What is Riemann zeta function: Definition and 48 Discussions

The Riemann zeta function or Euler–Riemann zeta function, ζ(s), is a mathematical function of a complex variable s, and can be expressed as:




ζ
(
s
)
=



n
=
1






n


s


=


1

1

s




+


1

2

s




+


1

3

s




+



{\displaystyle \zeta (s)=\sum _{n=1}^{\infty }n^{-s}={\frac {1}{1^{s}}}+{\frac {1}{2^{s}}}+{\frac {1}{3^{s}}}+\cdots }
, if



Re

(
s
)
>
1


{\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.

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  1. S

    I Geometry of series terms of the Riemann Zeta Function

    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...
  2. nomadreid

    I Hausdorff dimension of Riemann zeta function assuming RH

    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...
  3. nomadreid

    I No Way to Solve π(x) from Riemann's Zeta Function?

    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...
  4. binbagsss

    Zeros of Riemann zeta function, functional equation and Euler product

    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...
  5. Cathr

    A What if the (semi) field characteristic of N is not zero?

    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...
  6. Jason C

    Can Zeta ζ(½+it) be interpreted as a Wave function?

    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...
  7. jedishrfu

    I Quantum Mechanics does Riemann Zeta Function

    "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/
  8. D

    A Paper About the Riemann Zeta Function

    What do you think of the following paper about the Riemann Zeta Function? http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.118.130201
  9. binbagsss

    Riemann Zeta Function shows non-trival zeros critical-strip symmetry

    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...
  10. binbagsss

    Riemann Zeta Function showing converges uniformly for s>1

    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...
  11. A

    B Can Riemann zeta function be written as ##f(s)=u(s)+iv(s)##?

    I don't recall that I have seen Riemann zeta function put in the form of ##f(s)=u(s)+iv(s)##.
  12. L

    Functional equation Riemann Zeta function

    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...
  13. T

    Simple Riemann zeta function algebra help

    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
  14. V

    Physical applications of Riemann zeta function

    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.
  15. W

    Question about Riemann Zeta Function

    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) +...
  16. polygamma

    MHB The Euler Maclaurin summation formula and the Riemann zeta function

    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 -...
  17. D

    Inverse of the Riemann Zeta Function

    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 $$...
  18. chisigma

    MHB A curiosity about the Riemann Zeta Function....

    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) =...
  19. polygamma

    MHB Another integral representation of the Riemann zeta function

    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} +...
  20. polygamma

    MHB An integral representation of the Riemann zeta function

    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...
  21. L

    I'm trying to write a program that plots the riemann zeta function

    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...
  22. S

    Riemann Zeta Function Zeros

    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...
  23. C

    About interesting convergence of Riemann Zeta Function

    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...
  24. S

    Is Riemann Zeta function related to differential equations?

    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...
  25. M

    Euler's derivation of Riemann Zeta Function for even integers

    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
  26. K

    Riemann zeta function - one identity

    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...
  27. M

    Trivial zeros in the Riemann Zeta function

    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...
  28. R

    Programming details on the computation of the Riemann zeta function using Aribas

    (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...
  29. D

    Riemann Zeta function of even numbers

    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...
  30. K

    Riemann Zeta Function and Pi in Infinite Series

    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...
  31. T

    Derivative of Riemann zeta function

    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...
  32. H

    Unclear on Riemann Zeta Function

    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
  33. E

    Proof of Inf. Riemann Zeta Function Zeros at re(s)=1/2

    Does anybody know where I can find the proof that an infinite number of zeros of the riemann zeta function exist when re(s) = 1/2?
  34. D

    Riemann Zeta Function Z(z)

    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...
  35. S

    What is Riemann zeta function.

    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...
  36. M

    Trivial zeros of the Riemann zeta function

    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?
  37. M

    Riemann Zeta function zeros

    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...
  38. B

    Calculating Riemann Zeta function

    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...
  39. E

    Riemann zeta function

    \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...
  40. E

    More on riemann zeta function

    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...
  41. E

    Evaluating the Riemann Zeta Function: Step-by-Step Guide for \zeta(c + xi)

    Can someone show me the steps to evaluating \zeta(c + xi), where 0 \leq c<1?
  42. D

    Program for graphing Riemann zeta function

    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...
  43. Gib Z

    Ramanujan Summation & Riemann Zeta Function: Negative Values

    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?
  44. L

    Riemann zeta function generalization

    "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...
  45. P

    Understanding Zeros of the Riemann Zeta Function

    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...
  46. benorin

    Fractional Calculus and the Riemann Zeta function

    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...
  47. Jameson

    Riemann Zeta Function

    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
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