{\displaystyle \operatorname {Re} (s)>1}
.The Riemann zeta function plays a pivotal role in analytic number theory and has applications in physics, probability theory, and applied statistics.
Leonhard Euler first introduced and studied the function in the first half of the eighteenth century, using only real numbers, as complex analysis was not available at the time. Bernhard Riemann's 1859 article "On the Number of Primes Less Than a Given Magnitude" extended the Euler definition to a complex variable, proved its meromorphic continuation and functional equation, and established a relation between its zeros and the distribution of prime numbers.The values of the Riemann zeta function at even positive integers were computed by Euler. The first of them, ζ(2), provides a solution to the Basel problem. In 1979 Roger Apéry proved the irrationality of ζ(3). The values at negative integer points, also found by Euler, are rational numbers and play an important role in the theory of modular forms. Many generalizations of the Riemann zeta function, such as Dirichlet series, Dirichlet L-functions and L-functions, are known.
This is an Argand diagram showing the first 40,000 terms of the series form of the Riemann Zeta function, for the argument ##\sigma + i t = 1/2 + 62854.13 \thinspace i##
The blue lines are the first 100 (or so) terms, and the rest of the terms are in red. The plot shows a kind of approximate...
In several places, for example https://xxx.lanl.gov/pdf/chao-dyn/9406003v1, it is claimed that the Riemann zeta function is a fractal under the assumption of a positive result for the Riemann Hypothesis, because
(1) the Voronin Universality Theorem, and
(2) if the RH is true, then the zeta...
In the last part of https://en.wikipedia.org/wiki/Riemann_zeta_function#Mellin-type_integrals, I read two expressions of Riemann's zeta function ζ(s) in terms of s and of integrals of the prime-counting function π(x) (the second one using Riemann's prime-counting function J(x) from which, the...
Homework Statement
Question
Use the functional equation to show that for :
a) ##k \in Z^+ ## that ## \zeta (-2k)=0##
b) Use the functional equation and the euler product to show that these are the only zeros of ##\zeta(s) ## for ##Re(s)<0## . And conclude that the other zeros are all located...
By definition, the characteristic of a field is the smallest number of times one must use the ring's multiplicative identity element (1) in a sum to get the additive identity element (0). Can we use the same rule for the set of natural numbers?
If yes, I found a problem, that has something to...
In a recent article by BBM in Physical Review Letters highlights another approach to link QM to Zeta to Prove R.H. There approach proved unsuccessful. I want to ask professional Physicists if the following new approach have merit in connecting the Zeta function to QM? This new line of attack...
"Physicists are attempting to map the distribution of the prime numbers to the energy levels of a particular quantum system."
https://www.quantamagazine.org/20170404-quantum-physicists-attack-the-riemann-hypothesis/
1. Homework Statement
I want to show that the non-trival zeros of the Riemann Zeta function all lie in the critical strip ## 0 < Re(s) < 1## and further to this that they are symmetric about the line ##Re(s)= 1/2 ##
where ## \zeta(s) = \sum\limits^{\infty}_{n=1}n^{-s}##
With the functional...
Homework Statement
## g(s) = \sum\limits^{\infty}_{n=1} 1/n^{-s}, ##
Show that ##g(s)## converges uniformly for ## Re(s>1) ##
Homework Equations
Okay, so I think the right thing to look at is the Weistrass M test. This tells me that if I can find a ##M_{n}##, a real number, such that for...
There are two forms of Riemann functional equation. One is more symmetric and follows from the other and the duplication theorem of the Gamma function. At least, that's been claimed here...
Hi
It's just that last step I'm not getting, so you have:
[1 / Kz] - [1 / (2K)z]
= [ (2K)z - Kz ] / [(2K)z * Kz]
= [ (2)z - 1 ] / [(2K)z*]
Then what?
Thanks
Hi
I was wondering if there any observations that have only been described using the Riemann Zeta function? What would it mean in physics to assign a divergent series a finite value?
Thank you
Edit
Sorry I overlooked a thread just posted that asked about this so this might need to be deleted.
I understand how to calculate values of positive values ζ(s), it's pretty straightforward convergence. But when you expand s into the complex plane, like ζ(δ+bi), how do you assign a value with i as an exponent? Take for example
ζ(1/2 + i)
This is the sequence
1/1^(1/2+i) + 1/2^(1/2+i) +...
