Questions tagged [zeta-functions]
Questions on various generalizations of the Riemann zeta function (e.g. Dedekind zeta, Hasse–Weil zeta, L-functions, multiple zeta). Consider using the tag (riemann-zeta) instead if your question is specifically about Riemann's function.
820 questions
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Riemann Zeta function values at odd & even natural numbers [closed]
If there a general relationship between values of Riemann Zeta function at even natural numbers & odd natural numbers ? Can we get zeta function values of any odd natural numbers using values at ...
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0
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31
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Is there a popular treatise on Ihara-Bass formula?
It's basically in the title.
Recently I had to make a lot of use of Ihara Bass in my research. So I decided to communicate this result to a broader audience (maybe a wiki article or something). But ...
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1
answer
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Prove $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} \sum_{m=0}^{\infty} \frac1{n2^m+1}=1$
The First: I'm not sure if this identity has been asked before. I want to prove the summation:
$$\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} \sum_{m=0}^{\infty} \frac1{n2^m+1}=1$$
which I checked by ...
- 2,410
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Do there exist any analytical bounds on the Hurwitz Zeta function?
For the general real-valued Hurwitz Zeta Function
$$
\zeta\left(s,a\right),\quad s > 1,\ a > 0.
$$
Do there exist upper and/or lower bounds ?.
Even some ...
- 13
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111
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Is it possible to compute $\sum_{m=1}^{\infty} \frac{\coth(m\pi)}{m^9}$ in closed form?
I'm aware that it is possible to compute $$\sum_{m=1}^{\infty} \frac{\coth(m\pi)}{m^{4k-1}}$$
in closed form for $k = 1, 2, 3\ldots$ as demonstrated in the links below:
Cauchy-Ramanujan Formula $ \...
- 21
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Prove that the function $\bigl(2^{t+c}-1\bigr)\zeta(t)$ is logarithmically concave in $t\in[4,\infty)$
Let
\begin{equation}
c=\log_2\biggl(\frac{1}{16}\frac{60-\pi^2}{15-\pi^2}\biggr)=-0.711\dotsc
\end{equation}
For verifying that the sequence
\begin{equation}
\frac{1}{(2n+1)(n+1)}\frac{2^{2n+2+c}-1}{2^...
- 1,960
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2
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Sources to study the holomorphic continuation of $\zeta(2s)E(\tau,s)$, where $E(\tau,s)$ is the real-analytic Eisenstein series
I am studying the paper EISENSTEIN SERIES AND THE RIEMANN
ZETA-FUNCTION, by D. Zagier. He says that the function $\zeta(2s)E(\tau,s)$ has a holomorphic continuation to all $s$ except for a simple pole ...
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Polynomials over $\mathbb{F}_q$ with prescribed number of linear factors.
It can be shown that there are $q^n-q^{n-1}$ (monic) polynomials of degree $n$ in $\mathbb{F}_q[t]$ that are squarefree. The standard way to show this is using Zeta functions.
Suppose I wanted to ...
- 2,569
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Zeta function related double integral
I recently came across the following double Integral:
$$
I(a)=\int \limits_{0}^{1}\int \limits_{0}^{1}\frac{x^ay^a}{1-xy}dx\,dy
$$
which can be easily shown to be equal to:
$$
I(a)=\zeta(2,a)=\sum \...
- 129
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Dirichlet series with $ f(1-s) = f(s) $ or a proven critical line $\Re(s)=1/2$?
Context
I read this :
This paper has been withdrawn by Farid Kenas
[Submitted on 8 Mar 2024 (v1), last revised 25 Aug 2024 (this version, v2)]
Attempting to Prove the Riemann Hypothesis through the ...
- 17.1k
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Integral representation of Lambert Series.
Is there a general integral representation for Lambert series, similar to those that exist for various L-series and zeta functions?
For a sequence of complex numbers $\{a_n\}$, its corresponding ...
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On the Glaisher numbers and $6^5\sum_{n=0}^\infty\frac1{(6n+1)^5} = \frac{(2^5-1)(3^5-1)}2\,\zeta(5)+\frac{22}{\sqrt3}\,\pi^5$
I. Data
After some experimenting, we find for odd powers $s>1$,
\begin{align}
3^s\sum_{n=0}^\infty\frac1{(3n+1)^s} &= \frac{(3^s-1)}2\,\zeta(s)+\frac{G(s)}{2\, s!\sqrt3}\,(2\pi)^s\\[5pt]
4^s\...
- 56.2k
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Upper bound of Hurwitz zeta like function in two dimensions in terms of the parameter
Let $n>0$ a real number. I'm looking to bound the series
$$
\sum_{m_1,m_2 \in \mathbb{N}} \dfrac{1}{(m_1+n)^2+(m_2+n)^2}
$$
In terms of $n$. The series converges as
$$
\sum_{m_1,m_2 \in \mathbb{N}} ...
