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

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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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65 views

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 ...
2
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1answer
67 views

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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2answers
70 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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29 views

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 ...
2
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1answer
59 views

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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2answers
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): ...
3
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1answer
215 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 ...
2
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1answer
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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64 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 ...
8
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1answer
433 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) ...
7
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1answer
199 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 ...
3
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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
109 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 ...
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2answers
94 views

A convergent-everywhere expression for $\zeta(s)$ for all $1\ne s\in\Bbb C$ with an accessible proof

I'm looking for a way to define the Riemann zeta function $\zeta(s)=\sum_{n\in\Bbb N_0}n^{-s}$ on the whole complex plane, without having to use analytic continuation, or perhaps more accurately, in a ...
3
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1answer
184 views

Elementary bound on the Riemann zeta function

I am currently preparing for a course in analytic number theory and I wanted to get a heads start. In my preparation, I came across the following problem: Show that for $|y|\geq 2$, $|\zeta(1+iy)| ...
2
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0answers
72 views

Zeta Zeros and primes / prime powers

The plot $\Re\ x^{Zeta\ Zero} + \Im\ x^{Zeta\ Zero}$ for the first $1000$ Zeta Zeros up to $x = 30$ using the following Mathematica code: ...
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179 views

expressing $\log(\left \lfloor x \right \rfloor!)$ in terms of zeta-zeros

let $\psi(x)$ be the second Chebyshev Function. By the definition of this summatory function, and the fundamental theorem of arithmetic, we have the identity: $$\log(\left \lfloor x \right ...
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Zeta zeros by recurrence of zeta function, but this is useless isn't it?

One more useless question of mine can't do this site any harm. So here we go. The following Mathematica program converges to most of the riemann zeta zeros, by using an approximation as a starting ...
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4answers
174 views

Evaluation of Euler's Constant $\gamma$

Long back I had seen (in some obscure book) a formula to calculate the value of Euler's constant $\gamma$ based on a table of values of Riemann zeta function $\zeta(s)$. I am not able to recall the ...
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3answers
637 views

proving that $\sum_{n=1}^{\infty}\frac{(H_n)^2}{n^3}=\frac{7}{2}\zeta(5)-\zeta(2)\zeta(3)$

Prove that $$\sum_{n=1}^{\infty}\frac{(H_n)^2}{n^3}=\frac{7}{2}\zeta(5)-\zeta(2)\zeta(3)$$ ($H_n=\sum_{k=1}^{n}\frac{1}{k}$)
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1answer
71 views

Evaluating limit $\lim_{m\to{\infty}}\frac{\sum_{k=1}^m\cot^{2n+1}(\frac{k\pi}{2m+1})}{m^{2n+1}}$

How can I prove the following equality? $$\lim_{m\to{\infty}}\frac{\displaystyle\sum_{k=1}^m\cot^{2n+1}\left(\frac{k\pi}{2m+1}\right)}{m^{2n+1}}=\frac{2^{2n+1}\zeta(2n+1)}{\pi^{2n+1}}$$
6
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3answers
225 views

Question about Euler's approach to find $\sum_{n=1}^\infty\frac1{n^2}=\frac{\pi^2}6$

For a freshman calculus project, I used Euler's approach to find $\sum_{n=1}^\infty\frac1{n^2}=\frac{\pi^2}6$, and noted from Wikipedia's explanation that the infinite product representation of ...
5
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2answers
77 views

GCD and the Riemann zeta funtion

I'm completely stuck on this one, as I'm just starting with analytic number theory: How to write $$\sum_{a\in\mathbb{N}}\sum_{b\in\mathbb{N}}\frac{(a,b)}{a^sb^t}$$ in terms of the Riemann zeta ...
7
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1answer
92 views

closed form of $\frac{(-\pi)^m\zeta(m,\frac{x}{\pi})+\pi^m\zeta(m,1-\frac{x}{\pi})}{(-1)^{m}\pi^{2m}}$

I know that following equality is true. $$\sum_{n=-\infty}^{\infty}\frac{1}{(x+n\pi)^m}=\frac{(-\pi)^m\zeta(m,\frac{x}{\pi})+\pi^m\zeta(m,1-\frac{x}{\pi})}{(-1)^{m}\pi^{2m}}$$ But can we find the ...
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26 views

