Questions tagged [dirichlet-series]
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23
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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 $\forall \sigma >\sigma_a:$
$$\lim_{T\to\infty}\frac{1}...
- 6,708
14
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3
answers
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Convergence of L-series
I remember to have read that the L-function of an elliptic curve, which a priori only converges for $\Re s > \frac{3}{2}$ also converges at $s=1$ provided that the $L$-function
satisfies the ...
- 2,714
14
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1
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Mean value theorem for Dirichlet series - optimize?
Let $a_n\in \mathbb{C}$. We can prove a mean value theorem, meaning an inequality
$$\int_0^T \left|\sum_{n=1}^\infty a_n n^{-i t}\right|^2 dt \leq
\sum_{n=1}^\infty (c_0 T + c_1 n + c_2) |a_n|^2.$$
...
- 19.4k
0
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1
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Questions on analytic representations of the Kronecker delta function $\delta(x-1)$ and the Moebius function $\mu(n)$
This question is related to analytic formulas for $a(n)$ where $f_a(x)$ and $F_a(s)$ defined in formulas (1) and (2) below are the summatory function and Dirichlet series associated with $a(n)$.
$$...
- 1,061
15
votes
3
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858
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Tauberian theorem $\sum_{k=1}^{\infty}e^{-\lambda_{k}t}c_{k} \xrightarrow{t\to 0} \sum_{k=1}^{\infty}c_{k} $
I am trying to prove or disprove
$$\sum_{k=1}^{\infty}e^{-\lambda_{k}t}c_{k} \xrightarrow{t\to 0} \sum_{k=1}^{\infty}c_{k} ,$$
where $\sum c_{k}<\infty, \sum c_{k}^{2}<\infty\text{ and }\frac{\...
- 4,449
12
votes
1
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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 $x\geq |t|$,
$$\zeta(s) = \sum_{n\leq x} \frac{1}{n^s} + \frac{x^{1-s}}{s-1} + O\left(\...
- 19.4k
9
votes
2
answers
2k
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Divergence of Dirichlet series
Suppose $s$ is a complex number with $\Re(s) \in (0,1]$ and $\{a_n\}$ is a complex sequence converging to $a \neq 0$. Must the Dirichlet series $$\sum_{n=1}^\infty\frac{a_n}{n^s}$$ diverge?
I asked ...
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8
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1
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Dirichlet's theorem for number fields
I'd like to see a formulation of Dirichlet's theorem for number fields, i.e. some analogue of the assertion:
The number of primes less than $N$ congruent to $a \pmod{m}$ where $(a,m)=1$ is
$\frac{...
- 543
6
votes
0
answers
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Mean value theorem for Dirichlet series of prime support?
Let $\{a_n\}_{1\leq n\geq N}$, $a_n\in \mathbb{C}$. Let $F(s) = \sum_{n=1}^N a_n n^{-s}$. By a mean-value theorem (Montgomery-Vaughan, 1973),
$$\int_0^T |F(i t)|^2 = \sum_{n=1}^N |a_n|^2 (T + O(n)).$$
...
- 19.4k
5
votes
1
answer
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Does the Euler product for $L(s,\chi_4)$ also converge in the right half of the critical strip?
This question expands on this one from MSE.
In the literature about Dirichlet $L$-series, I found that their Euler products:
$$L(s, \chi) =\prod_p \bigg(\frac {1}{1-\frac{\chi(p)}{p^s}} \bigg)$$
...
- 4,169
5
votes
0
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Can we extend the twisted Poisson Summation formula with functions having a singularity in zero?
The following "twisted" Poisson Summation formula for $\chi$ primitive of conductor $q$ :
$$ \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)...
- 1,121
3
votes
1
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Special values of Hecke L-function
The Dedekind zeta function for a number field $K$ is defined as
$\zeta_K(s)=\sum_{I\subset O_K} (N_{K/\mathbb{Q}}(I))^{-s}$.
By attaching a Hecke character $\psi$, we can define $L(s,\psi)=\sum_{I\...
- 443
3
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1
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Multiplicative functions whose Dirichlet series have essential singularities
What can be said about the partial sums of a complex-valued completely multiplicative function, let's say bounded by 1 in absolute value, if its Dirichlet series has an essential singularity?
As a ...
- 1,671
2
votes
1
answer
265
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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\}...
- 271
2
votes
2
answers
604
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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 ...
- 7,272
2
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0
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171
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How to best approximate $1/\zeta(s)$ by a finite sum
I would like to approximate $1/\zeta(s)$ for $s=1+it$ by a finite sum:
$$\frac{1}{\zeta(1+it)} = \sum_n \frac{\mu(n)}{n} \eta\left(\frac{n}{x}\right) +
\epsilon(t)$$
with $\eta$ a function of compact ...
- 19.4k
2
votes
0
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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_{...
2
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1
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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} = ...
- 733
1
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1
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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 ...
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1
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1
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Generalizing closed form representations related to conjectured analytic formulas for $f_a(x)=\sum\limits_{n=1}^x a(n)$
Consider the summatory function $f_a(x)$ defined in formula (1) below where the related Dirichlet series $F_a(s)$ defined in formula (2) below converges for $\Re(s)\ge 2$.
$$f_a(x)=\sum\limits_{n=1}^...
- 1,061
0
votes
1
answer
284
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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
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1
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What is the critical exponent for irregular function in the Sobolev scale?
When I first saw the definition of general Sobolev spaces with real exponent I immediately got interested in the following problem: pick several of your favourite irregular functions/distributions and ...
- 9,150
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2
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385
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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 $...
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