All Questions
Tagged with cv.complex-variables dirichlet-series
14 questions with no upvoted or accepted answers
14
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0
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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
0
answers
104
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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
$$
D(s)=\sum_{\substack{k\in{\mathbb Z}^2}\\ k\ne 0}\frac{|k|}{g(k)}...
user1688
4
votes
0
answers
134
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Converse theorem for zeta universality
Voronin's Universality Theorem for $\zeta(s)$ is that the zeta function can uniformly approximate any non-vanishing holomorphic function to any degree of accuracy in the right-half of the critical ...
- 141
4
votes
0
answers
275
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Computing Bohr Radii
The Bohr radius $R$ for $\mathcal{H}(\mathbb{D})$ is defined as the radius $$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
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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
$\sigma_c(D) = \inf\{\sigma : D \text{ converges in } \mathrm{Re} s > \sigma \}$, ...
- 576
2
votes
0
answers
119
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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
0
answers
381
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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
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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 $$\varphi(s)=\sum_{n=1}^\infty\...
1
vote
0
answers
102
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Dirichlet series with an abscissa of absolute convergence $\sigma_{0}$, analytic in $\sigma > \sigma_{0} - \delta$
Suppose that a Dirichlet series $f(s)$ has the abscissa of absolute convergence $\sigma_{0}$ and is analytic in $\sigma > \sigma_{0} - \delta$ for some $\delta > 0$. For $\sigma > \sigma_{0}$,...
1
vote
0
answers
61
views
Which complex maps with branch cuts have a representation by Dirichlet series?
Which complex maps with branch cuts have a representation by Dirichlet
series?
I am aware of the work of A.F. Leont'ev on general Dirichlet series, and the theorems of representation of analytic ...
- 1,805
1
vote
0
answers
98
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Is there a general connection between value distribution and zero distribution for functions representable by Dirichlet series?
Some time ago I read part of a book in which the author made some conjectures outlining what kind of zero distribution is expected for functions representable by Dirichlet series with completely ...
- 2,480
0
votes
0
answers
151
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Abscissa of convergence of transformed Dirichlet series
Let
$$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},$$
where the infinite sum $\sum f(k)$ diverges, $f(k)$ and $g(k)$ are real numbers, $s$ a ...
- 3,098
-2
votes
0
answers
52
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Is the real and imaginary part of the Dirichlet eta function closest to its partial sums when trigonometric function changes signs?
To grasp the question we are concerned with three Theorems 1,2, and 3 in bold font below. First let us consider the Dirichlet eta function $\eta: \mathbb{C}\rightarrow \mathbb{R}$
$$
\eta(s) = \sum_{n=...
