Questions on the famed $\zeta(s)$ function of Riemann, and its properties.

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What is the Fourier transform of Riemann Zeta function?

All: Is there an explicit form of Fourier Transform of Riemann Zeta function ? Also, is there an discrete Fourier Transform (DFT) of Riemann Zeta function ? I remembered I had seen something like ...
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1answer
42 views

Does $\zeta(s)^2 \pm \zeta(1-s)^2$ have roots at the $\rho$s?

Maybe a strange (or stupid) question, but does $$\zeta(s)^2 \pm \zeta(1-s)^2$$ also have roots equal to the non-trivial zeros ($\rho$) ? At first sight you would expect so, however when I tried to ...
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1answer
35 views

How do I calculate the values of $\zeta(0.5+ie^x)$ for large $x$ ?

In wolfram alpha the values of $$\zeta(0.5+ie^x)$$ closed to zero then How do I know the real values of $\zeta(0.5+ie^x)$ for large real number $x$ ? Thank you for any help
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1answer
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Bibilography: Riemann's hypothesis and positive semi-definite billinear forms

This is a bibliography request: I remember browsing through a book, some years ago, in a library, in which Riemann's hypothesis was proved over some type of fields (I cannot remember what type), the ...
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1answer
51 views

How to get sine term in Analytical continuation of $\zeta(s)$

I am able to prove the symmetric functional equation that Riemann gives in his paper, using Poisson Summation and properties of $\theta(x)$. The functional equation is given like so, ...
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One-to-one correspondance between zeta zeros and the prime powers?

I have noticed an interesting property related to the Gibbs phenomenon for the Fourier transform of the zeta zeros in Riemann's explicit formula, namely that the rate at which $r\rightarrow 2 $ in the ...
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1answer
42 views

Does this product over the primes converge, and if so, to what?

I've been trying to play around with the product: $$\prod_{p \text{ prime}}\frac{1}{1-(-p)^{-1.5}}$$ Where the product runs over all the prime numbers. The product is similar in appearance to the ...
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1answer
40 views

Growth of the zeta function on the line $Re(s)=\frac{1}{2}$

I've seen that on the line $Re(s)=\frac{1}{2}$, $\zeta(s)=O(t^{\frac{1}{4}})$ where, as usual, $s=\sigma+it$. My teacher has told me that this can be derived directly from the functional equation of ...
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1answer
49 views

Estimating $\log|\zeta(\sigma+it)|$ for $\sigma$ sufficiently large

In a paper I am reading, I've come across the estimate $$ 2\pi\sum_{\substack{\sigma<\beta<\sigma_0\\ T<\gamma\leq ...
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2answers
76 views

Limit involving Zeta and Gamma function

Can someone help me evaluate this limit? $$\lim_{x\to +\infty}\frac {\zeta(1+\frac 1x)}{\Gamma(x)}$$ I never came across this kind of limit so I don't even know where to start.
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1answer
41 views

Limit of zeta function in $x = 1$

How can I prove that $\lim_{x \rightarrow 1}{\sum_{n=1}^{\infty}{\frac{1}{n^x}}} = \infty$? My idea is to show that we can exchange the positions of limit and sum, obtaining the harmonic sum, that we ...
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Proving the functional equation $\theta (x) = x^{-\frac{1}{2}} \theta (x^{-1})$ from the Poisson summation formula

We have the relationship $\theta (x) = x^{-\frac{1}{2}} \theta (x^{-1})$ Now I know one uses the Poisson summation formula to prove this. The Poisson summation formula comes from Fourier Transform ...
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0answers
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Solving summation; ( double sum).

I found the expression to the sum of powers long ago and ofcours I think it is true but i don't know for sure, the problem is, it's little though for me to test and try it out. Also i'd like to know ...
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1answer
42 views

Proving an inequality, involving the Riemann-$\zeta$-function

I'm studying a book, which wants to prove that for some constants $C,\kappa$, we have $$\left|\frac{\zeta ''(s)\zeta (s) -2\zeta '(s)}{\zeta (s)^2}\right|\le C*|t|^\kappa~~~~,\text{for ...
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3answers
565 views

Limit approach to finding $1+2+3+4+\ldots$

When exploring the divergent series consisting of the sum of all natural numbers $$\sum_{k=1}^\infty k=1+2+3+4+\ldots$$ I came across the following identity involving a one-sided limit: ...
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0answers
108 views

Zeta regulated product, solving without the zeta function.

Earlier i've asked about how to calculate divergent products, i got some directions which made me curious. Now i'm wondering is this correctly done. Divergent products. The most commen divergent ...
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2answers
46 views

What is the power series expansion for Riemann-Zeta at $0$?

What are the first few terms of the Laurent series expansion of $\zeta(0)$? It gets mentioned here but they only show the first term and I am kind of confused on how they got $-1/2$.
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Ways to prove Eulers formula for $\zeta(2n)$

I recently, out of interest, tried to prove Euler's formula $\zeta{(2n)}=(-1)^{n-1}\frac{(2\pi)^{2n}}{2(2n)!}B_{2n}$ for all $n\in\mathbb{N}$. I adapted Euler's original proof for ...
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1answer
45 views

Euler product of $\sum (2^k n + 1)^{-s}$

do we know for a given $k > 2$ the Euler product of $\ \displaystyle\sum_{n=0}^\infty (2^k n + 1)^{\textstyle-s} \ $ ? I saw that every prime numbers will appear in it, as well as some non-prime ...
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Can $\zeta(s)$ be written in the form $\zeta(s)=\Re(\zeta(s))+i·\Im(\zeta(s)) $ for some subset of $\mathbb{C}$?

