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

learn more… | top users | synonyms

4
votes
1answer
111 views

$0=\frac{13+13^2+13^3+\cdots}{1+2+3+\cdots}$ using infinite sums?

This is not homework, just curiosity. My question arose from the apparent absurdity that $\zeta(-1)=-\frac{1}{12}$, even though $\sum_{n=1}^\infty \frac{1}{n^z}$ only makes sense when $Re(z)>1$. ...
2
votes
1answer
97 views

Contradicting statements about the Riemann zeta function at positive odd integers

I have found two contradicting statements about the value of $\zeta(k)$ when $k=2n+1$ and $n\in\mathbb{Z_0^+}$. Which one is correct? "The Riemann zeta function for odd integers has no known ...
1
vote
0answers
33 views

Why the poisson summation formula works

If we put the function $ f(x)= |x|^{s-1} $ inside the Poisson sum formula and consider that $ \sum_{n=1}n^{z-1}= \zeta (1-s) $ then we can easily give a proof of Riemann's functional equation $$ ...
3
votes
3answers
66 views

How to prove that $ \lim_{u \downarrow 1} (u-1) \zeta(u) =1 $?

I would like to prove that $$ \lim_{u \downarrow 1} (u-1) \zeta(u) =1 \quad .$$ However, I am not sure which form of the Riemann-zeta function I ought to pick in order to compute this limit. I ...
3
votes
3answers
147 views

My (divergent) summation of the zetas with sets of cofactors give systematically errors of simple integer differences. What am I missing?

This is a "fiddling" in a small project of mine with which I'm concerned from time to time for three years now. I try to focus on the core of the problem, please ask if more context is needed. ...
3
votes
0answers
38 views

a question related to using Hurwitz theorem to bound the locations of zeros of Riemann zeta function

Let $F(s)=\pi^{-s/2}\Gamma(s/2)\zeta(s)$ be the Gamma-complete version of the Riemann $\zeta$ function. Let $f(z)=F(1/2+i z)$. So it is known that all zeros of $f(z)$ are in the strip $S_{1/2}$ ...
4
votes
1answer
103 views

Proof of $\sum_{n=1,3,5,\ldots}^{\infty}\frac{1}{n^4}=\frac{\pi^4}{96}$

I came across with the infinite series $$\sum_{n=1,3,5,\ldots}^{\infty} \frac{1}{n^4}= \frac{\pi^4}{96}$$ when calculating a problem about an infinite deep square well in quantum mechanics. ...
1
vote
1answer
70 views

Solving $\sum_{n=1}^{\infty} \frac{1}{n^2}$ using the fourier series.

Please do NOT solve the problem, I just need some help, not a full solution. I would like to try this myself. Find $\zeta(2) = \displaystyle \sum_{n=1}^{\infty} \frac{1}{n^2}$ The fourier series for ...
0
votes
0answers
61 views

An inverse Fourier transform of Riemann $\Xi(z)$ function

Riemann $\Xi(z)$ function is related to Riemann $\zeta(s)$ function via ($s=1/2+i z$): $$\Xi(z)=\frac{1}{2}s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\tag{1}$$ The functional equation for $\zeta(s)$ is ...
1
vote
2answers
78 views

Is there a formula for $k\pi ^n$, if $n$ is an odd number and $k$ is a rational number?

I always find the formula for $k\pi ^n$ when $n$ is an even number and $k$ is a rational number, but I did't find for an odd number.
5
votes
0answers
100 views

Value of a Sine-Like Infinite Product

Does the following infinite product have a "nice" closed form? $$ P = \prod_{k=2}^{\infty} \left(\left(1 - \frac{1}{k^2}\right)^\dfrac{(-1)^k}{k}\right) $$ I know that without the power one could ...
0
votes
0answers
40 views

Limit of Riemann Zeta Function as Imaginary Part tends to Infinity

Is it true that $$ \lim_{n\to \infty} \zeta(2+ni) =1 ?$$ If not, what is the value of the limit? What about the same but with other real parts other than 2?
4
votes
2answers
93 views

Looking for a closed form for $\sum_{k=1}^{\infty}\left( \zeta(2k)-\beta(2k)\right)$

For some time I've been playing with this kind of sums, for example I was able to find that $$ \frac{\pi}{2}=1+2\sum_{k=1}^{\infty}\left( \zeta(2k+1)-\beta(2k+1)\right) $$ where $$ ...
3
votes
1answer
76 views

Riemann Zeta Function at $s = 1 + 2 \pi i n / \ln 2$

I am aware that the function defined by $$ \zeta(s) = \frac{1}{1^s} + \frac{1}{2^s} + \frac{1}{3^s} + \cdots $$ for $Re(s)>1$ can be extended to a function defined for $Re(s)>0$ by writing $$ ...
1
vote
1answer
94 views

Why zeta(2) in these inifinite sums?