The Euler-Maclaurin summation formula and the Riemann zeta function
The Euler-Maclaurin summation formula states that if $f(x)$ has $(2p+1)$ continuous derivatives on the interval $[m,n]$ (where $m$ and $n$ are natural numbers), then
$$ \sum_{k=m}^{n-1} f(k) = \int_{m}^{n} f(x) \ dx -...
Homework Statement
I wish to prove that for s>1
$$
\sum\limits_{n=1}^{\infty}\frac{\mu(n)}{n^s}=\prod_{p}(1-p^{-s})=\frac{1}{\zeta(s)}.
$$
The Attempt at a Solution
(1) I first showed that
$$
\prod_{p}(1-p^{-s})=\frac{1}{\zeta(s)}.
$$
It was a given theorem in the text that
$$...
Recently some interesting material about the Riemann Zeta Function appeared on MHB and I also contributed in the post... http://mathhelpboards.com/challenge-questions-puzzles-28/simplifying-quotient-7235.html#post33008
... where has been obtained the expression...
$\displaystyle \zeta (s) =...
Here is another integral representation of $\zeta(s)$ that is valid for all complex values of $s$.
It's similar to the first one, but a bit harder to derive.$ \displaystyle \zeta(s) = 2 \int_{0}^{\infty} \frac{\sin (s \arctan t)}{(1+t^{2})^{s/2} (e^{2 \pi t} - 1)} \ dt + \frac{1}{2} +...
Show that $\displaystyle \zeta(s) = \frac{2^{s-1}}{1-2^{1-s}} \int_{0}^{\infty} \frac{\cos (s \arctan t)}{(1+t^{2})^{s/2} \cosh \left( \frac{\pi t}{2} \right)} \ dt $The cool thing about this representation is that it is valid for all complex values of $s$ excluding $s=1$.
This integral is...
I saw a picture of what it might look like when I was researching it, but I'm confused about something. The picture's caption said that the complex coordinates were darkened as their value got larger, leading to a helpful graph, but I do not understand what scale they used. For my program, I...
I was looking at the Wolfram Alpha page on the Riemann Zeta Function Zeros which can be found here, http://mathworld.wolfram.com/RiemannZetaFunctionZeros.html
At the top of the pag there are three graphs each with what looks to be a hole through the graph. Now I know the graph is an Argand...
Hi, I was playing with Riemann zeta function on mathematica. I encountered with a quite interesting result. I iterated Riemann zeta function for zero. (e.g Zeta...[Zeta[Zeta[0]]]...] It converges into a specific number which is -0.295905. Also for any negative values of Zeta function, iteration...
Hi. I just came back from my differential equation midterm and was surprised to see a problem with the Riemann-zeta equations on it. I think the problem went something like
"Prove that \pi/6 = 1 + (1/2)^2 + (1/3)^2 + ... "
The study guide did mention that "prepare for a problem or two...
So Euler derived the analytic expression for the even integers of the Riemann Zeta Function. I was wondering if there is a link to his derivation somewhere?
Also, is there anyone else who used a different method to get the same answer as Euler?
Thank you
Let p_n be number of Non-Isomorphic Abelian Groups by order n. For R(s)>1 with \zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^s} we define Riemann zeta function. Fundamental theorem of arithmetic is biconditional with fact that \zeta(s)=\prod_{p} (1-p^{-s})^{-1} for R(s)>1. Proove that for R(s)>1 is...
Hello, I have read in many articles that the trivial zeros of the Riemann zeta function are only the negative even integers (-2, -4, -6, -8, -10, ...).
The reason why these are the only ones is that when substituting them in the functional equation, the function is 0 because...
(1) Let s be a complex number like s = a + b i, then we define \zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}
Our aim:
to compute ζ(\frac{1}{2}+14.1347 i) with the help of the programming language Aribas
(2) Web Links
Aribas...
Given that
\zeta (2n)=\frac{{\pi}^{2n}}{m}
Then how do you find m with respect to n where n is a natural number.
For
n=1, m=6
n=2, m=90
n=3, m=945
n=4, m=9450
n=5, m=93555
n=6, m=\frac{638512875}{691}
n=7, m=\frac{18243225}{2}
n=8, m=\frac{325641566250}{3617}
n=9...
I was playing around with an infinite series recently and I noticed something peculiar, I was hoping somebody could clarify something for me.
Suppose we have an infinite series of the form:
\sum^_{n = 1}^{\infty} 1/n^\phi
where \phi is some even natural number, it appears that it is always...
