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

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Difference between Meromorphic and Analytic Continuation

We have that $\frac{X}{spec \mathbb{Z}}$, a scheme of finite type. Consider $\zeta(X,s)= \prod_{x\in{X}} \frac{1}{(1-Nx^{-s}}$, with $Nx$ the norm. I didn't catch what my teacher was saying and he is ...
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133 views

Information about Riemann Zeta function

I have general question on Riemann Zeta function. How can I improve knowledge on Riemann Zeta Function theory up to research? For example , what are the best books on Zeta ...
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54 views

Differential equation to model a pseudo-random behavior?

In the paper http://arxiv.org/abs/1401.3620, "The zeros of the Riemann zeta-function and the transition from pseudo-random to harmonic behavior", the author built a function based on a finite amount ...
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Does the Riemann zeta function tell us about the order theoretic properties of the natural numbers?

The classical Möbius function $\mu(n)$ fulfills the multiplicative inversion formula, e.g. see this thread. Now I see in the theory of posets, they generalize the concept of that function, see ...
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Zeta function universality: How to compute the shift parameter for simple functions?

I've come across Zeta function universality. For a nice function $f$ in a nice subset $U$ of the complex strip between real $0$ and $1$, one can find a real $t$, such the zeta function $\zeta$ shifted ...
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Ways to calculate $\int_0^1 \frac{-\log x}{1+x}\ \mathrm dx$

I came across the integral $$ \int_0^1 \frac{-\log x}{1+x}\ \mathrm dx = \frac{\pi^2}{12}, $$ which can be calculated as $\frac 1 2 \zeta(2)$ using analytic number theory. I'm interested if this ...
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$1 + 1 + 1 +\cdots = -\frac{1}{2}$ [closed]

The formal series $$ \sum_{n=1}^\infty 1 = 1+1+1+\dots=-\frac{1}{2} $$ comes from the analytical continuation of the Riemann zeta function $\zeta (s)$ at $s=0$ and it is used in String Theory. I am ...
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Are the zeros of the sum/difference of two reflexive, entire functions all on the line $\Re(s)=\frac12$?

Remove the order $1$ pole of $\zeta(s)$ at $s=1$, to create the following entire function: $$z(s):=\zeta(s)-\dfrac{1}{s-1}$$ I like to conjecture that all complex zeros of $z(s) \pm z(1-s)$ in the ...
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187 views

Calculating $\zeta(0)$ by the residue of $\zeta(1)$

$$\begin {aligned}\pi^{-s/2}\Gamma(s/2)\zeta(s)=&\pi^{-(1-s)/2}\Gamma((1-s)/2)\zeta(1-s) \\ \zeta(0) =&\frac{\pi^{-1/2}\Gamma(1/2)\zeta(1)}{\pi^{0}\Gamma(0)}=\frac{\zeta(1)}{\Gamma(0)}\end ...
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A connection between a sequence from the Collatz conjecture and a sequence of densities from $\zeta(k)-1$?

Just for grins, I created lists of first-entries of finite sequences of rank $r$ for the Syracuse problem (Collatz conjecture using only odd numbers) and found these sequences on OEIS. My sequences, ...
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91 views

Series involving the Riemann zeta function

Consider the series: $$\sum_{n=1}^{\infty}\frac{\zeta(2n+1)}{n(2n+1)}$$ We can easily prove that it's a convergent series. My question, is there a way to express this series in terms of zeta ...
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257 views

Integration method for $\int_0^\infty\frac{x}{(e^x-1)(x^2+(2\pi)^2)^2}dx=\frac{1}{96} - \frac{3}{32\pi^2}.$

The following definite integral is obtained directly from Hermite's integral representation of the Hurwitz zeta function. But is it possible to obtain the same result via the residue calculus or ...
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135 views

Closed form for $\sum_{k=1}^{\infty} \zeta(2k)-\zeta(2k+1)$

From WolframAlpha it seems that $$ \frac{1}{2}=\sum_{k=1}^{\infty} \zeta(2k)-\zeta(2k+1) $$ Could someone provide a proof for this? Thanks.
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1answer
150 views

Have all the zeros of the Riemann Zeta function real part smaller than 1?

