All Questions
Tagged with cv.complex-variables dirichlet-series
36 questions
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 $.
- 161
15
votes
1
answer
2k
views
Dirichlet series expansion of an analytic function
Let $F(s)=\sum_{n\geq 1}\frac{a_n}{n^s}$ be a Dirichlet series with (finite) abscissa of absolute convergence $\sigma_a$. It can be shown that $\forall \sigma >\sigma_a:$
$$\lim_{T\to\infty}\frac{1}...
- 7,127
14
votes
0
answers
1k
views
Analytic continuation of the Dirichlet generating series of the multiplicative partition function
Apologies for the lengthy question, but it seems it's the only way i can convey my thoughts. Consider the Dirichlet series:
$$\kappa(s)=\prod_{m=2}^{\infty}\frac{1}{1-m^{-s}}=\sum_{n=1}^{\infty}\frac{\...
- 181
11
votes
0
answers
530
views
Cesaro summation of a particular Dirichlet series associated with $\zeta(s)$
If you've investigated the error in Perron's formula in general, you've probably noticed that Cesaro summation $$\lim_{x\rightarrow\infty}\sum_{n\leq x} \left(1-\frac{\log n}{\log x}\right)\frac{a_n}{...
- 2,480
8
votes
2
answers
1k
views
Is the maximum domain to which a Dirichlet series can be continued always a halfplane?
Let $f(s)=\sum_n a_n n^{-s}$ be a Dirichlet series whose coefficients satisfy $\lvert a_n\rvert\leq n^{C}$. Then $f(s)$ converges absolutely in some halfplanes, and is conditionally convergent in (...
- 7,836
8
votes
1
answer
401
views
Is $\frac{1}{L(1+it)}$ unbounded?
Let $\chi$ be a Dirichlet character and $L(s, \chi)$ be the corresponding L-function. Is $$\frac{1}{L(1+it, \chi)}$$ unbounded for $t \in \mathbb{R}$? I'm aware that this is true if $L=\zeta$, but I'm ...
8
votes
0
answers
104
views
What is known about the following series?
For $k\in{\mathbb Z}^2$ write $|k|=\sqrt{k_1^2+k_2^2}$ for the euclidean norm. Then let $g(k)=gcd(k_1,k_2)$.
For $s\in\mathbb C$ let
$$
D(s)=\sum_{\substack{k\in{\mathbb Z}^2}\\ k\ne 0}\frac{|k|}{g(k)}...
user1688
6
votes
1
answer
247
views
Convergence and meromorphic continuation of a Dirichlet series under RH
Consider a Dirichlet series $F(s) = \sum_{n\geq1} \frac{a_{n}}{n^{s}}$ such that $a_{n} = O(1)$. Suppose the Dirichlet series
$$\zeta(s)F(s)=\sum_{n\geq1} \frac{(a\star1)(n)}{n^{s}}$$
converges ...
- 611
5
votes
1
answer
176
views
Frequency of large values of the Mertens function
It is known that with $M(x) = \sum_{n\le x}\mu(n)$, there are infinitely many $x$ s.t. $|M(x)|\ge x^{\frac{1}{2} - \varepsilon}$ (see Chapter 15 of Montgomery-Vaughan, for example). Is there any way ...
- 1,890
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)}...
- 433
4
votes
0
answers
134
views
Converse theorem for zeta universality
Voronin's Universality Theorem for $\zeta(s)$ is that the zeta function can uniformly approximate any non-vanishing holomorphic function to any degree of accuracy in the right-half of the critical ...
- 141
4
votes
0
answers
275
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)=\...
- 3,209
4
votes
0
answers
216
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 \}$, ...
- 576
3
votes
2
answers
451
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 ...
- 1,284
3
votes
2
answers
625
views
Continuation up to zero of a Dirichlet series with bounded coefficients
Let $a_n$ be a bounded sequence of positive real numbers. Is it the case that the Dirichlet series $\sum \frac{a_n}{n^s}$ can be meromorphically continued up to the right of zero, with at the most a ...
- 7,442
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 ...
- 203
2
votes
1
answer
178
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 ...
- 675
2
votes
1
answer
170
views
Is this a valid method of extending convergence of the Maclaurin series for $\frac{x}{x+1}$ from $|x|<1$ to $\Re(x)>-1$?
I originally asked this question on Math StackExchange a few months ago and no answers or even comments have yet been posted, so I'm asking this question again here on Math OverFlow.
This Math ...
- 1,126
2
votes
1
answer
275
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$$
And let $\left\{a_{n}\right\}...
