All Questions
Tagged with dirichlet-series analytic-number-theory
109 questions
15
votes
2
answers
728
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:
$$\zeta(s)=\frac{1}{1-2^{1-s}}\sum_{n=0}^\infty\frac{1}{2^{n+1}}\sum_{k=0}^n(-1)^k\ {n \choose k}...
- 1,126
0
votes
1
answer
239
views
Zeros of partial sums of the Ramanujan's zeta function
In this post we consider the Ramanujan tau function $\tau(n)$, see the Wikipedia Ramanujan tau function, and we consider partial sums of its corresponding Dirichlet series (see for example the article ...
- 791
12
votes
1
answer
663
views
Error term when truncating series for $1/\zeta(s)$
Let $s=\sigma+it$, $0\leq \sigma\leq 1$, $|t|\geq 1$, say. Using Euler-Maclaurin, one can easily show that, for $x\geq |t|$,
$$\zeta(s) = \sum_{n\leq x} \frac{1}{n^s} + \frac{x^{1-s}}{s-1} + O\left(\...
CommunityBot
- 1
0
votes
0
answers
199
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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 ...
- 790
1
vote
0
answers
92
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 ...
- 221
1
vote
1
answer
288
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 ...
- 105k
1
vote
0
answers
129
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 ...
4
votes
1
answer
244
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 $$\sum_{n=0}^\infty\frac{(-1)^n\mu(2n+1)}{...
- 105k
0
votes
0
answers
64
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Examples of geometrical interpretations for sequences of particular values of Dirichlet series
The remark [1] (in Spanish) shows a geometric interpretation (linking two sequences) of particular values of a given Dirichlet series, that are $\zeta(k)$ and $\zeta(2k)$. I wondered about if it is ...
3
votes
2
answers
451
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On the relation between the asymptotics of a Dirichlet series' coefficients and the series' analytic continuability
There is a wonderful series of articles by Flajolet et. al. about Mellin Transforms and the asymptotic analysis of generating functions. In particular, on page 45 of the article Mellin Transforms and ...
- 1,284
4
votes
1
answer
448
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The sign of an interesting sum involving a Dirichlet character
Let $\chi_{q}$ be a primitive Dirichlet character with modulus $q$ (see definition at wikipedia ).
For example for $q=5$ we have
\begin{equation}
\begin{aligned}
\chi_{5,1}&=(1, 1, 1, 1, 0),\\
...
- 105k
12
votes
3
answers
2k
views
Are L-functions uniquely determined by their values at negative integers?
Are L-functions uniquely determined by their values at negative integers? In another words, is there a sequence of integers $a_1, a_2, a_3, \cdots$ such that
the corresponding L-function
$$L_{\{a_n\}...
- 3,403
16
votes
1
answer
673
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 $.
- 43.7k
7
votes
2
answers
940
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 ...
- 91.8k
0
votes
2
answers
392
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What is the relationship between the abscissa of holomorphy and abscissa of convergence of a Dirichlet series
Given a Dirichlet series $$\phi(s)=\sum_{n\ge1}\frac{a_n}{n^s}$$
let $\sigma_{\text{conv}}\in\bar{\mathbb{R}}$ its abscissa of convergence, then we know that $\phi(s)$ is holomorphic on the half-plan $...
- 63.9k
4
votes
0
answers
216
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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
$\sigma_c(D) = \inf\{\sigma : D \text{ converges in } \mathrm{Re} s > \sigma \}$, ...
- 63.9k
3
votes
0
answers
97
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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. ...
- 356
4
votes
2
answers
566
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, ...
- 5,089
8
votes
4
answers
3k
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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
1
answer
178
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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
1
answer
387
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)}...
- 11.1k
2
votes
1
answer
454
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 ...
- 105k
3
votes
0
answers
63
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^...
- 131
1
vote
1
answer
419
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 ...
- 11.1k
3
votes
1
answer
164
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
\[
\sum_{n \geq 1}a(n)e^{-ny} \ll y^{-A}, \quad y \to 0^{+}.
\]...
- 41.1k
0
votes
1
answer
290
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 ...
- 148k
1
vote
1
answer
230
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 \& ...
CommunityBot
- 1
6
votes
1
answer
350
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 ...
- 36
2
votes
0
answers
81
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 ...
- 501
-3
votes
1
answer
250
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 ...
- 105k
2
votes
1
answer
150
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 ...
- 131
8
votes
0
answers
104
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
$$
D(s)=\sum_{\substack{k\in{\mathbb Z}^2}\\ k\ne 0}\frac{|k|}{g(k)}...
CommunityBot
- 1
3
votes
1
answer
560
views
The abscissa of convergence of the real part of a Dirichlet series
Let $L(s)=\sum_{n\ge1}\frac{a(n)}{n^s}$ be a Dirichlet series with a finite abscissa of convergence $\sigma_c.$ My question is the following :
On what condition the abscissa of convergence of $\sum_{...
CommunityBot
- 1
6
votes
1
answer
510
views
If a Dirichlet series converges Conditionally, how can I apply Euler product?
In 1737, Euler discovered that if $ f(n) $ is multiplicative and $ \sum f(n)/n^{s} $ converges absolutely for ${\rm Re}(s) > \sigma_a$ then we have
\begin{equation}
\sum_{n=1}^{\infty} \frac{f(n)}{...
- 79.9k
2
votes
0
answers
451
views
Analytic continuation of "composite" zeta function
Let us define the Dirichlet series $$\mathcal C(s):=\sum_{n\text{ composite}}\frac{1}{n^s},\quad P(s):=\sum_{p\text{ prime}}\frac{1}{p^s}.$$
They are absolutely convergent in the half-plane $\sigma>...