- 2,730
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1
answer
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Why $\sum_{n=0}^{\infty} \frac{1}{(6n+1)^3}=\frac{1}{6^3}(91\zeta(3)+2\sqrt3\pi^3)$ is just the tip of the iceberg (Part 2)
(Continued from this post. Update: Section III below now includes the Glaisher H-numbers.)
I. Definitions
As before, given the Hurwitz zeta function,
$$\zeta(s,a) = \sum_{k=0}^\infty \frac1{(k+a)^s}$$
...
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Why $\sum_{n=0}^{\infty} \frac{1}{(6n+1)^3}=\frac{1}{6^3}(91\zeta(3)+2\sqrt3\pi^3)$ is just the tip of the iceberg (Part 1)
While trying to solve this post (an identity also mentioned by Ramanujan), I noticed it had two parts of interest. This is the first one.
I. Definitions
Given the Hurwitz zeta function,
$$\zeta(s,a) = ...
- 56.2k
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vote
1
answer
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A limit involving the Hurwitz Zeta function and binomial coefficients
Let $n \in \mathbb Z_{\geq 3}$. While playing around in Mathematica I noticed the following limit $$B_n(x) \searrow 0 \qquad \text{as }\, x\to\infty,$$
where $B_n(x)$ is an alternating binomial-type ...
- 1,746
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answer
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Bounds for the Prime Zeta function
The Prime Zeta function is defined as
$$\zeta_{P}(s) = \sum_{p\in \mathbb P} \frac 1{p^s}$$
where $\mathbb P$ is the set of primes.
In an answer to this question, TravorLZH shows
$$\sum_{p>x}p^{-s}=...
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Expansion and Simplification of a Hurwitz Zeta Function
On one math page I saw this simplification being made involving the Hurwitz Zeta Function easily simplifying into a normal Zeta Function form: $$\zeta(s,\frac{1}{2})=2^{s}\sum_{k=0}^{\infty}\frac{1}{(...
- 103
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Evaluating/Simplifying a Series involving Zeta Functions
I was trying to learn how to evaluate this expression: $$\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(2n+1)^{3}}$$ And I learned that it can be easily simplified to: $$\frac{1}{64}(\zeta(3,\frac{1}{4})-\zeta(3,...
- 103
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1
answer
74
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Equivalent infinite series to Riemann Zeta function
I was studying the analytic continuation of Riemann Zeta function. There was this relationship given-
$$\zeta(s)= \sum_{n=1}^{\infty}{1\over n^s}=\sum_{n=1}^{\infty}n[{1\over n^s}-{1\over(n+1)^s}]$$
...
- 55
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votes
2
answers
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What is the identity of this zeta function?
There are a Riemann zeta function, a Hurwitz zeta function, and many different types of zeta functions. However, I saw the zeta function below in a Japanese blog.
$$\zeta(s)=\frac{1}{1-2^{1-s}}\sum_{m=...
- 665
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Constant term in the Euler-Maclaurin expansion of $s_n=\sum_{k=1}^n \tfrac{1}{k+1/2}$
This question is a followup from my previous post based on the Euler-Maclaurin formula: How to find the correct constant term with Euler-Maclaurin formula, $\sum_{j=1}^n j\log j$.
This time I am ...
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is this solution correct $\frac {\partial}{\partial x} \int_0^∞ \frac{\sin((x+it)\arctan(t))}{((1+t^2)^{(x+it)/2} (e^{2\pi t} -1))} dt =0 $?
when I was reading about the Riemann zeta function I found out this integral $\ \frac {\partial}{\partial x} \int_0^∞
\frac{\sin((x+iy)\arctan(t))}{((1+t^2)^{(x+iy)/2} (e^{2\pi t} -1))} dt $
and ...
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Periodic zeta function
Let $e(x)=e^{2\pi ix}$ and let $$F(x,s)=\sum _{n=1}^\infty \frac {e(nx)}{n^s}$$ be the periodic zeta function.
What is the functional equation for the periodic zeta function ?: I can find a statement ...
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answer
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Finding a closed form for $\sum^{\infty}_{n=1} \frac{1}{(n+1)n^\alpha}$ [duplicate]
I encountered the following sum in my work and I was wondering if it has a known closed form:
$$
\sum^{\infty}_{n=1} \frac{1}{(n+1)n^{\alpha}} \quad , \quad 0 < \alpha < 1 \; , \; \alpha \in \...
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"Are there any simple groups that appear as zeros of the zeta function?" by Peter Freyd; why is this consternating to mathematicians?
