(s,3)|^2=\frac{1}{2}$?

Helmut Hasse has proved that for $s \in \mathbb{C}-\{1\}$ the Riemann zeta function can be written as: $$\zeta(s)=\frac{1}{1-2^{1-s}}\sum_{n=0}^\infty\frac{1}{2^{n+1}}\sum_{k=0}^n(-1)^k\ {n \choose k}...
12
votes
1answer
425 views

Error term when truncating series for $1/\zeta(s)$

Let $s=\sigma+it$, $0\leq \sigma\leq 1$, $|t|\geq 1$, say. Using Euler-Maclaurin, one can easily show that, for $x\geq |t|$, $$\zeta(s) = \sum_{n\leq x} \frac{1}{n^s} + \frac{x^{1-s}}{s-1} + O\left(\...
2
votes
0answers
161 views

Dirichlet series of powers of the prime omega function

Let $\omega(n)$ denote the number of distinct prime factors of a positive integer $n$. I was wondering what is known about the dirichlet series $$\sum_{n=1}^{\infty}\frac{\omega(n)^k}{n^s},$$ in ...
3
votes
1answer
203 views

Approximation of $\sum_{\rho}\frac{1}{|\rho|^2}$, over the non-trivial zeros of the Ramanujan's zeta function

I would like to know if it in the literature an approximation for $$\sum_{\rho}\frac{1}{|\rho|^2}\tag{1}$$ where the sum is over all of the non-trivial zeros of the Ramanujan's zeta function (also ...
1
vote
0answers
94 views

Characterization of turning points for the Ramanujan's zeta function in the spirit of a definition by Arias de Reyna and van de Lune

In [1] the authors provided a definition and characterization of turning points for the Riemann's zeta function. In this post I denote the Ramanujan's zeta function as $$\varphi(s)=\sum_{n=1}^\infty\...
0
votes
1answer
142 views

Zeros of partial sums of the Ramanujan's zeta function

In this post we consider the Ramanujan tau function $\tau(n)$, see the Wikipedia Ramanujan tau function, and we consider partial sums of its corresponding Dirichlet series (see for example the article ...
0
votes
0answers
159 views

List of properties of Twin primes Dirichlet series

In a paper R. Arenstorf - There are infinitely many prime twins he stated the following Dirichlet series : $$ T(s) = \sum_{n=1}^\infty \frac{\Lambda(n)\Lambda(n+2)}{n^s} $$ Question : What are ...
1
vote
0answers
65 views

convergence abcissa for Mellin transforms

Where can I find the theory of abcissa of convergence for integrals necessary to understand ChenClass answer to On the integral $I_s =\int_{1}^{\infty} (\pi(x)-Li(x))x^{-s-1} dx$ ? Note that the ...
0
votes
0answers
59 views

Examples of geometrical interpretations for sequences of particular values of Dirichlet series

The remark [1] (in Spanish) shows a geometric interpretation (linking two sequences) of particular values of a given Dirichlet series, that are $\zeta(k)$ and $\zeta(2k)$. I wondered about if it is ...
1
vote
0answers
115 views

Asymptotic of $\sum_{1\leq n\leq x}a_n$ where $\exp(\sum_{n=1}^\infty\alpha\operatorname{rad}(n)n^{-s})=\sum_{n=1}^\infty\frac{a_n}{n^s}$

Yesterday I tried to study the article [1] in wich were showed incredible expressions related to Dirichlet series. In the same way I wondered about next question. We denote for integers $m>1$ the ...
3
votes
2answers
362 views

On the relation between the asymptotics of a Dirichlet series' coefficients and the series' analytic continuability

There is a wonderful series of articles by Flajolet et. al. about Mellin Transforms and the asymptotic analysis of generating functions. In particular, on page 45 of the article Mellin Transforms and ...
1
vote
1answer
190 views

Bounding Coefficients of Dirichlet Series

Consider the exponentiated Riemann-Zeta function $\zeta(s)^p$. If it is represented as $$\zeta(s)^p = \sum_{n=1}^\infty\frac{a_n}{n^s}$$ Is there any upper bound we can put on $|a_n|$ in terms of ...
4
votes
1answer
351 views

The sign of an interesting sum involving a Dirichlet character

Let $\chi_{q}$ be a primitive Dirichlet character with modulus $q$ (see definition at wikipedia ). For example for $q=5$ we have \begin{equation} \begin{aligned} \chi_{5,1}&=(1, 1, 1, 1, 0),\\ ...
4
votes
1answer
200 views

The function $\sum_{n=0}^\infty\frac{(-1)^n\mu(2n+1)}{(2n+1)^s}$: reference request or particular values at integers and abscissa of convergence

We denote for integers $m\geq 1$ the Möbius function as $\mu(m)$. With the help of a CAS, Wolfram Alpha online calculator, I was calculating certain values of $$\sum_{n=0}^\infty\frac{(-1)^n\mu(2n+1)}{...
12
votes
3answers
2k views

Are L-functions uniquely determined by their values at negative integers?