- 181
2
votes
1
answer
186
views
Existence of analytic continuation of Dirichlet series corresponding to the indicator sequence of a complement of a special multiplicative set
Let $K/ \mathbb Q $ be a finite Galois extension and let $X$ be a proper non-empty subset of the Galois group $G=Gal(K/ \mathbb Q)$ that is closed under conjugation. Consider a set of integer primes $...
- 739
2
votes
0
answers
119
views
An analytic function, asymptotically expandable in a Dirichlet series, is the sum of this series
Let there be a function $F(s)$ that is analytic in some half-plane $\sigma>\sigma_0$ (where $s=\sigma + it $). Let the function $F(s)$ have an asymptotic expansion of the form $F(s)\sim\sum\limits_{...
1
vote
2
answers
336
views
Abscissa of convergence for a very specific Dirichlet series / Euler product
I am interested in the convergence of the following Euler product:
$$
\prod_p \frac{1}{1-\chi(p)\cdot p^{-s}}.
$$
The product is over all primes (in increasing order), with $\chi(p)=+1$ if $p \bmod 4 =...
- 3,098
1
vote
2
answers
310
views
Dirichlet Series that fail to be L-functions
For $\sigma \in \mathbb{R}$, let each $\mathbb{C}_\sigma = \{s \in \mathbb{C} : \Re(s) > \sigma\}$. For a sequence $a_n \in \mathbb{C}$, consider the Dirichlet series $D(s) = \sum_{n\ge 0} a_n n^{-...
- 1,104
1
vote
0
answers
381
views
Analytic continuation of Euler product $\phi(s)=\prod_p(1+p^{-s})^{-1}$
I am actually interested in the analytic continuation of $\phi_w(s)=\prod_p(1+w\cdot p^{-s})^{-1}$. Here $w$ is rational, or the imaginary unit multiplied by a rational. Consider for now that $w=1$. ...
- 3,098
1
vote
0
answers
102
views
Characterization of turning points for the Ramanujan's zeta function in the spirit of a definition by Arias de Reyna and van de Lune
In [1] the authors provided a definition and characterization of turning points for the Riemann's zeta function. In this post I denote the Ramanujan's zeta function as $$\varphi(s)=\sum_{n=1}^\infty\...
1
vote
0
answers
102
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 $\sigma > \sigma_{0}$,...
1
vote
0
answers
61
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 ...
- 1,805
1
vote
0
answers
98
views
Is there a general connection between value distribution and zero distribution for functions representable by Dirichlet series?
Some time ago I read part of a book in which the author made some conjectures outlining what kind of zero distribution is expected for functions representable by Dirichlet series with completely ...
- 2,480
0
votes
2
answers
392
views
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 $...
- 93
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 ...
0
votes
1
answer
121
views
Dirichlet series without order term
is there a name in use for Dirichlet series without the order term, analogously to Laurent or Puiseux polynomials? Is there work known about such expressions?
$D(s) = \sum_{0<n<N}a_n/n^s$
The ...
- 155
0
votes
0
answers
151
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 ...
- 3,098
-1
votes
1
answer
109
views
Analyzing a Dirichlet series with log-oscillating terms via Fourier methods
I am investigating the series $S(z)$ defined as follows:
$$
S(z) = \sum_{n=1}^{\infty} n^{-a}\cos(b\ln(n)),
$$
where $z = a + bi \in \mathbb{C}$, with $0 < a < 1$, and $b \in \mathbb{R}$.
I want ...
- 375
-2
votes
1
answer
139
views
Convergence of scrambled product for Dirichlet-$L$ function with modulo 4 character
A Dirichlet-$L$ function is typically defined by its series, and its Euler product is a consequence of the definition. Here my approach is the other way around.
I define the function
$$
L_4^*(s) = \...
- 3,098
-2
votes
1
answer
80
views
when does $h$ exist?
Let $\zeta(s)$ denote the Reimann zeta function in the critical strip. It is easy to see that $$ \zeta(s) = 0 \Longleftrightarrow \Re(\zeta(s))+\Im(\zeta(s)) = 0 ~~~~ \text{and} ~~~~~~ \Re(\zeta(s)) \...
-2
votes
0
answers
52
views
Is the real and imaginary part of the Dirichlet eta function closest to its partial sums when trigonometric function changes signs?
To grasp the question we are concerned with three Theorems 1,2, and 3 in bold font below. First let us consider the Dirichlet eta function $\eta: \mathbb{C}\rightarrow \mathbb{R}$
$$
\eta(s) = \sum_{n=...