Questions on the famed $\zeta(s)$ function of Riemann, and its properties.

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Mobius function related series and Dirichlet summation

A well known identities of rare beauty is that: $$\sum_{n=1}^\infty \frac{\mu(n)}{n^s}=\frac 1{\zeta(s)}$$ Where $\mu(n)$ is the Mobius function and $\zeta(s)$ is the Riemann Zeta function. So, in ...
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1answer
23 views

On $\zeta(s)=h^2(s)$ when we presume that the Riemann zeta function has no zeros for $\Re s>\frac{1}{2}$

By specialization of a theorem from complex analysis, one has that on assumption that the Riemann zeta function $\zeta(s)$ has no zeros with $\Re s>\frac{1}{2}$, then there exists an analytic ...
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10answers
12k views

Nice proofs of $\zeta(4) = \pi^4/90$?

I know some nice ways to prove that $\zeta(2) = \sum_{n=1}^{\infty} \frac{1}{n^2} = \pi^2/6$. For example, see Robin Chapman's list or the answers to the question "Different methods to compute ...
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2answers
83 views

Finding a closed form for $\sum_{k=1}^{\infty}(-1)^{n-1}\frac{\zeta(2n+1)}{2^{2n+1}}$

I've been playing with series involving odd values of the zeta function. Some time ago I found the following closed form $$ \sum_{k=1}^{\infty}\frac{\eta(2k+1)}{2^{2k+1}}=\frac{1}{2}-\ln(2) $$ and ...
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0answers
54 views

$\zeta(1)=\frac12\ln(2)$? Did I do something wrong?

I attempted to calculate $\zeta(1)$ and I got $\frac12\ln(2)$. $$\zeta(1)=\lim_{\epsilon\to0}\frac{\zeta(1+\epsilon)+\zeta(1-\epsilon)}2$$ ...
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0answers
22 views

How is justified the derivation under the integral sign $\frac{d}{d\sigma} \left( \Re\frac{1}{\zeta(s)} \right) $?

Taking $\sigma=\Re s>1$ (this is we take $s=\sigma+it$, $\sigma$ and $t$ real numbers) then the using theknown integral representation for $\frac{1}{\zeta(s)}$, where $\zeta(s)$ is the Riemann ...
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3answers
186 views

Geometric Interpretation of the Basel Problem?

I'm probably asking a question no one knows the answer to, but everyone must mentally ask at some point. Does the Pi in the solution to the Basel problem have any geometric significance? Every time ...
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1answer
39 views

Particular values of the Riemann zeta function.

On the wikipedia, near the bottom of the "Specific Values" section, there is a statement that bothers me. $$\zeta(-13)=\zeta(-1)$$ Firstly, it is well noted that the summations must be evaluated ...
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1answer
56 views

Evaluate the indefinite integral $\int \frac{t\sin at}{b^2+t^2}dt$

It is known DLMF (25.2.8) that for $\Re s>0$ and for integers $N\geq 1$ $$\zeta(s)=\sum_{k=1}^N\frac{1}{k^s}+\frac{N^{1-s}}{s-1}-s\int_{N}^\infty \frac{x-\lfloor x \rfloor}{x^{s+1}} dx,$$ where ...
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1answer
85 views

How to find the bound of this sum?

Let $t>0,a(t)=\arg(\Gamma(1/4+it))$,$\kappa(n)=\frac{1}{2}x\pi n^2$,we need to calculate the bound,$A(x)$, of the following finite sum: $$ S(x)=\sum_{1\le n\le ...
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4answers
4k views

Riemann zeta function at odd positive integers

Starting with the famous Basel problem, Euler evaluated the Riemann zeta function for all even positive integers and the result is a compact expression involving Bernoulli numbers. However, the ...
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0answers
57 views
+50

Is right this application of Hadamard three-lines theorem for $ \frac{\zeta(s)}{s}- \frac{d\zeta(s)}{d\sigma}$?

Let the complex variable $s=\sigma+it$, then from the following identity valid for $\sigma=\Re s>1$ $$\zeta(s)=s\int_1^\infty \frac{[x]}{x^{s+1}}dx$$ where $\zeta(s)$ is the Riemann Zeta function, ...
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0answers
22 views

Connection between the area of a n-sphere and the Riemann zeta function?