The infinite sum of the reciprocals of these two sequences have zeta(2) in the result. The value is not in OEIS. A000326 A002411 Edit---rolled back the changes. Both $\frac{1}{2}$ and $2$ are ...
1
vote
0answers
18 views

q question regarding the numerical zero finding of Riemann zeta function and actual proof of Riemann Hypothesis

(1)Suppose that people verified in 2004, that all zeros of $\zeta(\sigma+i t)$ with $0<t<T<=10^{22},0<\sigma<1$ are on the critical line ($\sigma=1/2$). (2)Suppose Bob proved in ...
0
votes
0answers
64 views

How i could show that this inequality true or false: $|\zeta(s)| \leq |2^s \dfrac{1}{1-2^{-s}}|\leq {1}$ for $0< \sigma <1$?

Is this inequality true: $|\zeta(s)| \leq |2^s \dfrac{1}{1-2^{-s}}|\leq {1}$ for $0< \sigma <1$ ? note :$s=\sigma + it$, where $\sigma, t\in \mathbb{R}$. I would be interest for any replies ...
0
votes
0answers
38 views

A Contradiction of Riemann Zeta Residues

We can show (1+2+3...+n)^2 = 1^3 + 2^3 + ... +n^3, which holds for any finite n, shouldn't this imply Z(-1)^2 = Z(-3)? However, this does not hold if we look at the residues of the zeta function ...
1
vote
1answer
31 views

zero distribution of the Fourier kernel $\Phi(u)$ for Riemann $\Xi(z)$ function

Riemann $\Xi(z)$ function is related to Riemann $\zeta(s)$ function via ($s=1/2+i z$): $$\Xi(z)=\frac{1}{2}s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)$$ The functional equation for $\zeta(s)$ is equivalent ...
3
votes
0answers
74 views

How prime numbers are related to special functions?

We know that the Riemann zeta function is defined as $$\zeta(s)=\sum_{n=1}^\infty\frac{1}{n^s},$$ for all $\Re(s)>1$. Because of Euler product formula we also know that $$\zeta(s) = ...
-2
votes
1answer
82 views

the definition of Riemann zeta function

I've read that the Riemann zeta function for $0<s<1$ is defined : $$\lim_{x\rightarrow\infty} \left(\sum_{n \leq x}\frac{1}{n^s}- \frac{x^{1-s}}{1-s}\right)$$ I don't know how to prove that ...
0
votes
1answer
45 views

Plot zeros of partial sum of zeta Riemann with Maple

I want to plot the zeros of the partial sum of the Riemann zeta function with Maple. Some hint?? Thanks!!!
2
votes
1answer
59 views

Zeros of Zeta function and exact roots

Are there exact roots to any of the Zeta zeroes? For example the first one 1/2 +14.134725I, is there a nice looking polynomial that has an exact solution? I would assume if there is an exact value, ...
0
votes
0answers
45 views

Singularities of zeta function

I have to prove (if $\gamma \ne 0$) that there is a analytic continuation for $\Re s >0$ of the function $$f(s)=\frac{\zeta (s)^2 \zeta(s-i\gamma )\zeta(s+i\gamma ) }{\zeta(2s)} $$ and that this ...
1
vote
0answers
69 views

Riemann Zeta Function equation

So I'm a amateur mathematician and I was working with the Riemann Zeta Function and I was able to proof this identity. So I'm just wondering if this has already been proven before. For S>0 So I ...
7
votes
1answer
148 views

