1
vote
2answers
50 views

Riemann zeta, why are the residues either zero or one?

One more question, probably equally simple to answer but I don't know how this is true either: Why is the residue of Riemann zeta zero - trivial or non-trivial: $$\text{residue}\left(\zeta ...
2
votes
1answer
40 views

Pole of Riemann zeta and Riemann zeta zeros, prove this relation.

Prove this relation: $$\displaystyle \lim_{s\to 1} \, \left(\zeta (s)-\frac{\zeta '(s-1+\rho _n)}{\zeta \left(s-1+\rho _n\right)}\right)=\gamma -\frac{\zeta ''(\rho _n)}{2 \zeta '(\rho ...
5
votes
1answer
102 views

Riemann Zeta Function Non-Vanishing on the Line $\mathrm{Re} \; z = 1$

The result quoted in the title is usually a stepping stone in the proof of the prime number theorem and I am familiar with the usual argument for this result. The other day my professor was telling ...
2
votes
1answer
43 views

Continuation of the Riemann Zeta Function

I am actually aware of the argument showing $\zeta$ has a meromorphic extension to $\mathbb{C}$ with a single pole at $z = 1$. On a recent number theory exam, however, one of the questions asked to ...
1
vote
0answers
27 views

Upper bound on $\zeta(s)$

I'd like to know an upper bound for $\zeta(s)$ in the critical strip, and hopefully one that is not too difficult to prove. For instance, ...
0
votes
0answers
51 views

Specific form of integral representation of the Riemann zeta function

Is there an integral represenation of the Riemann zeta function of the form: $$\zeta(s) = f(s)+c\int_a^b\frac{g(x)}{x^{p(s)}}dx,$$ where $a,b,c\in\mathbb{R}$ with $a\neq b$, $p(s)$ is some ...
5
votes
2answers
91 views

$\prod_{i=1}^{\infty}{1+(\frac{k}{i})^3}$ for integer k

Can anyone compute $$\prod_{i=1}^{\infty}{1+(\frac{k}{i})^3}$$ for integer k? Can it be done in closed form, using only elementary functions, without the use of the Gamma function? For k=1, the closed ...
2
votes
1answer
102 views

Integral representation of the Riemann zeta function

I've come across the following integral representation for the Riemann zeta function, $$\zeta(s) = \frac{s+1}{2(s-1)} + \frac{s}{8} - \frac{s(s+1)}{2\pi^2}\int_1^\infty \frac{(\tan^{-1}\cot(\pi ...
6
votes
2answers
86 views

Ways to calculate $\int_0^1 \frac{-\log x}{1+x}\ \mathrm dx$

I came across the integral $$ \int_0^1 \frac{-\log x}{1+x}\ \mathrm dx = \frac{\pi^2}{12}, $$ which can be calculated as $\frac 1 2 \zeta(2)$ using analytic number theory. I'm interested if this ...
3
votes
1answer
161 views

Elementary bound on the Riemann zeta function

I am currently preparing for a course in analytic number theory and I wanted to get a heads start. In my preparation, I came across the following problem: Show that for $|y|\geq 2$, $|\zeta(1+iy)| ...
29
votes
3answers
577 views

proving that $\sum_{n=1}^{\infty}\frac{(H_n)^2}{n^3}=\frac{7}{2}\zeta(5)-\zeta(2)\zeta(3)$

Prove that $$\sum_{n=1}^{\infty}\frac{(H_n)^2}{n^3}=\frac{7}{2}\zeta(5)-\zeta(2)\zeta(3)$$ ($H_n=\sum_{k=1}^{n}\frac{1}{k}$)
5
votes
2answers
71 views

GCD and the Riemann zeta funtion

I'm completely stuck on this one, as I'm just starting with analytic number theory: How to write $$\sum_{a\in\mathbb{N}}\sum_{b\in\mathbb{N}}\frac{(a,b)}{a^sb^t}$$ in terms of the Riemann zeta ...
5
votes
0answers
129 views