The Riemann Xi-Function is defined as $$ \xi(s) = \tfrac{1}{2} s(s-1) \pi^{-s/2} \Gamma\left(\tfrac{1}{2} s\right) \zeta(s) $$ and it satisfies the reflection formula $$ \xi(s) = \xi(1-s). $$ But the ...
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0answers
88 views

Analytic Bound on The Riemann Zeta Function

Given the canonical infinite product representation (Weierstrass form) of the gamma function, $$\Gamma(z)= \left [ze^{\gamma z}\prod_{m=1}^{\infty} \left ( 1+ \frac{z}{m} \right)e^{-z/m} \right ]^{-1} ...
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4answers
194 views

Limit of $\sqrt{\frac{\pi}{1-x}}-\sum\limits_{k=1}^\infty\frac{x^k}{\sqrt{k}}$ when $x\to 1^-$?

I am trying to understand if $$\sqrt{\frac{2\pi}{1-x}}-\sum\limits_{k=1}^\infty\frac{x^k}{\sqrt{k}}$$ is convergent for $x\to 1^-$. Any help? Update: Given the insightful comments below, it is ...
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0answers
125 views

Why is Fourier Analysis effective for studying uniform distributions

On his great expository article about the naturality of the Zeta function in number theory, Tim Gowers makes the following claim: When it comes to the primes, we find that we do not have a good ...
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2answers
43 views

$\zeta(0)=-\frac{1}{2}$

How can I proove $\zeta(0)=-\dfrac{1}{2}$ only with the fact $\zeta(s)=\sum_{n=1}^{\infty}n^{-s}$ for $\mathbb{R}\ni s>1$? Does Euler's formula for $\zeta(2n),~\mathbb{N}\ni n>0$ holds also for ...
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1answer
36 views

Calculate $\zeta(s) = \frac{1}{\Gamma(s)} \sum_{n=0}^\infty \frac{B_n}{n!} \int_0^\infty x^{s + n - 2} dx$

By Taylor expanding $$\frac{x}{e^{x}-1} = \sum_{n=0}^\infty \frac{B_n}{n!}x^n$$ in the Zeta function $$\zeta(s) = \frac{1}{\Gamma(s)} \int_0^\infty x^{s-2} (\frac{x}{e^{x}-1})dx$$ we find ...
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0answers
21 views

representation of $\zeta(s)$ with line integral

Let $0<\varepsilon<2\pi$ and $\gamma_{\varepsilon}:\mathbb{R}\rightarrow\mathbb{C}, \gamma_{\varepsilon}(t)=\begin{cases}-1-t+\varepsilon i&t\le-1\\\varepsilon\exp(\pi ...
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1answer
183 views

Closed form for $ S(m) = \sum_{n=1}^\infty \frac{2^n \cdot n^m}{\binom{2n}n} $ for integer $m$?

What is the (simple) closed form for $\large \displaystyle S(m) = \sum_{n=1}^\infty \dfrac{2^n \cdot n^m}{\binom{2n}n} $ for integer $m$? Notation: $ \dbinom{2n}n $ denotes the central binomial ...
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0answers
23 views

Pole Of Zeta function extension

Please advise on how to proceed with this? I don't know where to begin? Thanks. Show that the Riemann-Zeta function has a pole of order 1 at 1 after it has been extended to a holomorphic function on ...
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0answers
38 views

Deriving $ \frac{1}{n^s} = \frac{1}{\Gamma(s)} \int_0^{\infty}t^{s-1} e^{-nt} \, dt $ Backwards?

Is it possible to start with $\dfrac{1}{n^s}$ and then, without knowing the Gamma function in advance, naturally (with reasons!) derive that $$ \frac{1}{n^s} = \frac{1}{\Gamma(s)} ...
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0answers
19 views

A new formula relating the factorial and Riemann Zeta function resp. Bernoulli numbers?

I proved the following identities involving the factorial and Riemann's Zeta function respectively the Bernoulli numbers: $$\sum _{k=1}^{i}-{\frac {{\pi }^{-2\,k}\zeta \left( 2\,k \right)\left( -1 ...
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1answer
62 views

Is the given expression convergent as $n\to\infty$?