I think that all the zeros of the Riemann-Zeta function ${\zeta}( z ) = \frac{1}{1-2^{1-z}} \sum_{n = 0}^{\infty} \frac{1}{2^{n+1}} \sum_{k = 0}^{n} (-1)^k \binom{n}{k} (k+1)^{-z}$ have real part ...
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251 views

Does $\zeta(-1)=-1/12$ or $\zeta(-1) \to -1/12$? [duplicate]

I saw NumberPhile channel on Youtube, and they proved $1+2+3+\cdots=-1/12$. Also, I read This. So, which one is correct $$\zeta(-1)=-1/12\\ \text{or} \\\zeta(-1) \to -1/12$$ Equivalent to: ...
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Does $\sum _{n=1}^{\infty } \frac{\sin(\text{ln}(n))}{n}$ converge?

Does $\sum _{n=1}^{\infty } \dfrac{\sin(\text{ln}(n))}{n}$ converge? My hypothesis is that it doesn't , but I don't know how to prove it. $ζ(1+i)$ does not converge but it doesn't solve problem here. ...
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113 views

Computing Zeta(2)

It is well known that $$\int_0^1\frac{\log{x}}{1 - x}\,\mathrm{d}x = -\frac{\pi^2}{6} $$ This is generally proved by expanding the geometric series and then using $\zeta(2) = \pi^2/6$. My question ...
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Can you prove that the $13$ first zeros of $(r\zeta(r+\Im i))^2$ have real part $r=\frac{1}{2}$, assuming LeClaire's approximation?

Can you prove using double series reversion that the $13$ first zeros of $(r\zeta(r+\Im i))^2$ have real part $r=\frac{1}{2}$ (as their convergent), with initial guess for the real part $r$ to be ...
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135 views

Value of Riemann zeta function at $-1$

This claim is false $\sum_{n=1}^{\infty}n=\sum_{n=1}^{\infty}n^{-(-1)}= \zeta(-1)=-1/12$. The error is that we should $\sum_{n=1}^{\infty}n=\sum_{n=1}^{\infty}(1/n ^1)^{-1}=(0)^{-1}$. Am I correct? ...
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180 views

what the RH equivalent for Riemann prime formula $\Pi(x)$?

Question follow the one answered already, zeros about Riemann Zeta function and some L-function Let's me try my best to make it clear on what I am asking. In his 1859 paper "On the Number of Primes ...
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483 views

Formula for $\zeta(3)$ -verification

By simple manipulating with some series I have found the following formula for $\zeta(3)$: $$\zeta(3)=\frac27\sum_{k=0}^{\infty}(-1)^kB_{2k}\frac{\pi^{2k+2}}{(2k+2)!},$$ where $b_k$ are Bernoulli ...
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125 views

Closed form for $\sum\limits_{k=1}^{\infty}\zeta(4k-2)-\zeta(4k)$

I am looking for a closed form of the expression $$ \sum_{k=1}^{\infty}\zeta(4k-2)-\zeta(4k) $$ Closed form would be something in terms of constants such as $\pi$, $\gamma$, $e$, etc.
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138 views

Is this closed form expression of $\zeta(3)$ correct?

In this paper on pages 6 & 7. Page 6 lists the variables used in the equation on page 7. The author claims a closed form expression of $\zeta(3)$ (he also goes on to claim a closed form ...
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82 views

Is there a name for this constant?

$$\prod_{n=2}^\infty \zeta(n)=2.294856591673313794183$$ Is there a name for this constant and what are some if its properties?
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355 views

Proving that $\frac{\pi^{3}}{32}=1-\sum_{k=1}^{\infty}\frac{2k(2k+1)\zeta(2k+2)}{4^{2k+2}}$

After numerical analysis it seems that $$ \frac{\pi^{3}}{32}=1-\sum_{k=1}^{\infty}\frac{2k(2k+1)\zeta(2k+2)}{4^{2k+2}} $$ Could someone prove the validity of such identity?
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Proving that $\left(\frac{\pi}{2}\right)^{2}=1+\sum_{k=1}^{\infty}\frac{(2k-1)\zeta(2k)}{2^{2k-1}}$.