Riemann zeta function and Bernoulli function

I encountered the following problem: Show that $$\zeta(2n+1)=\frac{(-1)^{n+1}(2\pi)^{2n+1}}{2(2n+1)!}\int_0^{1}B_{2n+1}(x)\cot({\pi}x)dx$$ where $B_{2n+1}(x)$ is the Bernoulli polynomial. This ...
1
vote
1answer
70 views

An upper bound for $-\frac{\zeta'}{\zeta}(s)-\frac{1}{s-1}$

Let $\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}$. We have $\frac{\zeta'}{\zeta}(s) = \sum_{n=1}^\infty \frac{\Lambda(n)}{n^s}$ for $s>1$, where $\Lambda$ stands for the von Mangoldt function ...
0
votes
1answer
91 views

Is there only one analytic continuation of the Riemann zeta function?

If I were to manipulate the zeta function in a 'new way' would I end up with an analytic continuation that is equal to the one know or something completely new for values less than 1 and complex ...
6
votes
1answer
127 views

Questions regarding the Riemann-Siegel $\theta$ Function

My questions are a request, please, for help in understanding some comments in the wikipedia article discussing the Riemann-Siegel $\theta$ function ...
2
votes
0answers
103 views

Integer values of the Riemann function - II

For what value of $n \ge 2$ can we have an real $x > 0$ such that both the numbers $$ \zeta\Big(1+\frac{1}{x}\Big) \text{ and } \zeta\Big(1+\frac{1}{nx}\Big) $$ are positive integers.
4
votes
3answers
103 views

Two questions regarding $\mathrm {Li}$ from “Edwards”

I would appreciate help understanding a relation in Edwards's "Riemann's Zeta Function." On page 30 he has: $$\int_{C^{+}} \frac{t^{\beta - 1}}{\log t}dt = \int_{0}^{x^{\beta}}\frac{du}{\log u}= ...
3
votes
2answers
197 views

Some identities with the Riemann zeta function

Can someone either help derive or give a reference to the identities in Appendix B, page 27 of this, http://arxiv.org/pdf/1111.6290v2.pdf Here is a reproduction of Appendix B from Klebanov, Pufu, ...
2
votes
0answers
390 views

Proofs of trivial zeros of zeta function?

I know that the trivial zeros of zeta function are negative even integers . I have seen the wiki-proof using the functional equation of zeta function, I might have seen a proof using Bernoulli ...
3
votes
1answer
117 views

Convergence of the Zeta and Phi functions

I want to show that the following functions (in the picture) are absolutely and locally uniformly convergent if real part of complex number $s$ is bigger than 1. Absolute part for zeta function is ...
5
votes
2answers
69 views

Prove that $\sum_{n=1}^\infty \frac{\sigma_a(n)}{n^s}=\zeta(s)\zeta(s-a)$

I would appreciate a hint concerning how to surpass the roadblock I've encountered in my attempt at a proof below. A nicer proof than mine would also help (Edit: The latter part is now done by Gerry ...
0
votes
1answer
64 views

To which extent distribution of Riemann non-trivial zeros follow a gauss process?

I am trying to clearer and preciser understand to which extent the distribution of the non-trivial zeros of the Riemann $\zeta$-function follow a Gauss process? Yet, what I figured out from ...
2
votes
4answers
2k views

What is the analytic continuation of the Riemann Zeta Function

I am told that when computing the zeroes one does not use the normal definition of the rieman zeta function but an altogether different one that obeys the same functional relation. What is this other ...
7
votes
2answers
315 views

Tying some pieces regarding the Zeta Function and the Prime Number Theorem together

I came across two remarks that I would appreciate help in making the connections. I) In Riemann's Explicit Formula: for $x > 1$ $\Pi = Li(x) - \sum_{\rho:\zeta(\rho)=0}Li (x^{\rho})- \log(2) +$ ...
14
votes
1answer
239 views

What is a zeta function?