Are the solutions to $\zeta \left(\frac{1}{2}+i n a\right)=a$ the same as fixed points? $n=1,2,3,…$

Are the solutions to $\zeta \left(\frac{1}{2}+i n a\right)=a$ called fixed points? $n=1,2,3,...$ $\zeta$ is Riemann zeta function. Mathematica program for finding $\zeta \left(\frac{1}{2}+i 3 ...
5
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0answers
143 views

Riemann zeta function and Bernoulli function

I encountered the following problem: Show that $$\zeta(2n+1)=\frac{(-1)^{n+1}(2\pi)^{2n+1}}{2(2n+1)!}\int_0^{1}B_{2n+1}(x)\cot({\pi}x)dx$$ where $B_{2n+1}(x)$ is the Bernoulli polynomial. This ...
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2answers
142 views

Proof of my conjecture on closed form of $\int _{0}^{\infty}\frac{x^{a-1}e^{-mbx}}{1-e^{-bx}}$

Let $a$, $b\in \Bbb R^+$ and $m \in \Bbb N$ then My conjectural closed form is $$\int _{0}^{\infty}\frac{x^{a-1}e^{-mbx}}{1-e^{-bx}}\,{\rm d}x = ...
4
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1answer
139 views

Is $\frac{\zeta (m+n)}{\zeta (m)\zeta (n)}$ a rational number for $m,n\ge 2\in\mathbb N$?

Question : Is $$\frac{\zeta (m+n)}{\zeta (m)\zeta (n)}$$ a rational number for $m,n\ge 2\in\mathbb N$ where $\zeta (s)=\sum_{n=1}^{\infty}\frac{1}{n^s}$? Motivation : We know that $$\zeta ...
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0answers
508 views

The log gamma integral $\int_{0}^{z} \log \Gamma (x) \ \mathrm dx$

One way to evaluate $ \displaystyle\int_{0}^{z} \log \Gamma(x) \ \mathrm dx $ is in terms of the Barnes G-function. $$ \int_{0}^{z} \log \Gamma(x) \ \mathrm dx = \frac{z}{2} \log (2 \pi) + ...
6
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1answer
112 views

The use of log in the Mean density of the nontrivial zeros of the Riemann zeta function (part 2)

As part of my MSc I am reviewing a paper. The paper is a review on the statistical distribution of the unfolded zeros (see below) of the Reimann functional equation. In the paper there is a sentence: ...
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1answer
78 views

An upper bound for $-\frac{\zeta'}{\zeta}(s)-\frac{1}{s-1}$

Let $\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}$. We have $\frac{\zeta'}{\zeta}(s) = \sum_{n=1}^\infty \frac{\Lambda(n)}{n^s}$ for $s>1$, where $\Lambda$ stands for the von Mangoldt function ...
6
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2answers
236 views

Can we prove that all zeros of entire function cos(x) are real from the Taylor series expansion of cos(x)?

Q1: Can we prove that all zeros of cos(x) are real from the following Taylor series expansion of cos(x)? $$ \cos(x) = \sum_{n=0}^\infty \frac{(-1)^k}{(2k)!}x^{2k} $$ The Riemann $\xi(z)$ function is ...
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0answers
142 views

Is there a elementary way to prove $\zeta(2)=\frac{\pi^2}{6}$

The proof in the Wikipedia is still much complicated, can any one provide a really simple way to prove this.
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1answer
56 views

How does $\zeta(i\pi)$ converge?

$$\zeta(i\pi) = \sum_{r=1}^{\infty}r^{-i\pi} = \sum_{r=1}^{\infty}e^{-i\pi \ln(r)} = \sum_{r=1}^{\infty}\operatorname{cis}(-\pi\ln(r))$$ Did I mess up somewhere in the steps above? I can't see how ...
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2answers
224 views

A series related to $\zeta (3)$.

I'm not really up to date on the current status of $\zeta (3)$ but I was messing around the other day with Fourier series and found that $$\sum_{n=1}^{\infty} \frac{(-1)^n}{(2n-1)^3} = ...
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1answer
79 views

Closed Forms of Certain Zeta constants?