- 17.6k
1
vote
0
answers
325
views
Extension of Heath-Brown's fourth moment $\sum_ \chi \lvert L (s, \chi) \rvert^4$ to complex characters
Let $L(s, \chi)$ denote the Dirichlet $L$-function associated to the character $\chi$.
In his paper A Mean Value Estimate for Real Character Sums, Heath-Brown proves a mean value bound for $L(s,\chi)$...
CommunityBot
- 1
0
votes
1
answer
356
views
Meromorphic continuation of a Dirichlet series
I asked this question in SEM but I got no answer, so I'm trying my luck here.
Let the Dirichlet series $\phi(s)=\sum_{n\ge 1}\frac{a(n)}{n^s}$ be absolutely convergent for $\Re(s)>1$ and extend to ...
CommunityBot
- 1
2
votes
0
answers
425
views
Analytically continuing the limit of this series?
Main Question
I believe the following formula gives the right answer:
$$ \lim_{k \to \infty} \lim_{n h \to k} \left( \sum_{r=1}^n c_r f(hr) h\right) = \int_0^\infty f(x) \, dx \times \sum_{r=1}^\...
- 89
3
votes
1
answer
1k
views
Two Dirichlet's series related to the Divisor Summatory Function and to the Riemann's zeta-function
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 $...
- 105k
5
votes
1
answer
202
views
Sufficient conditions for $\sum_{n \ge 1} a_n e^{-(a_1+\cdots+a_n) s} \sim \frac{1}{s}$ as $s \to 0^+$
Let $(a_n)_{n \ge 1}$ be a sequence of non-negative real numbers such that $\sum_{n \ge 1} a_n = \infty$, and set $\lambda_n := a_1 + \cdots + a_n$ for each $n$. Then the (generalized Dirichlet) ...
- 109k
3
votes
0
answers
192
views
Determining coefficients of a Dirichlet series based on values on a vertical line
Let us suppose we have a Dirichlet series
$$ D(s) = \sum_{n \geq 1} \frac{a(n)}{n^s},$$
and that we know the values of $D(\tfrac{1}{2} + im)$ for $m \in \mathbb{Z}$. Can we recover the coefficients $a(...
- 1,570
5
votes
1
answer
1k
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Does the Euler product for $L(s,\chi_4)$ also converge in the right half of the critical strip?
This question expands on this one from MSE.
In the literature about Dirichlet $L$-series, I found that their Euler products:
$$L(s, \chi) =\prod_p \bigg(\frac {1}{1-\frac{\chi(p)}{p^s}} \bigg)$$
...
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- 1
1
vote
0
answers
113
views
Extracting information from $\sum_{n \leq X} a(n) (X-n)^d$
Allow me to give context to the question, which appears in the box at the bottom.
A very general hope from the theory of Dirichlet series is to try to extract information about the coefficients of a ...
- 1,570
4
votes
0
answers
318
views
Equivalence of Euler products of Dirichlet series and Meromorphic continuation
Suppose $(f_n(s))_n$ and $(g_n(s))_n$ are two sequences of Dirichlet series with positive coefficients such that $\exists \alpha\in\mathbb R$ such that for all $s\in\mathbb C$ with $\Re(s)>\alpha$ ...
- 993
5
votes
0
answers
326
views
Can we extend the twisted Poisson Summation formula with functions having a singularity in zero?
The following "twisted" Poisson Summation formula for $\chi$ primitive of conductor $q$ :
$$ \sum_{n\in\mathbb{Z}}\chi(n)f\left(\frac{nx}{\sqrt{q}}\right) =
\frac{A}{x}\sum_{n\in\mathbb{Z}}\bar\chi(n)...
- 1,199
12
votes
2
answers
382
views
Vanishing of certain periodic series: A question related to $L(1 , \chi) \neq 0$.
Fix $q$ to be a positive integer. Let $$f : \mathbb{N} \to \{-1 ,0, 1\}$$ be a $q$-periodic arithmetic function such that $$\sum_{n = 1}^q f(n) = 0.$$ If $f$ is not identically zero, is it true that $...
- 13.8k
3
votes
1
answer
963
views
Abscissa of absolute convergence of the product of two Dirichlet series
I first asked the following question on Mathematics StackExchange (a few weeks ago), since the content of MathOverflow is mostly above my academic level. I didn't want to bother people on this forum ...
- 7,024
9
votes
2
answers
559
views
Recovering $\sum_{n \leq x} a(n)$ from $\sum_{n \leq x} a(n)e^{-n/x}$
In the theory of automorphic forms and multiple Dirichlet series, we often take inverse Mellin transforms of Dirichlet series to come up with Tauberian theorems, like the Ikehara Tauberian method. In ...
- 5,037
4
votes
1
answer
441
views
A lower bound on the $L^2$ norm of a Dirichlet polynomial
The Question. Suppose $0 < \alpha < \beta$ are fixed, and $a_n$ is an arbitrary sequence of real numbers. Is it known how to bound from below
\begin{equation*}
\int_0^{T} \Big| \sum_{\alpha T &...
- 43.7k
2
votes
1
answer
1k
views
On the convergence of Dirichlet series over the Mobius Mu function
It is known that if $\sum_{k=1}^{\infty} \frac{\mu(k)}{k^s} = \frac{1}{\zeta(s)}$ for $\Re(s) > 1/2$ then RH holds. My question is:
Under RH why is it not $\sum_{k=1}^{\infty} \frac{\mu(k)}{k^s} = ...
CommunityBot
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