I want to know whether the following expression is convergent as $n\to\infty$ $$\frac{1}{n}\sum\limits_{k=1}^{\infty}\frac{|\ln n-\ln k|}{k^{(1+1/n)}}\cdot$$ With use of Riemann zeta function ...
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0answers
105 views

Finding the singularities and residues of a Gamma/Riemann Zeta function.

The function I have is $f(z)=\zeta(z)\Gamma(z − 1)\sin(\pi z)$ and I need to find all singularities and their residues so I can evaluate a clockwise contour integral for the contour $\left\lvert ...
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1answer
96 views

How many elementary ways are there to prove that $\displaystyle\left. \frac d {ds}\,\frac 1 {\zeta(s)} \right|_{s=1} = 1 $?

In a comment under this answer, a user boldly asserts that there is ONLY ONE way to prove that $$ \left. \frac d {ds}\,\frac 1 {\zeta(s)} \right|_{s=1} = 1 $$ where $\zeta$ is Riemann's zeta function. ...
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1answer
49 views

Signature of The Riemann Zeta Function

Let $\zeta\colon\mathbb{C}\to\mathbb{C}$ denote the Riemann zeta function, analytically defined on $\mathbb{C}$ by meromorphic continuation, where $z \in D$. Similarly, let $G$ represent the co-domain ...
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3answers
626 views

Conditional convergence of Riemann's $\zeta$'s series

Do Riemann's zeta-function's partial sums $\sum_{n=1}^N n^{-s}$ converge conditionally for some value $s=\sigma+it$ with $\sigma\le 1$? (We must at least have $t\ne 0$ of course.) Partial summation ...
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0answers
26 views

Relationship between Riemann Zeta function and Prime zeta function

In his paper, Daniel Grunberg shows a relationship between the Stirling Numbers of the first kind and the Harmonic numbers via series of partitions (see Equation 3.1 on Page 5 in the link above). If ...
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1answer
34 views

On $-\frac{\zeta'(x)}{x\zeta(x)}$ and von Mangoldt function

I believe that it is possible show the following Fact. For real $x>e$ then $$-\frac{\zeta'(\log x)}{x\zeta(\log x)}=\sum_{n=1}^\infty\frac{\Lambda(n)}{n^{\log x}},$$ where $\zeta(x)$ is the ...
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0answers
94 views

For which $s$ is defined $\frac{1}{\zeta(s)}\sum_{n=1}^\infty\frac{n}{ \left( \zeta(s) \right) ^n}$, where $\zeta(s)$ is the Riemann Zeta function?

If there are no mistakes in my words, if one of the factors $\sum_{n=1}^\infty a_n$ of a convolution product or Cauchy product converges, and the other $\sum_{n=1}^\infty b_n$ corverges absolutely ...
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1answer
63 views

Is it known if $\frac{\zeta(3)}{\pi^3}\in\mathbb{Q}$? [duplicate]

Is it known if $\frac{\zeta(3)}{\pi^3}\in\mathbb{Q}$? It is obvious that $\frac{\zeta(2n)}{\pi^{2n}}\in\mathbb{Q}$, but since there is no closed form for the odd values, are we left to be unable ...
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1answer
47 views

The theory of riemann zeta function titchmarsh page 15 question in the proof of the functional equation

I am currently reading Titchmarsh's book about the Riemann Zeta function and came across a problem in a proof of the functional equation that I cannot solve. To be precise, I am referring to this ...
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1answer
107 views

The solution to $\sum_{n=1}^\infty\frac1{n^3}$ in a form not considered closed form?

Is there a solution to $\sum_{n=1}^\infty\frac1{n^3}$ that isn't in the standard closed form? I was wondering if a looser definition of "solution" could allow someone to solve this or if the solution ...
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2answers
54 views

Significance of the Riemann hypothesis to algebraic number theory?

Of course, the truth of the Riemann hypothesis is a central question in analytic number theory. Does its truth/falsehood have important consequences in purely algebraic number theory as well? ...
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15answers
26k views

Why does $1+2+3+\cdots = -\frac{1}{12}$?