Are L-functions uniquely determined by their values at negative integers? In another words, is there a sequence of integers $a_1, a_2, a_3, \cdots$ such that the corresponding L-function $$L_{\{a_n\}...
16
votes
1answer
632 views

Dirichlet series with a single zero

I need to find a Dirichlet series f that has the following property. f is zero in only one point s such that Re(s) > $\sigma_c $.
5
votes
2answers
452 views

Extracting Dirichlet series coefficients

Cauchy's integral formula is a powerful method to extract the $n$'th power series coefficient of an analytic function by evaluating a single complex integral. Is there any such analytic method to ...
3
votes
0answers
87 views

Supremum of certain modified zeta functions at 1

Let $D$ be an integer number and let $\chi$ be the Dirichlet character defined by $$\chi(m) = 0 \text{ if $m$ even, } \chi(m) = (D/m) \text{ if $m$ odd,}$$ where $(D/m)$ denotes the Jacobi symbol. ...
4
votes
2answers
240 views

Real non trivial zeros of Dirichlet L-functions

When dealing with the prime number theorem in arithmetic progressions, one cannot exclude the possible presence of a real zero close to $1$ for at most one real character mod $q$. On the other hand, ...
10
votes
0answers
365 views

Additive and multiplicative convolution deeply related in modular forms

From the fact spaces of modular forms are finite dimensional, from the decomposition in Hecke eigenforms, and from the duplication formula for $\Gamma(s)$ there are a lot of identities mixing additive ...
7
votes
4answers
2k views

Modern Algebraic Geometry and Analytic Number Theory

I am currently discovering the algebraic geometry of Grothendieck. I have the impression that this theory, which leads to categories, schemas, topos etc. alone can encompass all modern mathematics (...
2
votes
1answer
100 views

meromorphic extension of dirichlet series

Suppose $\{a(n)\}_{n\ge 1}$ is a bounded complex sequence. Let $\phi(s)=\sum_{n\ge 1} \frac{a(n)}{n^s}$. Obviously, the Dirichlet series $\phi(s)$ is absolutely convergent for $\mathcal{R}(s)>1$. I ...
4
votes
1answer
233 views

Zeros of derivatives of Dirichlet Eta function

Let $$ \eta^{(d)}(z) = \sum_{n=1}^\infty \dfrac {(-1)^d(-1)^{n-1}\ln(n)^d} {n^z} $$ be the derivative of Dirichlet Eta function of order $d$. Does it exist any known or not known zero of $\eta^{(d)}...
4
votes
0answers
263 views

Computing Bohr Radii

The Bohr radius $R$ for $\mathcal{H}(\mathbb{D})$ is defined as the radius $$R = \sup\limits_{0<r<1} \Bigl\{ r\ \Big|\sum\limits_{k=0}^{\infty}|a_k|r^k \leq |f|_\mathbb{D}, \text{ for all }f(z)=\...
2
votes
1answer
355 views

Does $\sum_{n=1}^\infty \frac{\mu(n)}{n^s}$ converge for $\sigma > \frac{1}{2}$?

Looking at @Lucia's answer to this question it appears $\sum_{n=1}^\infty \frac{\mu(n)}{n^s}$ converges for $\sigma > \frac{1}{2}$. Can someone point me to a proof or provide proof for this? If I ...
5
votes
0answers
135 views

Dirichlet eta function and Stirling Permutations

The Stirling permutations of order $k$ is a permutation of the multiset $1, 1, 2, 2, ..., k, k$. The Dirichlet $\eta$-function is a function closely related to the Riemann $\zeta$-function. According ...
3
votes
0answers
55 views

Analytic continuation of a Dirichlet series with several complex variables

For $w_1,w_2,z_1,z_2\in\mathbb{C}$ with $\operatorname{Re}(w_1)>0$ and $\operatorname{Re}(w_2)>0$, define \begin{equation*} U(w_1,w_2;z_1,z_2):=\prod_{p}\left(1-\frac{e^{z_1}}{p^{1+w_1}}-\frac{e^...
1
vote
1answer
276 views

On a certain integral representation for Dirichlet L-functions

It is an ancient result of Jensen that $$(s-1)\zeta(s)=\frac{\pi}{2} \int_{-\infty}^{\infty} \frac{(1/2+it)^{1-s}}{\cosh^{2}\pi t} \mathrm{d}t$$ where $\zeta$ denotes the Riemann zeta function. Is ...