The Riemann Zeta function $\zeta(s)=\sum_{n=1}^\infty \frac{1}{n^s}$ converges for $Re(s)>1$. I am interested in some particular "irrational " Values of it. Like $\zeta(\pi)=1.176241738...$ ...
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0answers
55 views

Zeros of “nearby” holomorphic functions

I would appreciate help in how to show that On a compact subset of {$z \in \mathbb{C}: 1/2 < \Re (z) < 1$}: "Given a holomorphic function with an isolated zero, any "nearby" holomorphic ...
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0answers
351 views

I can Euler-sum $\sqrt{-\ln(1)}-\sqrt{-\ln(2)}+\sqrt{-\ln(3)}-\cdots$. But how can I do $\sqrt{-\ln(1)}+\sqrt{-\ln(2)}+\sqrt{-\ln(3))}+\cdots$?

This is also related to an older thread in MSE ("what is the half derivative of zeta at zero?") . One of the possible steps in the problem of that thread was to evaluate the series ...
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6answers
967 views

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

‎If $f\left(z \right)=\sum_{n=2}^{\infty}a_{n}z^n$ and $\sum_{n=2}^{\infty}|a_n|$ converges then‎, $$\sum_{n=1}^{\infty}f\left(\frac{1}{n}\right)=\sum_{n=2}^{\infty}a_n\zeta\left(n\right)‎.$$ ‎Since ...
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80 views

How should I prove that Zeta'(x)/Zeta(x)+1/(x-1) is strictly monotonously decreasing on the real line (for x>=0)?

The Riemann Zeta function, like most other complex functions, are much easier to deal with in the real line, since the values are also real, and definitions can be used in a more straightforward ...
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2answers
73 views

How we can calculate this derivative in despite that $ζ(s)$ is defined in the half-plane $α>1$?

The Riemann zeta function is the function of the complex variable $s=α+iβ$, defined in the half-plane $α>1$ by the absolutely convergent series $$ζ(s)=\sum_{n=1}^\infty \frac{1}{n^s}$$ In many ...
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1answer
136 views

Why is the difference between these two infinite series equal to $\frac12$?

A follow up on this MO question: Take $s \in \mathbb{C}$, $\Re s \gt 1$, and the two infinite series: $$Z_1(s) = \sum _{n=1}^{\infty } (-1)^n \left( \frac12 + n \right) \left( {\frac ...
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1answer
103 views

Is there only one analytic continuation of the Riemann zeta function?

If I were to manipulate the zeta function in a 'new way' would I end up with an analytic continuation that is equal to the one know or something completely new for values less than 1 and complex ...
6
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1answer
138 views

Questions regarding the Riemann-Siegel $\theta$ Function

My questions are a request, please, for help in understanding some comments in the wikipedia article discussing the Riemann-Siegel $\theta$ function ...
3
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1answer
66 views

A double sum and its relation to a simple sum, is this an identity for any complex number $S=a+i b$ and any integers n and t?

Does: $$\sum _{m=1}^t \lim_{s\to \text{S}} \, \zeta (s) \sum _{k=(m-1) n+1}^{m n} \frac{1-\text{If}[k \bmod n=0,n,0]}{k^{s-1}}$$ equal: $$\lim_{s\to \text{S}} \, \zeta (s) \sum _{k=1}^{n t} ...
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0answers
105 views

Mandelbrot set and riemann hypothesis

Has anyone tried to make a connection between the Mandelbrot set and the non-trivial zeros the zeta function? Looking at the Mandelbrot set, it appears that all points are to the left of the line 0.5 ...
4
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2answers
239 views

How to Compute $\zeta (0)$?

Ultimately, I am interested in analytically continuing the function $$ \eta _a(s):=\sum _{n=1}^\infty \frac{1}{(n^2+a^2)^s}, $$ where $a$ is a non-negative real number, and calculating $\eta _a$ and ...
2
votes
1answer
71 views

Show in between steps in this Riemann zeta function equivalence/reduciton

In the answer chosen by the OP in this question I had trouble understanding the steps taken to get the equivalences/reduce the zeta function into another one. Can somebody show me the steps to go from ...