Wolfram$\alpha$ says that we have the following identity $$ \left(\frac{\pi}{2}\right)^{2}=1+2\sum_{k=1}^{\infty}\frac{(2k-1)\zeta(2k)}{2^{2k}} $$ but, how does one prove such identity?
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Unconventional way, how to expand to Maclaurin series

Let's have function $f$ defined by: $$f(x)=2\sum_{k=1}^{\infty}\frac{e^{kx}}{k^3}-x\sum_{k=1}^{\infty}\frac{e^{kx}}{k^2},\quad x\in(-2\pi,0\,\rangle$$ My question: Can somebody expand it into a ...
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201 views

Evaluating $\sum_{n=1}^\infty\frac{\zeta(3n)}{2^{3n}}$

I want to find $$\sum_{n=1}^\infty\frac{\zeta(3n)}{2^{3n}}$$ I let $f(z)=\sum_{n=1}^\infty\frac{1}{2^{3n}}z^{3n}$ and now $$\sum_{n=1}^\infty ...
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140 views

$(1-2^{1-s})\zeta(s)$ is an entire function

Show that $(1-2^{1-s})\zeta(s)$ is an entire function, which is represented by the series $$(1-2^{1-s})\zeta(s)=1-\dfrac{1}{2^s}+\dfrac{1}{3^s}-\dfrac{1}{4^s}+\cdots$$ for $\Re{s}>1$. From the ...
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208 views

Prove $\sum_{n=1}^\infty\frac{H_{2n+1}}{n^2}=\frac{11}{4}\zeta(3)+\zeta(2)+4\log(2)-4$

How can I prove that $$\sum_{n=1}^\infty\frac{H_{2n+1}}{n^2}=\frac{11}{4}\zeta(3)+\zeta(2)+4\log(2)-4$$ I think this post can help me, but I'm not sure. Mathematica can provide a closed form for ...
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267 views

Closed form of $\int_0^\frac{1}{2}x^n\cot(\pi x)\,dx$

What is the closed form of the following integral $$\int_0^\frac{1}{2}x^n\cot(\pi x)\,dx,n\in\mathbb{N}$$ By Mathematica I saw that $$\int_0^\frac{1}{2}x\cot(\pi x)\,dx=\frac{\log(2)}{2\pi}$$ ...
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56 views

Series divergence - how precise should the answer be

Morning. I've written down some of my reasoning and arguments as to why the series diverges, however I am not certain I can safely conclude it diverges to $\infty$. Would you give it a look, please? ...
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How generalize the alternating Möbius function?

Here is what I want to do, I have this matrix: $$\displaystyle T = \begin{bmatrix} +1&+1&+1&+1&+1&+1&+1 \\ +1&-1&+1&-1&+1&-1&+1 \\ ...
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Limit $(t-1)\zeta(t)$ as $t\rightarrow 1^+$

Show that $\lim_{t\rightarrow 1^+}(t-1)\zeta(t)=1$. For $t>1$, we can use the definition $\zeta(t)=\sum_{n=1}^\infty \dfrac{1}{n^t}$, so it is approximately $\int_1^\infty \dfrac{1}{x^t}dx$. ...
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1answer
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Reflections of zeros of zeta function in the critical strip

Show that if $a$ is a zero of the zeta function in the critical strip, then so are $\bar{a}$, $1-a$, and $1-\bar{a}$. The definition of $\zeta$ is ...
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Does the funcitonal equation of the zeta function apply for all reals?

If I plug in 4 for s into: $$\begin{equation} \zeta(s)=2^s\pi^{s-1}\Gamma(1-s)\sin\left(\frac{\pi s}{2}\right)\zeta(1-s). \end{equation}$$ Doesn't $$\zeta(4) = 0 $$ because of the sine function? ...
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How to prove that there are $O(T\ln T)$ zeros in the critical strip of the Riemann zeta function?