In my readings, I've come across a wide variety of objects called zeta functions. For example, the Ihara zeta function, Igusa local zeta function, Hasse-Weil zeta function, etc. My question is simple: ...
3
votes
0answers
114 views

question about riemann zeta function

How can one prove that $$\zeta (2n)=\frac{(-1)^{n-1}2^{2n-1}\pi ^{2n}B_{2n}}{(2n)!}$$ where $n\in N$ and how can one prove that $$\zeta (2n)=\frac{(-1)^{n}2^{2n-2}\pi ...
4
votes
0answers
93 views

Inverting the Riemann zeta function in $s>1$

Let $s>1$ be a positive real and the Riemann zeta fucntion be defined for $s>1$ as $$ \zeta(s) = \sum_{n=1}^{\infty}\frac{1}{n^s}. $$ I am looking for an inversion formula for the zeta ...
0
votes
0answers
44 views

How to introduce an integer function into $\zeta$ function instead of $n$

I have a problem that I am struggling with since long and probably it is simple but I can not get through. So your help would be very welcome. Known that Riemann $\zeta$ function is defined as sum ...
2
votes
1answer
100 views

The relation of $\zeta$-function and $p^k$ for $Re(s) \le 1$?

The von Mangoldt function: $$\Lambda(n) = \begin{cases} \log p &; \mbox{if }n=p^k \mbox{ for some prime } p \mbox{ and integer } k \ge 1, \\ 0 &; \mbox{otherwise.} \end{cases}$$ establishes a ...
2
votes
2answers
112 views

Analytic continuation of Riemann Zeta funtion

I am reading about zeta function from book by Ingham. In that book the following theorem is given. I am unable to understand what does he mean by finite part of plane in the statement.
2
votes
0answers
249 views

Riemann $\zeta(s)$ non-trivial zeros on $Re(s)=1/2$ and the “spin” of the primes? [closed]

Can this conjecture be true? Let me intro before getting to the conjcture, but for those who like to go straight, see the bold paragraph at the end. Recently I was reading through the well known ...
5
votes
2answers
211 views

Generating functions and the Riemann Zeta Function

The generating function for the terms of the harmonic series: $\frac{1}{n}$ is $-\ln(1 - x)$. Does an ordinary generating function exist for the terms of the zeta function $\zeta(s) = ...
0
votes
4answers
104 views

Let ${P_n}$ be the sequence of all consecutive prime numbers. Is $\sum_{n\geq 1} \frac{1}{p_n}$ convergent? [duplicate]

Let ${P_n}$ be the sequence of all consecutive prime numbers. Is $\sum_{n\geq 1}\frac{1}{p_n}$ convergent?
8
votes
1answer
430 views

how to understand $\log\zeta(s)$ (Riemann zeta function)?

I know that if a function $f$ is analytic and has no zeros in a simple connected region, then we can define $\log{f}$ making it analytic in that region. Let's assume $Re(s)>1$. Is $\zeta(s)$ ...
7
votes
2answers
268 views

Intuitive explanation with rigorous details why zeta has infinitely many zeros?

I have seen a proof outline that $\zeta$ has infinitely many zeros on the critical line here but what I really want is: Simplest possible (least "magic") argument that explains why zeta has ...
5
votes
3answers
449 views

Why does zeta have infinitely many zeros in the critical strip?

I want a simple proof that $\zeta$ has infinitely many zeros in the critical strip. The function $$\xi(s) = \frac{1}{2} s (s-1) \pi^{\tfrac{s}{2}} \Gamma(\tfrac{s}{2})\zeta(s)$$ has exactly the ...
20
votes
1answer
1k views

Books about the Riemann Hypothesis

I hope this question is appropriate for this forum. I am compiling a list of all books about the Riemann Hypothesis and Riemann's Zeta Function. Here is my list: The Riemann Hypothesis: A Resource ...
1
vote
2answers
147 views

Sum of Stieltjes constants

Does anyone know of any papers or resources dealing with the following question: For which values of $s=\sigma+it$ does the following sum of Stieltjes constants hold, ...
7
votes
1answer
412 views

Derivative of the Riemann zeta function for $Re(s)>0$.