An Inequality Invollving The Riemann Zeta Function

I'm having trouble proving the following inequality for $2<r<3$: $$(1+2^{-r})\frac{(3^r+1)^2}{3^{2r}+1}>\frac{\zeta(r)}{\zeta(2r)}.$$ I can easily plot the graph, and the inequality clearly ...
1
vote
1answer
59 views

alternating sum of zeta functions minus one is one half

During my work on a different infinite series I happened to prove that $\displaystyle\sum_{k=2}^{\infty}(-1)^k (\zeta(k)-1)=\frac{1}{2}$ where $ ...
1
vote
1answer
53 views

Riemann Zeta of 1/2 $\zeta(\frac{1}{2})$

This may be a silly question, but I need to figure out how to evaluate the value of $\zeta(\frac{1}{2})$. In wikipedia, it says: $\zeta(1/2) \approx -1.4603545$. I am interested to know how this value ...
0
votes
0answers
78 views

Question about Riemann zeta function + my proof

First let me say that I am 16 years old so I am not very professional in math. English is also a second language so I apologize for any mistakes. Now i have been reading about the Riemann zeta ...
2
votes
2answers
89 views

Does the series $\sum_{n=1}^\infty (-1)^n \frac{\cos(\ln(n))}{n^{\epsilon}},\,\epsilon>0$ converge?

Numerical results showed that the series $$Q=\sum_{n=1}^{\infty} (-1)^n \frac{\cos(\ln(n))}{n^{\epsilon}},\epsilon>0\tag{1}$$ with $\epsilon=10^{-6}$ converged to $-0.53259554096828...$ We can ...
7
votes
2answers
188 views

Does the series $\sum_{n=1}^\infty (-1)^n \frac{\cos(\ln(n))}{\sqrt{n}}$ converge?

Numerical results for $m=1$ to $2000$ showed that the series $$Q(m)=\sum_{n=1}^m (-1)^n \frac{\cos(\ln(n))}{\sqrt{n}}$$ converged to $-0.63986...$ Does the series $$\sum_{n=1}^{\infty} (-1)^n ...
3
votes
1answer
47 views

A question about the convergence of partial products of zeta of one.

Recently I've been toying around with the Totient function and the Prime Number Theorem and came up with the odd result that the following limit $$\lim_{n\to\infty}\frac{\pi(n)m_n}{\phi(m_n)n}$$ ...
2
votes
0answers
46 views

how to prove that an entire function is positive on the real axis

The error function $\mathrm{erf}(x)$ is defined as: $$\mathrm{erf}(x):=\frac {2}{\sqrt{\pi}} \int_{0}^{x} e^{-t^{2}}dt\tag{1}$$ Let us define the following 3 functions: ...
1
vote
2answers
59 views

Asymptotics for zeta zeros?

What are the best known asymptotics for the nth zeta zero (imaginary part)? Is there anything similar to $p_n\sim n\log n$, ie where $\rho$ is in form $\sigma+it$, $t_n\sim\dots?$
0
votes
0answers
57 views

Analytic Continuation of the zeta function

Is the analytic continuation of the Riemann zeta function to the upper half plane unique? I don't know much complex analysis, so I can't see why that is the case.
1
vote
0answers
40 views

Zeta zero sum & reciprocals of prime powers

Below is a plot of $$\dfrac{1}{x}\sum_{n=1}^{75}2\Re\left(\operatorname{Ei}\left(\rho_n\log\left(x\right)\right)\right)$$ where $\rho_n$ is the $n$th zeta zero, with grid lines at primes and prime ...
0
votes
0answers
35 views

Question about the zeros of $\zeta_{H}(s,a) \pm \zeta_{H}(s,1-a)$.

Assume $\zeta_{H}(s,a)$ is the Hurwitz Zeta function. Note that for $a=\frac13,\frac14,\frac16$ the zeros of: $$\zeta_{H}(s,a) + \zeta_{H}(s,1-a)$$ are the same as the non-trivial zeros $\rho$ of ...
1
vote
1answer
57 views

What is the number $A$? And the function $G$?

In one of my screenshots I found the following equality : ...
7
votes
4answers
526 views

Usage of Complex Numbers in the Riemann Hypothesis.