Define $F(T)$ as the number of solutions to $\zeta(a+ ti) =0$ for $0\le t\le T$ and $0<a<1$. How to show that $F(T)= O(T\ln T)$? For clarity, $\zeta$ is the Riemann zeta function, $i$ is the ...
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The multiplication formula for the Hurwitz/generalized Riemann zeta function

I'm having a difficult time showing that $$ \displaystyle \zeta(s,mz) = \frac{1}{m^{s}} \sum_{k=0}^{m-1} \zeta \left(s,z+\frac{k}{m} \right) $$ A couple of authors referred to it as an obvious fact. ...
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69 views

How does one derive this formula $\zeta(-n) =\frac{B_{n+1}}{n+1}$?

$$\zeta(-n) =-\frac{B_{n+1}}{n+1}$$ What is the motivation or the derivation of this formula?
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Dirichlet series minus Riemann zeta

Suppose $\{a_n\}$ is a sequence of complex numbers such that the sums $A_n=a_1+\cdots+a_n$ satisfy $$|A_n-nb|\leq Cn^{\sigma}$$ for all $n$, where $b\in\mathbb{C},C>0,0\leq\sigma<1$. Consider ...
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What does $\lim_{\epsilon\to 0} \frac{\zeta(1+\epsilon) + \zeta(1-\epsilon)}{2} =\gamma$ really mean?

$$\lim_{\epsilon\to 0} \frac{\zeta(1+\epsilon) + \zeta(1-\epsilon)}{2} =\gamma$$ I am somewhat familiar with the zeta function, but have not taken complex analysis, yet. I saw this on Wikipedia and ...
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135 views

Have I found correct formula? $\zeta(3)$

Have I found the correct formula? Or is this only numerical aproximation? $\zeta(3)=\frac{2{\pi}^2}{7}(\ln 2-\frac{4}{15})$ Reedited: I add another aproximation(may be better): ...
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1answer
209 views

Analytic continuation of the Riemann zeta function using contour integration

To find the analytic continuation of the Riemann zeta function using contour integration one can integrate $\displaystyle f(z) = \frac{z^{s-1}}{e^{-z}-1}$ around a contour that consists of rays just ...
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58 views

Is this formula for $\zeta(2n+1)$ correct or am I making a mistake somewhere?

I am calculating $\zeta(3)$ from this formula: $$\zeta(2n+1)=\frac{1}{(2n)!}\int_0^{\infty} \frac{t^{2n}}{e^t -1}dt$$ From Grapher.app, I get $\int_0^{\infty} \frac{x^{2}}{e^x -1}dx = .4318$ ...
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61 views

Fox H-function and the Riemnann zeta function

What is the link between the Fox's H-function and the Riemnann zeta function or the polylogarithmic function? PS: I would be glad if someone could provide me references about this gadget ...
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429 views

Different methods to prove $\zeta(s)=2^s\pi^{s-1}\sin\left(\frac{s\pi}{2}\right) \Gamma (1-s) \zeta (1-s)$.

I've recently encountered this strangely attractive equation (Riemann's functional equation), along with Riemann's original proof. $$\displaystyle\zeta(s)=2^s\pi^{s-1}\sin\left(\frac{s\pi}{2}\right) ...
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1answer
197 views

Serie for $\text{Arg}( \zeta (z))$ [closed]

Though I don't know if the formula I've found is useful, I decided to publish it anyway. $$ \text{Arg}( \zeta (z)) = -\sum_ {k = 1}^{\infty}\sum _ {q = 1}^{\infty}\frac {1} {k P_q^{k x}}\text {Sin}( k ...
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1answer
83 views

Sum of reciprocal of zeroes of zeta function

The sum of reciprocal of zeroes of riemann zeta function converges conditionally that if they are paired as $\rho $ and $1-\rho$ My question is if the sum still converges if they are paired as $\rho$ ...
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1answer
104 views

Relation between the prime density and Riemann's zeros?

Soft question: Where is the connection between the zeros of Riemann's $\zeta$-funciton and the density of prime numbers? Is there a short answer to this question, to get the overview? I once had a ...
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1answer
42 views

Can we determine which statements are incomplete due to Godel?

Due to Godel's incompleteness theorems we know that there are true statements in a system that cannot be proven with that system. My questions are 1) can we tell which statements in a system are the ...