Questions tagged [dirichlet-series]

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92 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.
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1answer
74 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 $...
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1answer
147 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,...
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2answers
108 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=...
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2answers
549 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
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1answer
396 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(\...
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103 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 ...
1
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1answer
160 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 ...
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67 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\...
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1answer
126 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 ...
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0answers
145 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 ...
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59 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 ...
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82 views

What should $N(T)$ (cf. $N(T,\chi)$) mean?

Should $N(T)$ mean the number of zeros of $\zeta(s)$ with $0<\Re(s)<1$ and $0<\Im(s)\leq T$, $0\leq \Im(s)\leq T$ or $|\Im(s)|\leq T$? There does seem to be a tendency towards one of the ...
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0answers
49 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 ...
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110 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 ...
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2answers
343 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 ...
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1answer
173 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
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1answer
333 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
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1answer
193 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)}{...
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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\}...
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1answer
624 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 $.
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2answers
386 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 ...
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0answers
85 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. ...
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2answers
192 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, ...
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291 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 ...
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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 (...
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1answer
86 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 ...
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1answer
203 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)}...
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0answers
258 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)=\...
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1answer
321 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 ...
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0answers
130 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 ...
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0answers
51 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
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1answer
248 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 ...
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0answers
62 views

Some theoretical question on Euler product

It is very rare that the Euler product \[ \lim_{X \to \infty}\prod_{p \leq X}(1 + a(p)p^{-s}) \] conditionally converges for $\sigma > A$ with some $0 < A \leq 1$ when $|a(n)| = 1$. Suppose ...
3
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1answer
150 views

Recovering information for $\sum_{n \leq x}a(n)$ from $\sum_{n \geq 1}a(n)e^{-nx}$

I am wondering if I could deduce the bound for the partial sums \[ \sum_{n \leq x}a(n) \ll x^{A}, \quad x \to \infty \] from the relation \[ \sum_{n \geq 1}a(n)e^{-ny} \ll y^{-A}, \quad y \to 0^{+}. \]...
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0answers
79 views

Dirichlet series with an abscissa of absolute convergence $\sigma_{0}$, analytic in $\sigma > \sigma_{0} - \delta$

Suppose that a Dirichlet series $f(s)$ has the abscissa of absolute convergence $\sigma_{0}$ and is analytic in $\sigma > \sigma_{0} - \delta$ for some $\delta > 0$. For $\sigma > \sigma_{0}$,...
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1answer
134 views

Analytic continuation of Euler product $\prod_{p} (1 - e^{-2 \pi i p \alpha}p^{-s})^{-1}$

Is anything useful known about the function defined by \[ f(s, \alpha) = \prod_{p} (1 - e^{-2 \pi i p \alpha}p^{-s})^{-1} \quad ? \] Here, $\alpha$ is real. When $\alpha = 1$, this is certainly the ...
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1answer
167 views

Asymptotic for a number theoretic sequence and its Dirichlet series' convergence

I would like to know the asymptotic behaviour at large $n$ for $t\in\mathbb{R}$, $t\neq0$ of the following function: \begin{align*} A_n(t)&=\sum_{q=\frac{a}{b}\in \mathbb{Q}^+|\gcd(a,b)=1 \& ...
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0answers
44 views

Which complex maps with branch cuts have a representation by Dirichlet series?

Which complex maps with branch cuts have a representation by Dirichlet series? I am aware of the work of A.F. Leont'ev on general Dirichlet series, and the theorems of representation of analytic ...
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0answers
69 views

Question on a generalized Dirichlet series

Given the generalized Dirichlet series $$S(x) =\sum_{(n,m)\in \mathbb{Z}^2}e^{-x\sqrt{n^2+m^2}} $$ is there any way to solve the equation $$2S(2x)=S(x)$$ for $x\in\mathbb{R}$? I am only interested in ...
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1answer
194 views

Twin prime based Dirichlet series

Assuming there are infinitely many twin primes, one can consider a Dirichlet series $ \sum_{n>0}a_{n}{n^{-s}} $ and replace the sequence of positive integers with the sequence of twin primes. That ...
2
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1answer
121 views

Residue of the following variant of Dirichlet function [closed]

I am working with the Piltz divisor problem where the number of ways in which a number $n$ can be written as a product of $k$ is of the form $$D_{k}(x)=xP_{k}(\log x)+\Delta _{k}(x)$$ where $P_k$ is ...
6
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1answer
186 views

Counting smooth numbers in short intervals

I am reading a few papers about counting smooth numbers in the interval $[x, x+\sqrt{x}]$, including the work of Harman, and Matomaki. Both authors mentioned that the Dirichlet polynomial techniques ...
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1answer
162 views

Calculating a Dirichlet Character

How would I calculate a function such as $$\sum_{n \leq x}r(n) = L(1, \chi) \cdot x + O(x^{1-\eta}),$$ where $r(n) = \sum_{d|n} \chi(d)$? The part I'm having difficulty calculating is the L-function ...
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0answers
115 views

Analytic continuation of $\alpha(s)=\sum_{n=a+1}^\infty\frac{1}{(n^2-a^2)^s}$. Possibly related to Riemann Zeta function $\zeta(s)$?

I'm trying to find the analytic continuation for $\alpha(s)=\sum_{n=a+1}^\infty\frac{1}{(n^2-a^2)^s} ,$ with $a\in \mathbb{N^+}$ and $s<1$. I need most likely only the values for $s=\frac{1}{2}-m$...
4
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0answers
137 views

Gap Between Abscissae of Conditional Convergence and Holomorphicity for Dirichlet Series

For a Dirichlet series, $D = \sum_n a_n n^{-s}$ we may define the abscissae, in (non-strictly) increasing order $\sigma_c(D) = \inf\{\sigma : D \text{ converges in } \mathrm{Re} s > \sigma \}$, ...
7
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0answers
99 views

What is known about the following series?

For $k\in{\mathbb Z}^2$ write $|k|=\sqrt{k_1^2+k_2^2}$ for the euclidean norm. Then let $g(k)=gcd(k_1,k_2)$. For $s\in\mathbb C$ let $$ D(s)=\sum_{\substack{k\in{\mathbb Z}^2}\\ k\ne 0}\frac{|k|}{g(k)}...
2
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1answer
222 views

critical line inequality concerning the square of the modulus of a Dirichlet polynomial

I am currently studying the following inequality involving the square of the modulus of a specific Dirichlet polynomial: $$\left( \sum_{1}^{N}\frac{1}{n} \right)^2 \ \ - \ \left| \sum_{1}^{N}\frac{(...
14
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3answers
671 views

Tauberian theorem $\sum_{k=1}^{\infty}e^{-\lambda_{k}t}c_{k} \xrightarrow{t\to 0} \sum_{k=1}^{\infty}c_{k} $

I am trying to prove or disprove $$\sum_{k=1}^{\infty}e^{-\lambda_{k}t}c_{k} \xrightarrow{t\to 0} \sum_{k=1}^{\infty}c_{k} ,$$ where $\sum c_{k}<\infty, \sum c_{k}^{2}<\infty\text{ and }\frac{\...
2
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0answers
96 views

Bound for partial sums of $ L(1/2+it,\chi)$

Let $\chi$ be a primitive Dirichlet character of conductor $q>1$. One may use partial summation to prove an upper bound of the form (I hope I am right) $$ \sum_{n\leq X} \chi(n)n^{-1/2-it} \ll \...