Zeta function Definition and 111 Threads

i have a question about the relation between the riemann zeta function and the prime counting function . one starts with the formal definition of zeta : \zeta (s)=\prod_{p}\frac{1}{1-p^{-s}} then : ln(\zeta (s))= -\sum_{p}ln(1-p^{-s})=\sum_{p}\sum_{n=1}^{\infty}\frac{p^{-sn}}{n} using the...

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...

I was reading through the first chapter of Edwards' book on the zeta function, and I'm a little confused about Riemann's original continuation of zeta to all of the complex plane... The zeta function is supposed to be defined for all s in the set of complex numbers by \zeta \left( s \right) =...

Help with Mobius Inversion in "Riemann's Zeta Function" by Edwards (J to Prime Pi) Can someone please add more detail or give references to help explain the lines of math in "Riemann's Zeta Function" by Edwards. At the bottom of page 34 where it says "Very simply this inversion is effected...

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...

Hello to all. This could be quite long. Apologies. I am a physics student trying to understand the Zeta function and the Riemann hypothesis. Its not on my coursework, but I am interested in pure mathematics. I have a few questions. Perhaps you can help me out. Thank you. My questions are...

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...

Does anyone know where I can download or purchase this paper? I can't find it anywhere... http://edoc.mpg.de/39003 New points of view on the Selberg zeta function Authors: Zagier, Don Language: English Research Context: research report Publisher: Ryushi-do Place of Publication: Osaka...

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...

Homework Statement An infinite number of incoherent sources of sound are located on the x-axis at positions given by n2 (in meters) with n= 1,2,3,4,5... If all the sources emit with a power of 10.0W, calculate the sound level of the total sound wave at the origin. Prove your answer using the...

It has been determined that the square of zeta can be written in terms of the divisor function. {\zeta ^2}(s) = \sum\limits_{n = 1}^\infty {\frac{{d(n)}}{{{n^s}}}} Being a first semester student in complex variables, I have only recently started looking at zeta. But I have deduced an...

I just started reading the book by Edwards. I am currently in a complex variables course so i figured that I would give zeta a shot. I realize that there are easier ways to evaluate it for integer values, namely, the infinite sum. But I trying to at least evaluate it so that I could at least...

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

Can anybody of you people recommend me the best, most pedagocical, clearest, easiest, but detailed enough explanation of how to analytical continue the zeta function to the whole complex plane (except 1, of course!)? In a book, notes on the net, whatever! thank you

In "A Prime Case of Chaos" (http://www.ams.org/samplings/math-history/prime-chaos.pdf), the author states that "Physicists ... believe the zeroes of the zeta function can be interpreted as energy levels..." I have two problems with this: (1) the non-trivial zeroes of the zeta function are...

I was looking at a paper about strong-coupling expansion (N. F. Svaiter, Physica (Amsterdam) 345A, 517 (2005) ) and it claims that -\int d^d x \int d^d y (-\Delta + m^2)\delta^d(x-y) = \textbf{Tr} I + \left.\frac{d}{ds}\zeta(s)\right|_{s=0} where \zeta(s) is the spectral zeta function, and I...

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?

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?

What about the Riemann Zeta function makes it so difficult to prove that all the zeros have real part 1/2? Is it that we lack the discoveries and tools necessary, or we just aren't creative enough, or maybe both? Same question for Goldbach's. Fermat's seemed to rely on elliptic curves which have...

Homework Statement Prove that sum(n=0 to infty, (zeta(it))^(n)) equals zero when the variable (it) is the imaginary part of the nontrivial zeros of the Riemann zeta function that have real part 1/2. For example, it=14.134i. Note: n represents the nth derivative of the zeta function...

Homework Statement When extended to the complex plain, does the zeta function approach zero as the number of derivatives of it approaches infinity? Homework Equations The Attempt at a Solution

Homework Statement Find the numerical value of \sum_{k=0}^{\infty} (\zeta(-k)) Homework Equations The Attempt at a Solution I have no idea how to get a numerical value for this sum.

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...

given the function Z(s)= \prod _{k=0}^{\infty}\zeta (s+k) with \zeta (s) being the Riemann Zeta function the idea is if ALL the roots have real part (i mean Riemann Hypothesis) is correct, then what would happen with the roots of Z(s) ?? what would be the Functional equation relating...

