What is Zeta function: Definition and 111 Discussions

In mathematics, a zeta function is (usually) a function analogous to the original example, the Riemann zeta function




ζ
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n
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{\displaystyle \zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}.}
Zeta functions include:

Airy zeta function, related to the zeros of the Airy function
Arakawa–Kaneko zeta function
Arithmetic zeta function
Artin–Mazur zeta function of a dynamical system
Barnes zeta function or double zeta function
Beurling zeta function of Beurling generalized primes
Dedekind zeta function of a number field
Duursma zeta function of error-correcting codes
Epstein zeta function of a quadratic form
Goss zeta function of a function field
Hasse–Weil zeta function of a variety
Height zeta function of a variety
Hurwitz zeta function, a generalization of the Riemann zeta function
Igusa zeta function
Ihara zeta function of a graph
L-function, a "twisted" zeta function
Lefschetz zeta function of a morphism
Lerch zeta function, a generalization of the Riemann zeta function
Local zeta function of a characteristic-p variety
Matsumoto zeta function
Minakshisundaram–Pleijel zeta function of a Laplacian
Motivic zeta function of a motive
Multiple zeta function, or Mordell–Tornheim zeta function of several variables
p-adic zeta function of a p-adic number
Prime zeta function, like the Riemann zeta function, but only summed over primes
Riemann zeta function, the archetypal example
Ruelle zeta function
Selberg zeta function of a Riemann surface
Shimizu L-function
Shintani zeta function
Subgroup zeta function
Witten zeta function of a Lie group
Zeta function of an incidence algebra, a function that maps every interval of a poset to the constant value 1. Despite not resembling a holomorphic function, the special case for the poset of integer divisibility is related as a formal Dirichlet series to the Riemann zeta function.
Zeta function of an operator or spectral zeta function

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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. bhobba

    A Does The Use Of The Zeta Function Bypass Renormalization

    I am trying to figure out if the use of the Zeta function allows renormalization to be bypassed. I have formed a preliminary view but would like to hear what others think: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.570.4579&rep=rep1&type=pdf Thanks Bill
  3. 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...
  4. JorgeM

    I Does the Integral of Riemman Zeta Function have a meaning?

    I have been trying to use numerical methods with this function but now I realize that I if I could suggest a Polynomial in theory, I could get some value for the Integral at least in any interval. In general, does the Integral of the Riemman dseta function has a meaning by itself?
  5. 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...
  6. 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...
  7. H

    B U-238 spacings and Zeta function zeroes?

    I read some old unpublished student notes where the student in an undergrad project was looking at general patterns in QM results. Seemed arbitrary to claim anything but the student did show that spacings in EV between transitions in U238 followed the same general form of zero spacings in the...
  8. P

    A String theory and zeta function z(-1)

    Over the last couple of years there has been allot of traffic on youtube about the sum of all positive integers being equal to -1/12 as is explained in numberphile video. Some argue that their calculations are wrong and the sum really is infinity. In their original video numberphile shows a...
  9. 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/
  10. 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
  11. M

    B Does the Riemann Hypothesis Follow from the Prime Number Theorem?

    There are fewer primes for larger n. The n^1/2 just makes the larger n have less impact on the result. So the riemann hypothesis follows from pnt. Does this make any sense to you?
  12. 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...
  13. M

    B Behavior of the zeta function

    Consider Z(s)=Sum(1/N^s) For n=1 to infinity. Let s=(xi+1/2). The divisor is then: N^(xi+1/2) This is equivalent to (N^xi)(N^(1/2)) As n increases, the n^1/2 term will more greatly slow the increase of the divisor and accelerate z(s) away from zero. This means that zeroes will occur less...
  14. 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...
  15. M

    A Has anything similar to the Riemann hypothesis ever solved

    Has anything similar to the Riemann hypothesis ever been solved? Specifically, has anyone proven that the real part of a result of some particular function always assumes a particular value?
  16. 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)##.
  17. nomadreid

    I Riemann zeta: regularization and universality

    I am not sure which is the appropriate rubric to put this under, so I am putting it in General Math. If anyone wants to move it, that is fine. Two questions, unrelated except both have to do with the Riemann zeta function (and are not about the Riemann Hypothesis). First, in...
  18. ognik

    MHB Zeta function integral test

    Hi - just done the integral test on the Riemann zeta series, came out to $\frac{1}{p-1}$ I can clearly see it therefore converges for P > 1, is singular for p=1, but for p < 1 I can't see why it diverges? In the limit p < 1 just gets smaller? Would also like to check about p = 1, all I need...
  19. 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...
  20. nomadreid

    Distribution of the zeros of the zeta function

    In http://www.americanscientist.org/issues/pub/the-spectrum-of-riemannium/5, the author mentions that the function P(x) = 1-(sin(πx)/(πx))2 seems to be, assuming the Riemann Hypothesis is true, to the two-point correlations of the zeros of the Riemann zeta function. Going by...
  21. 6c 6f 76 65

    Calculating the zeta function over a hypersurface in project

    Homework Statement Calculate the zeta function of x_0x_1-x_2x_3=0 in F_p Homework Equations Zeta function of the hypersurface defined by f: \exp(\sum_{s=1}^\infty \frac{N_s u^s}{s}) N_s is the number of zeros of f in P^n(F_p) The Attempt at a Solution My biggest struggle is finding N_s...
  22. 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
  23. ddd123

    Why are Ramanujan sums the same as the complex Zeta values?