$\displaystyle\sum_{n=1}^\infty \frac{1}{n^s}$ only converges to $\zeta(s)$ if $\text{Re}(s) > 1$. Why should analytically continuing to $\zeta(-1)$ give the right answer?
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A generalization of an integral related with $\zeta(2)$

It is pretty well-known (and not difficult to prove) that: $$ \int_{0}^{+\infty}\frac{x}{e^{x}-1}\,dx = \zeta(2) = \sum_{n\geq 1}\frac{1}{n^2} \tag{1}$$ but what is known about $$ I_2 = ...
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2answers
636 views

A new formula for Apery's constant and other zeta(s)?

I recently found these Plouffe-like formulas using Mathematica's LatticeReduce. Has anybody seen/can prove these are indeed true? $$\begin{aligned}\frac{3}{2}\,\zeta(3) &= ...
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2answers
54 views

Analytic continuation of Riemann zeta-function

I showed that $$\zeta(s)=\frac1{1-2^{1-s}}\sum_{n=1}^\infty (-1)^{n-1}\frac1{n^s}.$$ Right hand side should be analytic when Re$(s)>0$ and $s\neq 1$, but there are problems at points ...
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2answers
110 views

An integral related with the Riemann $\zeta$ function

I have to prove that: $$ \forall s>1,\qquad\int_0^\infty \sum_{k=1}^{\infty}\frac{1}{(k^s+1)^x+k^s}dx=\zeta(s). $$ I how do I find the closed form for this sum? $$ ...
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1answer
49 views

Riemann Zeta Function for $\Re(s)=0$

All the sources I have read talk about continuation from $Re(s)>1$ to $Re(s)>0$ then $Re(s)<0$ $(s\neq 1)$. What about $Re(s)=0$? Where does that go?
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1answer
42 views

Difference between Eulers product and Zeta Function at a finite values

So a very important formula proven by Euler is that is equal to Of course these formulas give you the same value when they reach infinity, but my question is that say $s=1$. What would be the ...
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2answers
5k views

Show how to calculate the Riemann zeta function for the first non-trivial zero

I have very little understanding on how complex functions work but was wondering if someone could show what the summation of the zeta function simplifies to when $s$ is the first non-trivial zero of ...
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3answers
844 views

Sum of divergent series

I saw a lot of article in Math SE like Why does 1+2+3+⋯=−1/12? and S=1+10+100+100+10000+…=−1/9? How is that and lot of others. Also I saw this one of Ramanujan summation but I do not get the ...
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0answers
22 views

Generalisation of the sum operator for the divergent geometric series [duplicate]

I define a generalization of the sum operator as a linear operator from $C^N$ to C that matches with the already known operators (like the zeta regularization). With this technique, one can calculate ...
3
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1answer
59 views

Show that $\zeta'(0)=-\frac{1}{2}\ln(2\pi)$

I started with the functional equation which was derived in class, $$ \zeta(s)=2^s\pi^{s-1}\sin(\frac{\pi s}{2})\Gamma(1-s)\zeta(1-s) $$ and took the logarithmic derivative of both sides to get $$ ...
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0answers
17 views

Show that $(1-2^{1-z})\zeta(z)=\sum\limits_{s=1}^\infty \frac{(-1)^s}{s^z}=$ for Re$(z)>0$. [duplicate]

Show that $(1-2^{1-z})\zeta(z)=\sum\limits_{s=1}^\infty \frac{(-1)^s}{s^z}=\frac{1}{\Gamma(z)}\int\limits_{0}^\infty \frac{t^{z-1}}{e^t+1}dt$ for Re$(z)>0$. Not sure how to get started on this, we ...
16
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4answers
5k views

Factorial of infinity

So, I've read in this article that: $$\zeta'(0) = \log\sqrt\frac{1}{2\pi}$$ And that: $$e^{-\zeta'(0)} = 1\cdot2\cdot3\cdot\ldots\cdot\infty = \infty! = \sqrt{2\pi}$$ I found this result very ...
3
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2answers
89 views

Prove that the value of the constant $C$ must be $1$

After proving the prime number theorem in class, our professor directs us to a remark by Lagrange that for large values of $x$, $\pi(x)$ is approximately equal to $$ \frac{x}{\log x - B}. $$ (This is ...
3
votes
4answers
249 views

regularization of sum $n \ln(n)$

I was testing out a few summation using my previous descriped methodes when i found an error in my reasoning. I'm really hoping someone could help me out. The function which i was evaluating was ...