It seems like zeta(n) = (pi)^n / (some number), for even integers n. Can anyone point me to a proof of this?

Hi, I'm Yr 13 and just wanted to do some further reading/exploring. So i understand that the zeta function is something to do with summing up like this: 1/ (1^s) + 1/(2^s) etc etc Now, I just want to know what are non-trivial zeros and trivial zeros? I just want to be able to understand this...

can we really give a definition of \delta (x-a-ib) a,b real and 'i' means the square root of -1 if i try it in the sense of generalized function for any x a and b i get the result oo unless b is zero

I need to know if the following series converges: ∑(k=1 to k=oo)[(((-1)^k) ζ(k))/(e^k)] The problem is that zeta(1)=oo; however, the equation satisfies the conditions of convergence for an alternating series [the limit as k->oo=0 and each term is smaller than the last.] Any thoughts?

where could i get some info about the function \sum_{p} p^{-s}=P(s) * the functional equation relating P(s) and P(1-s) * the relation with Riemann zeta

Riemann says that the zeta function doesn't have zeros on the half plane \{z\in\mathbb{C}\;|\;\textrm{Re}(z)>1\}, because the sum \log(\zeta(z)) = \log\Big(\frac{1}{\underset{p\in\mathbb{P}}{\prod}\big(1 - \frac{1}{p^z}\big)}\Big) = -\sum_{p\in\mathbb{P}}\log\big(1 - \frac{1}{p^z}\big)...

how do i calculate values of the riemann zeta function in the critical strip? because if you only know zeta as a series: \zeta(s) = \sum 1/n^s and the functional equation \zeta(s) = 2^s\pi^{s-1}\sin\left(\frac{\pi s}{2}\right)\Gamma(1-s)\zeta(1-s)...

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...

it is true in general that the sum (density of states for a physicst) \sum_{n=0}^{\infty} \delta (x- \gamma _{n}) is related to the value \frac{ \zeta '(1/2+is)}{\zeta (1/2+is)}+\frac{ \zeta '(1/2-is)}{\zeta (1/2-is)} here the 'gamma' are the imaginary parts of the non-trivial...

I was doing some work with the zeta function and have a question. I am aware that the Riemann Hypothesis claims that all of the critical zeros of the analytically continued zeta function have a real part Re(z)=1/2. My question is, does the concept apply only to the complex zeros, or the...

So we were going over geometric series in my calc class (basic, I know), however I was intrigued by one point that the prof. made during lecture \frac{\pi^2}{6} = \sum^{\infty}_{n=1}\frac{1}{n^2} = \zeta (2) That's amazing (at least to me). Looking for the explanation for this, I found a...

\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...

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

I am wondering what are the prerequisites required for learning the theory behind Riemann's zeta function, starting from a base of mathematics that an average physics graduate might have. In particular, I want to be able to understand a book like this...

Let be H an Schrodinguer operator so H \phi =E_n \phi then we have the identity \sum_ {n} E_{n}^{-s} = \frac{1}{\Gamma (s)} \int_{0}^{\infty} dt t^{s-1} Tr[e^{-tH}] the problem is , that to define the Trace of an operator i should know the Eigenvalues or the Determinant of the...

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...

Let Euler's zeta function be given by \sum_{n=1}^{\infty}1/n^s Is there an exponent L which limits the finiteness of (\sum_{n=1}^{\infty}1/n^s)^L for the case where s=1?

...on the off chance anyone knows this, I'm trying to get from: V=\frac{1}{2A}Tr Log(\frac{-\Box}{\mu^2}) to V=\frac{(-1)^{\eta-1}}{4\pi^\eta\eta!}\frac{\pi}{L}^{D-1}\zeta'(1-D) I know this is a shot in the dark, but in case anyone has experience. The paper I'm reading explains...

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...

I'm experimenting with zeta function right now, and I assume there must be some kind of patter in zetas of consecutive (even) numbers. For example when we do, \zeta(2)=\pi^2 /6 \zeta(4)=\pi^4/90 \zeta(6)=\pi^6/945 \zeta(8)=\pi^8/9450 However, \zeta(12)=691\pi^{12}/638512875 So, Can...