The Riemann zeta function can be analytically continued to $Re(s)>0$ by the infinite sum $$\zeta(s)=\frac{1}{1-2^{1-s}}\sum_{n=1}^\infty\frac{(-1)^{n-1}}{n^s}.$$ Can we differentiate this with ...
1
vote
1answer
150 views

Two Representations of $\log \zeta$

I was looking for representations of $\log \zeta$ and found these two: $ \displaystyle \log\zeta(s)=\color{red}{s}\sum_{n>0} \frac{P(ns)}{n\color{red}{s}}$ from here [$\color{red}{s}$ inserted ...
3
votes
3answers
310 views

Other functional equations for $\zeta(s)$?

For the Riemann zeta function, we know of the standard functional equation that relates $\zeta(s)$ and $\zeta(1-s)$. I wanted to know whether there are functional equations that relates $\zeta(s)$ ...
14
votes
1answer
312 views

What is the binomial sum $\sum_{n=1}^\infty \frac{1}{n^5\,\binom {2n}n}$ in terms of zeta functions?

We have the following evaluations: $$\begin{aligned} &\sum_{n=1}^\infty \frac{1}{n\,\binom {2n}n} = \frac{\pi}{3\sqrt{3}}\\ &\sum_{n=1}^\infty \frac{1}{n^2\,\binom {2n}n} = ...
28
votes
5answers
2k views

What is so interesting about the zeroes of the $\zeta$ function

The Riemann $\zeta$ function plays a significant role in number theory and is defined by $$\zeta(s) = \sum_{n=0}^\infty \frac{1}{n^s} \qquad \text{ for } s > 1 \text{ and } s= \sigma + it$$ The ...
6
votes
3answers
2k views

Calculating the Zeroes of the Riemann-Zeta function

Wikipedia states that The Riemann zeta function $\zeta(s)$ is defined for all complex numbers $s \neq 1$. It has zeros at the negative even integers (i.e. at $s = −2, −4, −6, ...)$. These are ...
9
votes
2answers
545 views

How to show that the Laurent series of the Riemann Zeta function has $\gamma$ as its constant term?

I mean the Laurent series at $s=1$. I want to do it by proving $\displaystyle \int_0^\infty \frac{2t}{(t^2+1)(e^{\pi t}+1)} dt = \ln 2 - \gamma$, based on the integral formula given in Wikipedia. ...
1
vote
1answer
342 views

about the riemann zeta function and the prime counting function

i have posted this question on MO, and they referred me to post here . one starts with the formal definition of zeta : $$\displaystyle \zeta (s)=\prod_{p}\frac{1}{1-p^{-s}} $$ then : $ \ln(\zeta ...
27
votes
1answer
1k views

Are these zeros equal to the imaginary parts of the Riemann zeta zeros?

Edit 8.8.2013: See this question also. The Fourier cosine transform of an exponential sawtooth wave times $e^{-x/2}$: $$\operatorname{FourierCosineTransform}(\operatorname{SawtoothWave}(e^x)\cdot ...
4
votes
1answer
182 views

Zeta function identity

How does one prove the zeta function identity $$\sum_{s=2}^{\infty}\left(1-\sum_{n=1}^{\infty}\frac{1}{n^s}\right)=-1 \;?$$
19
votes
1answer
491 views

Upper bound on differences of consecutive zeta zeros

The average gap $\delta_n=|\gamma_{n+1}-\gamma_n|$ between consecutive zeros $(\beta_n+\gamma_n i,\beta_{n+1}+\gamma_{n+1}i)$ of Riemann's zeta function is $\frac{2\pi}{\log\gamma_n}.$ There are many ...