I would like to understand the "upsetting"-to-mathematicians nature of this question Freyd poses to demonstrate that "any language sufficiently rich that to be defined necessarily ...
- 258
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1
answer
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Closed form for $A = \sum_{a>1,b>1}\dfrac{1}{(2 a^2 + 3 b^2)^2}$?
Is there a closed form for
$A = \sum_{a>1,b>1}\dfrac{1}{(2 a^2 + 3 b^2)^2}$
??
We know
$$ \sum_{m,n = - \infty}^{\infty} \frac{(-1)^m}{m^2 + 58 n^2} = - \frac{\pi \ln( 27 + 5 \sqrt {29})}{\sqrt {...
- 17.1k
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How does Wolfram Alpha know this closed form?
I was messing around in Wolfram Alpha when I stumbled on this closed form expression for the Hurwitz Zeta function:
$$
\zeta(3, 11/4) = 1/2 (56 \zeta(3) - 47360/9261 - 2 \pi^3).
$$
How does WA know ...
- 5,373
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answers
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views
Extensions of Hardy's Inequality for Tail Sums
I have been studying various formulations of Hardy's inequality, inspired by notable theorems from works like those of Paul Richard Beesack and others. A particular theorem that caught my attention ...
- 1,023
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63
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What's the point of the local zeta function?
I'm currently reading through Ireland and Rosen's "A Classical Introduction to Modern Number Theory", and I feel I'm missing the point of Chapter 11 on (local) zeta functions. When I see ...
- 702
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Estimation of the absolute value of the $n$th non-real zero of the Riemann zeta function
Recently, I have been studying the oringinal proof of the prime number theory by Hadamard. I didn't get it on the estimation of the absolute value of the $n$th non-real zero of the $\zeta$ function by ...
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the zero's of $f(s,a) = \sum_{n=1}^{a-1} n^{-s} $
I was looking at the zero's of
$$f(s,a) = \sum_{n=1}^{a-1} n^{-s} $$
for integer $a>3$ in the strip $0 < \operatorname{Re}(s) < 1$.
Now this clearly relates to the Riemann zeta:
$$f(s,a) + \...
- 17.1k
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Some curios sums of Hurwitz zeta-function and Lehmer's totient problem [closed]
For all squarefree $k\in \mathbb N$
$$\left|\frac{\sum_{n=1}^{k-1}\zeta_H(-1,n/k)}{\sum_{n=1}^{k-1}\chi_0(n)\zeta_H(-1,n/k)}\right|=\left|\frac{\sum_{n=1}^{k-1}1}{\sum_{n=1}^{k-1}\chi_0(n)}\right|=\...
- 67
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0
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Question about a limit of the euler-product of the riemann zeta function
The limit $$\lim_{s\to 1} (s-1)\zeta(s)=1=\lim_{s\to 1}(s-1)\prod_{p\in\mathbb P} (1-p^{-s})^{-1}$$
is well-known.
Consider that there are infinitely many distinct subsets $\mathbb P_{k}\subsetneq\...
- 67
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Showing that $\int_0^1 \frac{\text{d}x}{\zeta(x)\Gamma(x)}<0<1<\int_0^\infty \frac{\text{d}x}{\zeta(x)\Gamma(x)}$ in $3$ minutes, without a calculator
The following question is to be solved within $3$ minutes, without a calculator.
$$\text{Let }I=\int_0^\infty \frac{\text{d}x}{\zeta(x)\Gamma(x)}\text{, and let }J=\int_0^1 \frac{\text{d}x}{\zeta(x)\...
- 5,480
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Estimates of the derivatives of $\Xi(s)$
The $\Xi$ Function is defined by $\Xi(s)=\xi(\frac{1}{2}+is)$, where $\xi(s)=\frac{1}{2}s(s-1)\pi^{-\frac{s}{2}}\Gamma(\frac{s}{2})\zeta(s)$.
This is a problem from my homework: since we can write it ...
- 81
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0
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53
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Zeta Hurwitz function bounds of summation
I have been trying to derive the following equality
$$
\frac{1}{2} \sum_{n=1}^{\infty} \sum_{m=0}^{p-1} \frac{1}{n^s}\left[\cos \left(2 \pi \frac{m q}{p}\left(q^*-1+n\right)\right)+\cos \left(2 \pi \...
- 1
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1
answer
84
views
How can I evaluate $1 / \Gamma (-1)$?
I understand intuitively that the Gamma function diverges at all negative integers. This leads me to believe that the inverse Gamma function would have zeros at all negative integers. However, I’m ...
