All Questions
Tagged with dirichlet-series analytic-continuation
19
questions
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$ f(s) = 1 + \sum_{n=2}^{\infty} \frac{n^{-s}}{p_n}= 0$?
Let $p_n$ be the $n$ th prime number.
Let $f(s)$ be a Dirichlet series defined on the complex plane as :
$$f(s) = 1 + \sum_{n=2}^{\infty} \frac{n^{-s}}{p_n}= 1 + \frac{2^{-s}}{2}+ \frac{3^{-s}}{3} + \...
- 16.4k
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1
answer
99
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can we have a Taylor series expansion of the function $\frac{1}{\zeta(s)}$?
can we have a Taylor series expansion of the function $\frac{1}{\zeta(s)}$at $s=1$?
if so how can we determine the radius of convergence of this expansion without assuming the truth of the riemann ...
- 43
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67
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Question on convergence of product and Dirichlet series representations of a function
Consider the following two representations of $f(s)$
$$f(s)=
\underset{K\to\infty}{\text{lim}}\left(\prod\limits_{k=1}^K \left(1-\frac{2}{\left.p_k\right.^s}\right)\right)\tag{1}$$
$$f(s)=\underset{N\...
- 7,631
2
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82
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Question on analytic continuation of functions related to $\log\zeta(s)$ and $\frac{\zeta'}{\zeta}(s)$ that converge for $|\Re(s)|>1$
Consider the following two Dirichlet series
$$C_3(s)=\underset{N\to\infty}{\text{lim}}\left(\sum\limits_{n=1}^N 1_{n=p^k}\, \log(n)\, n^{-s}\right),\quad\Re(s)>1\tag{1}$$
$$K_\Omega(s)=\underset{N\...
- 7,631
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57
views
Analytic continuation of Dirichlet series with sine in numerator
The following Dirichlet series converges for $\Re(s)\in (1,\infty)$ (constant $k\in\mathbb{R}$), which is evident since sine function in numerator is in interval $[-1,1]$.
$$f_1(s)=\sum _{n=1}^{\infty ...
- 1,206
1
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2
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342
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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
1
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3
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52
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How does this identity so elegantly combine an infinite sum in $\eta$ and an improper integral in $\Gamma$?
This is all well and good, but where did this come from?
In the article on the Gamma function, Wikipedia shows most of its alternate definitions with clear proofs, yet in the article on the Dirichlet ...
- 42.7k
1
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109
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Does my odd proof for the Abel sum for $\eta(-2)$ work?
EDIT: The correct answer to the Abel sum of $\eta(-2)$ has been given by the comments under this post. The focus of the question is now whether there is any sense to my method and my "proof" ...
- 42.7k
1
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1
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64
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Analytic continuation of a certain family of $L$ series
Consider the function
$$
G(s,x) : = \sum_{n=1}^\infty \frac{\exp(2\pi i nx)}{n^s}
$$
where $x\in[0,1)$ and $s\in\mathbb{C}$. The series is absolutely convergent for $Re(s) > 1$.
If $x\in\mathbb{Q}\...
- 991
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votes
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answers
72
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Where does $s-1$ come from in this $\zeta(s)$ equation?
I have been working my way through this Arxiv paper concerning the analytic continuation of the zeta function. I don't understand the first equality in equation (19), page 6.
In equation (11), the ...
1
vote
1
answer
123
views
Analytic continuation of a Dirichlet series
Suppose we have a Dirichlet series
$$
D(s) = \sum_{n=1}^\infty \frac{a(n)}{n^s}
$$
which we know is absolutely convergent for $Re(s)>1$. Suppose that we prove that $\lim_{s\to 1^+}D(s) < \infty$....
- 991
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31
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Regularity of Epstein zeta Z|h,0|(A; s) in h
My question considers the regularity of the Epstein zeta function in one of it modules. Let $h\in \mathbb R^d$ and $A\in \mathbb R^{d\times d}$ positive definite (we can also assume $A=I$ for ...
- 1
1
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50
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Generalized Dirichlet Series
I am looking into Dirichlet series book by Mandelbrojt but unfortunately I don't find results regarding boundary behavior of the type of series $\sum a_ne^{-\lambda_ns}$ in which the density ($\limsup ...
- 185
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61
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An $L-$function and a $J-$function. Related?
Consider a Dirichlet series for a non real character of modulus $q$
$$ L(s,\chi)=\sum_{n=1}^\infty \frac{\chi(n)}{n^s} $$
and $s\in\Bbb C$ with real part greater than one.
Consider a $J$-series $$ J(s,...
- 756
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139
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Why do Dirichlet series always have infinitely many nontrivial nonreal zero's?
This is a follow-up question to this :
Why does $\sum a_i \exp(b_i)$ always have root?
question in link :
Let $z$ be complex.
Let $a_i,b_i$ be polynomials of $z$ with real coefficients.
Also the $a_i$ ...
- 16.4k