    Possibly a difficult question, but I've never found a discussion on the topic. Thanks
  24. 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.
  25. S

    Generating function for the zeta function of the Hamiltonian

    Given a Hamiltonian ##H##, with a spectrum of eigenvalues ##\lambda##, you can define its zeta function as ##\zeta_H(s) = tr \frac{1}{H^s} = \sum_{\lambda}^{} \frac{1}{\lambda^s}##. Subsequently, the log determinant of ##H## with a spectral parameter ##m^2## acts as a generating function for...
  26. 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) +...
  27. David Carroll

    Conjecture about the Prime Zeta Function

    I was fooling around with the Prime Zeta Function just recently. Prime Zeta Function, P(s), is defined as Σ1/(p^s), where p is each successive prime. When inputting various positive integer values for (s) on wolfram alpha, I noticed a pattern. Well, an approximate pattern, I should say. My...
  28. B

    MHB Inequality involving Zeta Function

    Prove that for $r>2$ we have $$\frac{\zeta\left(r\right)}{\zeta\left(2r\right)}<\left(1+\frac{1}{2^{r}}\right)\frac{\left(1+3^{r}\right)^{2}}{1+3^{2r}}.$$ I've tried to write Zeta as Euler product but I haven't solve it.
  29. S

    Zeta function regularisation

    Homework Statement Hi I need to regularize \sum_{r \in Z+1/2} r In my opinion there are two ways of going about it either re-express it as \sum_{r \in Z+1/2} r = \sum_{r =1} r - \frac{1}{2} \sum_{r =1} = \zeta (-1) - \zeta (0) = \frac{1}{6} or \sum_{r \in Z+1/2} r =...
  30. DreamWeaver

    MHB Derivatives and Integrals of the Hurwitz Zeta function

    Initially, the purpose of this tutorial will be to explore and evaluate various lower order derivatives of the Hurwitz Zeta function. In each case, the Hurwitz Zeta function will be differentiated with respect to its first parameter. A little later on - although this will take some time! - these...
  31. C

    Reimann Zeta Function Error

    I am currently writing a c++ program to calculate the Value of the Reimann Zeta Function, The problem is At the state its at, when you input a number a + bi it only gives the correct answer to values of a > 1. Can anyone show or link me Riemann's Analytic continuation of the Function so it...
  32. N

    Zeta Function -1 1/2 and prime numbers

    I talked with an old friend of mine. We discussed prime numbers and Ulams Spiral, and the mathematical patterns that surround us all. He brought up something called the Zeta-Function and something about -1 1/2 and how this all related to prime numbers. I did a google search and found some...
  33. DreamWeaver

    MHB Trigonometric series related to the Hurwitz Zeta function

    This thread is dedicated to exploring the trigonometric series shown below. This is NOT a tutorial, so all and any contributions would be very much welcome... (Heidy)\mathscr{S}_{\infty}(z)= \sum_{k=1}^{\infty}\frac{\log k}{k^2}\cos(2\pi kz) This series can be expressed in terms of the...
  34. 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 -...
  35. 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 $$...
  36. 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) =...
  37. polygamma

    MHB An integral representation of the Hurwitz zeta function

    For $ \text{Re} (a) >0$ and $\text{Re} (s)>1$, the Hurwitz zeta function is defined as $ \displaystyle \zeta(s,a) = \sum_{n=0}^{\infty} \frac{1}{(a+n)^{s}} $. Notice that $\zeta(s) = \zeta(s,1)$. So the Hurwitz zeta function is a generalization of the Riemann zeta function. And just like the...
  38. 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} +...
  39. 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...
  40. mathworker

    MHB Zeros of the Zeta Function: Exploring $\rho$ Values

    In an article it is given that, \zeta(s)=\text{exp} (\sum_{n=1}^\infty\frac{\Lambda{(n)}}{\text{log}(n)}n^{-s}) $\zeta(s)$ has pole at $s=1$ and zeroes at several $s=\rho$. here i think he considered the function inside the exponential rather than whole exponential to obtain poles and zeroes...
  41. 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...
  42. P

    Zeta function for Debye Low Temp. Limit

    Hello, In the low temp. limit of Debye law for specific heat, we encounter the following integral: ∫(x4 ex)/(ex-1)2 dx, from 0 to ∞. The result is 4∏2/15. I have searched and found this to be related to zeta function but zeta functions do not have ex in the numerator so I am unable...
  43. 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...
  44. 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...
  45. nomadreid

    Riemann's zeta function fractal because of Voronin?

    Riemann's zeta function "fractal" because of Voronin? I am not sure which rubric this belongs to, but since the zeta function is involved, I am putting it here. I noticed a comment (but was in too much of a hurry to remember the source) that, because of the "universality" of the Riemann zeta...
  46. 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...
  47. Karlx

    Discovering Prime Numbers & Riemann's Zeta Function

    Hi everybody. I would like to find a book about the Distribution of Prime Numbers and the Riemann's Zeta Function. I know about the "classical" books: 1) Titchmarsh's "The Theory of the Riemann Zeta-Function" 2) Ingham's "The Distribution of Prime Numbers" 3) Ivic's "The Riemann...
  48. 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
  49. K

    Which book is the best for understanding the Riemann hypotheses?

    I'm looking for a semi-popular book on the Riemann hypotheses, on the scale of Derbyshire's book. It seems that Derbyshire and Edwards are the best in this area. Which one should I go for? The content must be quite extensive. Please suggest other ones too...
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