0
votes
1
answer
106
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Showing $\sum_{n=1}^{\infty }\left ( \sum_{j=1}^{\infty }\frac{x^{(n-j)^2}-x^{(n+j-1)^2}}{(2n-1)(2j-1)} \right ) = \frac{\pi^2}{8}$
Show that
$$\sum_{n=1}^{\infty }\left ( \sum_{j=1}^{\infty }\frac{x^{(n-j)^2}-x^{(n+j-1)^2}}{(2n-1)(2j-1)} \right) = \frac{\pi^2}{8}$$
I liked this problem because the result is a final answer, and ...
- 1,541
3
votes
2
answers
244
views
Is there a closed form expression for $ \sum_{n=2}^\infty \frac{1}{n \sqrt{n^2-1}} $?
I tried to find a closed form for the series
$$ \sum_{n=2}^\infty \frac{1}{n \sqrt{n^2-1}} $$
I got another form for the series by using the known series
$$\frac{1}{\sqrt{n^2-1}}=\frac{1}{n\sqrt{1-n^{-...
- 1,979
5
votes
1
answer
285
views
Relationship between $\zeta(3)$ and ordinary logarithm function $\text{log}(\text{x})$
While working on another problem, I came up with the following expression. This involved many manual definite integral evaluations and not at all elegant. So I am not going into the details of the ...
- 1,301
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votes
0
answers
63
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Evaluating the sum $\sum_{n=1}^\infty \frac{1}{n(n+a)^b}$ [duplicate]
I am looking for ways to simplify the sum
$$\sum_{n=1}^\infty \frac{1}{n(n+a)^b}, \quad a\in\mathbb{R}^+, b\in\mathbb{N}.$$
The first thought I had approaching this was to use Hurwitz and/or Zeta ...
- 45
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votes
0
answers
80
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Difficulty computing $\int_{0}^{\infty} \frac{\ln(x)}{e^x+1} dx$=$-\frac{1}{2}\ln^2(2)$ [duplicate]
Here some context , after computing some integral in the form $\int_{0}^{\infty} \frac{\ln(a^2+x^2)}{\cosh(x)+\cos(b)}dx \,\,\, , \int_{0}^{\infty} \frac{x\ln(a^2+x^2)}{\sinh(x)}dx \,\,\, , \int_{0}^{\...
- 167
2
votes
1
answer
115
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Limit of a Function Involving Hurwitz Zeta Function
I am trying to prove the following limit of a function involving the Hurwitz Zeta function:
$$
\lim_{N \to \infty} \frac{\zeta(-d, 1 + N) - \zeta(-d, 1 + p N)}{N^{1 + d}} = \frac{-1 + p^{1 + d}}{1 + ...
1
vote
2
answers
343
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Proper Way to Calculate Value of Riemann Zeta function?
I understand that an Analytic Continuation of a function will extend its domain into areas that it previously wasn't defined in.
I've been looking at one of the Analytic Continuations of the Zeta ...
- 5,957
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0
answers
129
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Is there other method to deal with $\int_0^1 \int_0^1\frac{\ln^m (1+x y)}{(1+x y)^n}d x d y,$ where $m\ge 3?$
After finding the exact value of the integral, $$\displaystyle \int_0^1 \int_0^1\left(\frac{\ln (1+x y)}{1+x y}\right)^2 d x d y=-\frac{\zeta(3)}{4}+\frac{1}{3} \ln ^3 2-\frac{\pi^2}{6}+\ln ^2 2+2 \...
- 25.7k
0
votes
0
answers
35
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Zeta functions on q-deformed compact Lie groups
I’m reading some of the recent works on representation zeta functions for groups. Along the way, I have also explored some of the remarkable properties of Witten zeta function. I’m wondering if there ...
- 1
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0
answers
44
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Relations between Dilogarithms and Imaginary part of Hurwitz-Zeta function
I'm working through a paper that involves a problem concerning the calculation of the Imaginary part of the derivative of the Hurwitz-Zeta function $\zeta_H(z,a)$ with respect to $z$, evaluated at a ...
1
vote
1
answer
117
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Connection between the polylogarithm and the Bernoulli polynomials.
I have been studying the polylogarithm function and came across its relation with Bernoulli polynomials, as Wikipedia site asserts:
For positive integer polylogarithm orders $s$, the Hurwitz zeta ...
- 945
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1
answer
58
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step function question: What tools can be used to study it?
Consider the step function
$$A(x)=\sum_{n=1}^\infty e^{\mathrm{floor}\bigg(\frac{\log n}{\log x}\bigg)+\mathrm{floor}\bigg(\frac{\log n}{\log (1-x)}\bigg)} = \prod_{\mathrm{ p~ prime}} \frac{1}{1-e^{\...
- 204