All Questions
Tagged with dirichlet-series complex-analysis
94
questions
5
votes
0
answers
101
views
Zeta Lerch function. Proof of functional equation.
so I'm trying to prove the functional equation of Lerch Zeta, through the Hankel contour and Residue theorem, did the following.
In the article "Note sur la function" by Mr. Mathias Lerch, a ...
2
votes
1
answer
88
views
A question about Landau’s theorem for Dirichlet series and integrals
A well known theorem of Landau’s for Dirichlet series and integrals goes as follows (I copy the theorem almost exactly as it appears in Ingham’s Distribution of Prime Numbers, Theorem H in Chapter V, ...
- 405
0
votes
1
answer
67
views
Asymptotics for the number of $n\le x$ which can be written as the sum of two squares. Is Perron's formula applicable?
For all $n\ge 1$, let
$$
a_n = \begin{cases}
1\quad&\text{if $n$ can be written as the sum of two squares;}\\
0&\text{otherwise}
\end{cases}
$$
I am interested in $A(x):=\sum_{n\le x}a_n$.
...
- 8,421
2
votes
1
answer
273
views
Dirichlet series with infinitely many zeros
Can a Dirichlet series have infinitely many zeros and be nonzero?
To be precise, by a Dirichlet series I mean a function of the form $s\mapsto \sum_{n\geq 1}\frac{a_n}{n^s}$ where the domain is the ...
- 165
0
votes
0
answers
26
views
How to construct a Dirichlet series that cannot be analytically continued beyond its abscissa of absolute convergence?
If I want a power series $\sum_n a_n \, z^n$ that cannot be analytically continued anywhere beyond its disk of convergence $|z| < R$, then I can use a lacunary series, e.g., $\sum_n z^{2^n}$.
Are ...
- 1,755
1
vote
0
answers
61
views
Asymptotic order of the square of the modulus of the second derivative of the Dirichlet kernel in zero
Consider the Dirichlet kernel $D_N(x)=\sum_{|k|\le N} e^{ikx}$. Its second derivative reads as $$D_N^{\prime\prime}(x) = -\sum_{|k|\le N} e^{ikx}k^2.$$
What is the asymptotic order of $|D_N^{\prime\...
- 11
0
votes
1
answer
99
views
can we have a Taylor series expansion of the function $\frac{1}{\zeta(s)}$?
can we have a Taylor series expansion of the function $\frac{1}{\zeta(s)}$at $s=1$?
if so how can we determine the radius of convergence of this expansion without assuming the truth of the riemann ...
- 43
1
vote
1
answer
49
views
inequality involving two dirichlet series
Let $f\left( s \right) = \sum\limits_{n = 1}^{\infty}\left[ a_{n} \cdot \left( {\frac{1}{n^{s}}-\frac{1}{n^{1 - \operatorname{conj}\left( s \right)}}} \right) \right]$ and Let $g\left( s \right) = \...
- 11
1
vote
0
answers
40
views
Question on conjectured method of extending convergence of Maclaurin series for $\frac{x}{x+1}$ from $|x|<1$ to $\Re(x)>-1$
The question here is motivated by this Math StackExchange question and this Math Overflow question which indicate the evaluation of the Dirchleta eta function
$$\eta(s)=\underset{K\to\infty}{\text{lim}...
- 7,631
12
votes
0
answers
473
views
How to interpret a strange formula about $\zeta'(s)/\zeta(s)$
I obtained a strange formula about $\zeta'(s)/\zeta(s)$
$$
\begin{split}
\frac{\zeta'(s)}{\zeta(s)}-(2\pi)^s&\sum_{\Im(\rho)>0} (-i\rho)^{-s}(2\pi)^{-\rho} e^{-i\pi \rho / 2} \Gamma(\rho)\;\;\...
- 78.4k
1
vote
0
answers
51
views
Why are these numbers close to $-\log(2)+\text{integer}\,i\pi$?
The following function $f(n)$ has been derived from the Dirichlet eta function:
$$f(n)=\log \left(\sum _{k=1}^n (-1)^{k+1} x^{c \log (k)}\right)-c \log (n) \log (x) \tag{$\ast$}$$
Let: $$s=\rho _1$$ ...
- 7,448
0
votes
0
answers
49
views
Uniformly convergent series manipulation
I get confused reading about L-series and there is a lemma on infinite series. The question should only concern about analysis and there should be no number theory involved. The lemma is below,
Let $\{...
- 558
1
vote
2
answers
343
views
Does every Dirichlet series admit an analytic continuation? If so, to what extent?
The identity theorem for analytic continuation shows that uniqueness of analytic continuations of functions is very easy to characterize. This helps us a lot when we are trying to extend functions ...
- 2,461
2
votes
1
answer
462
views
On the abscissa of convergence of a Dirichlet series.
I am trying to find the abscissa of convergence of the Dirichlet series for the arithmetic function $|\mu(n)|$.
I have managed to show that $$\sum_{n=1}^{\infty}\frac{|\mu(n)|}{n^s}=\frac{\zeta(s)}{\...
user1021241
1
vote
0
answers
28
views
Are there some natural bijections between general Dirichlet series and power series?
The theory of general Dirichlet series and the theory of power theory have some analogs:
The abscissa, line and half-plane of convergence of a Dirichlet series
are analogous to radius, boundary and ...
- 53