I don't have a very good understanding of the Riemann Hypothesis, however that being said, could someone explain to me why complex numbers are used, instead of just using real numbers? Everything I've ...
4
votes
2answers
132 views

Challenging Infinite summation involving the zeta function [duplicate]

Evaluate: $$\large\sum_{k=1}^{\infty}\left(\zeta{(2)}-\sum_{n=1}^{k}\frac{1}{n^2}\right)^2$$ MY ATTEMPT: Recognizing that $\zeta{(2)}-\sum_{n=1}^{k}\frac{1}{n^2}$ can be written as $ ...
19
votes
3answers
193 views

Integrals of integer powers of dilogarithm function

I'm interested in evaluating integrals of positive integer powers of the dilogarithm function. I'd like to see the general case tackled if possible, or barring that then as many particular cases as ...
8
votes
3answers
200 views

Hard Definite integral involving the Zeta function

Prove that: $$\displaystyle \int_{0}^{1}\frac{1-x}{1-x^{6}}{\ln^4{x}} \ {dx} = \frac{16{{\pi}^{5}}}{243\sqrt[]{{3}}}+\frac{605\zeta(5)}{54} $$ I was able to simplify it a bit by substituting ${y = ...
0
votes
1answer
42 views

Proof that $\zeta'(-2n)=(-1)^n\frac{(2n)!}{2(2\pi)^{2n}}\zeta(2n+1)$.

How would one prove the following statement which I found here, and/or does anyone know of a reference with a proof? $$\zeta'(-2n)=(-1)^n\frac{(2n)!}{2(2\pi)^{2n}}\zeta(2n+1).$$
5
votes
3answers
92 views

How would you show that the Riemann Zeta function, $\zeta(s) < 0$ for $s \in (0,1)$?

How would you show that the Riemann Zeta function, $\zeta(s) < 0$ for $s \in (0,1)$? So far I have that along the critical strip \begin{align} \zeta(s) &= ...
2
votes
2answers
116 views

Is it true that $\int_0^1 \lfloor x^{-1} \rfloor^{-1} x^n dx = \frac{1}{n+1}(\zeta(2)+\zeta(3) + \dots + \zeta(n+2) ) - 1$?

This question is inspired by the formula $$\displaystyle\int_0^1{\left\lfloor{1\over x}\right\rfloor}^{-1}\!\!dx={1\over2^2}+{1\over3^2}+{1\over4^2}+\cdots = \zeta(2)-1,$$ see for instance this ...
23
votes
1answer
340 views

Is this integral $\int_0^1\left(\left\{\frac1x\right\}-\frac12\right)\frac{\log(x)}{x}dx$ equal to zero?

My initial question was to find if this integral $$ \int_0^1 \left(\left\{\frac 1x\right\}-\frac12\right)\frac{\log(x)}{x}dx$$ is convergent or divergent. ($\left\{\frac 1x\right\}$ is the fractional ...
2
votes
0answers
50 views

Intuitive explanation for $\zeta (2)=\frac{\pi^2}{6}$ [duplicate]

Using $f(x)=x^2$ Fourier' series, the proof for $\zeta (2)=\frac{\pi^2}{6}$ is pretty straight forward. I'm wondering if there is a more intuitive explanation for the equality, one that a layman could ...
7
votes
3answers
193 views

Upper and lower bounds for the smallest zero of a function

The function $G_m(x)$ is what I encountered during my search for approximates of Riemann $\zeta$ function: $$f_n(x)=n^2 x\left(2\pi n^2 x-3 \right)\exp\left(-\pi n^2 x\right)\text{, ...
0
votes
1answer
41 views

Convergence of $\sum_{n=1}^{\infty} n$ and integral test [duplicate]

I have read online, that one can show that $\sum_{n=1}^{\infty}n = -\frac{1}{12}$. But isn't this a Riemann Series of the form $\sum_{n=1}^{\infty} \frac{1}{n^p}$, where $p=-1$. And if so, can't ...
10
votes
3answers
264 views

How to evaluate a certain definite integral: $\int_{0}^{\infty}\frac{\log(x)}{e^{x}+1}dx$

How can I show that: $$\int_{0}^{\infty}\frac{\log(x)}{e^{x}+1}dx=-\frac{\log^{2}(2)}{2}$$ EDIT: This is equivalent to showing that $\eta'(1)=-\ln2\gamma-\dfrac{\ln^2(2)}{2}$.