I'm trying to evaluate the derivative of the Riemann zeta function at the origin, \zeta'(0), starting from its integral representation
\zeta(s)=\frac{1}{\Gamma(s)}\int_0^\infty t^{s-1}\frac{1}{e^t-1}.
I don't want to use a symbolic algebra system like Mathematica or Maple.
I am able to...
After reading about the Riemann Zeta Function on Wolfram Alpha (http://mathworld.wolfram.com/RiemannZetaFunction.html), it's still unclear to me how the Euler product formula is essentially equal to the limit of a p-series.
Someone please enlighten me
I was wondering how do you calculate the Riemann value, of a Riemann Zeta Function, for example the riemann zeta function for n = 0, is -1/2, which envolves a bernoulli number (what is a bernoulli number and what roll does it play in the Riemann Zeta Function), can anyone explain that to me...
Could anyone tell me what is the Riemann zeta function. On Wikipedia , the definition has been given for values with real part > 1 , as :
Sum ( 1 / ( n^-s) ) as n varies from 1 to infinity.
but what is the definition for other values of s ? It is mentioned that the zeta function is the...
Clearly I am missing something obvious here, but how is it that negative even numbers are zeros of the Riemann zeta function?
For example:
\zeta (-2)=1+\frac{1}{2^{-2}}+\frac{1}{3^{-2}}+...=1+4+9+..
Which is clearly not zero. What is it that I am doing wrong?
Hi:
____________________________________________________________________
Added Nov.3, 2009
(For anyone who can't read the formula below (probably everyone) and who
might have an interest in the subject: - the derivation of two simple equations
that locate all the zeros of the zeta...
Homework Statement
Using method of Euler, calculate \zeta(4), the Riemann Zeta function of 4th order.
Homework Equations
\zeta(s)=\sum_{n=1}^\infty \frac{1}{n^s}
Finding \zeta(2):
\zeta(2)=\sum_{n=1}^\infty...
\zeta (s)= \frac{1}{(1-2^{1-s})} \sum_{n=0}^{\infty} \frac {1}{(2^{n+1})} \sum_{k=0}^{n}(-1)^k{n \choose k}(k+1)^{-s}
Is the main problem with trying to prove the hypothesis algebraically boil down to the fact that s is an exponent to a "k" term? Would a derivation of the function that had...
I still don't understand a few things.
Let's say we had a non-trivial zero counting function, Z_n(n), for the riemann zeta function. Couldn't we fairly easily prove the riemann hypothesis by evaluating \zeta (\sigma+iZ_n), solving for \sigma , then proving it for all n using induction...
Hello I plan on applying to the university of waterloo next year and due to the fact that many of my marks are not that great (failed gr 10 math) I decided to start a site to showcase my ability in math and programing.
For those of you who are interested I wrote a program to graph regions of...
I was wondering if anyone could tell me more about the Riemann Zeta function, esp at negative values. Especially when \sum_{n=1}^{\infty}n= \frac{-1}{12} R where R is the Ramanujan Summation Operator. Could anyone post a proof?
"Riemann zeta function"...generalization..
Hello my question is if we define the "generalized" Riemann zeta function:
\zeta(x,s,h)= \sum_{n=0}^{\infty}(x+nh)^{-s}
which is equal to the usual "Riemann zeta function" if we set h=1, x=0 ,then my question is if we can extend the definition...
They claim that the trivial 0s (zeta(z)=0) occur when z is a negative odd integer (with no imaginary component). But it seems obviously wrong.
Take z=-2
zeta(-2)=1+1/(2^-2)+1/(3^-2)...
=1+4+9...
Obviously this series will not equal 0.
Where have I gone wrong?
Have I misunderstood...
So it is well-known that for n=2,3,... the following equation holds
\zeta(n)=\int_{x_{n}=0}^{1}\int_{x_{n-1}=0}^{1}\cdot\cdot\cdot\int_{x_{1}=0}^{1}\left(\frac{1}{1-\prod_{k=1}^{n}x_{k}}\right)dx_{1}\cdot\cdot\cdot dx_{n-1}dx_{n}
My question is how can this relation be extended to...
I have read what MathWorld has to offer on this and I am extremely confused. Could someone please explain this as simply as possible? Or then again maybe MathWorld already did that. Also, why is this function so important?
Many thanks,
Jameson