Can $\zeta(s)$ be written in the form $\zeta(s)=f(s)+g(s) i $ for some subset of $\mathbb{C}$? I mean, is it possible to develop at least one of the formulas of $\zeta(s)$ so you get something like... ...
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Other way show that $\zeta(-2)=(-\frac{1}{12})\mod 2$? [closed]

Lemma: We knew that for any integer $a$ : ${a}^{p}=a \mod p$. Then $1^p=1\mod p ,2^p=2\mod p ,3^p=3\mod p , \ \dots,\ n^p=n\mod p $. Just to sum each term by term RHS and LHS we will get the ...
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Why $\zeta(-2) $ is not $\sum_{n=1}^{\infty}\frac{1}{n^{-2}}$? [duplicate]

Let $\zeta(s)= \sum_{n=1}^{\infty}\frac{1}{n^{s}}$ a standard formula. I'm confused if you tell me: does this series: $\sum_{n=1}^{\infty}\frac{1} {n^{s}}$ converge? I will answer you: this series ...
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Are all complex zeros of ${\frac {\zeta \left( s+1 \right) }{\zeta \left( s-1 \right) }}\pm\, 2\,\pi\frac{2-s}{s\,(s+1)}$ on the critical line?

From this question, it is easy to derive that a zero of $\xi(a+s)\pm \xi(a+1-s)$ should occur when: $$\displaystyle{\frac {\zeta \left( s+a \right) }{\zeta \left( s-a \right) }}=\pm{ \frac {{\pi ...
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1answer
74 views

Alternating series involving zeta function

Can anyone help me attain the result for the following series? $$\sum_{n=2}^{\infty} \frac{(-1)^n \zeta(n)}{n(n+1)}= \frac{1}{2} \left( \log 2 + \log \pi +\gamma -2 \right)$$ I don't know how to ...
2
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1answer
59 views

Riemann zeta function, representation as a limit

is it true that $$\displaystyle \zeta(s) = \ \lim_{\scriptstyle a \to 0^+}\ 1 + \sum_{m=1}^\infty e^{\textstyle -s m a } \left\lfloor e^{\textstyle(m+1)a} - e^{\textstyle m a} \right\rfloor$$ my proof ...
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1answer
49 views

can you integrate the dirac delta over any domain?

At eq 28 the following author takes it from 1 to infinity. I've never seen this before and i'm not sure his equation is correct. ...
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2answers
107 views

Would the Riemann Hypothesis being false affect how frequently primes occur in the number system?

I want to know that if Riemann hypothesis is false (big assumption) would that lead to any effect in how frequently primes occur . Well I got this half cooked information from here: ...
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2answers
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Regularizing the $\log\log n$ series

The divergent series $$\sum_{n=1}^\infty\log n$$ can be regularized using the derivative of the Riemann zeta function at $s=0$: ...
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3answers
63 views

Showing a series based on the Riemann zeta function converges (uniformly)

This question comes from Lars Ahlfors' complex analysis (page 178). We define $\zeta(z) = \sum_{n=1}^{\infty} n^{-z}$. This is just the Riemann zeta function. I am struggling however to prove that ...
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Is there an easy way to calculate the elementary symmetric functions?

Hello I am interested in the question of what, generally, is the sum of the series of reciprocals of a series of numbers we know its sum. I have particular interest in the Zeta function, which I ...
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1answer
79 views

infinity series of Riemann zeta function at odd integers

Properties of Riemann zeta function at odd and even integers diverge dramatically, which can be proved by many evidences. I once found an infinity series in wikipedia, it reads $$ ...
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1answer
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Sum of reciprocals of primes diverges

I can show that $$\log(\zeta (s)) = \sum _{p\in\Bbb P} \frac{1}{p} + R(s)$$where $$R(s) = \sum _{m\geq 2} \sum_{p\in\Bbb P} \frac{1}{m} \frac{1}{p^{ms}}$$ where $\Bbb P$ is the set of all primes, ...
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How the steps followed help me to understand

Few days before I posted one problem on MSE here in which it was asked to determine the residue. I understood almost entire but got stuck the following Note that after step 4, in step 5 it is ...
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1answer
95 views

What is the Möbius analoge for Ihara's $\zeta$ function?

The Dirichlet series that generates the Möbius function is the (multiplicative) inverse of the Riemann zeta function; if s is a complex number with real part larger than 1 we have $$ ...
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1answer
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Probabilistic interpretation for representation of unity using the zeta function

There's a cute identity, I believe due to Borwein, Bradley and Crandall (Section 4): $$1=\sum_{n=2}^\infty (\zeta(n)-1).$$ There are some generalizations in the linked paper as well. Question: Is ...
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3answers
50 views

Does the sum of the Zeta function taken on natural numbers converge?

Does this series \begin{equation*} \sum_{n\ge2} \zeta(n) \end{equation*} converge? If yes is it easy to prove ?
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On Zero-Free Regions for $\zeta(s)$ and $L(s,\chi)$ with $|t| \le 2$

I'm reading the proof from Hildebrand that for some $c_1 > 0$, the Riemann zeta function $\zeta(s)$ has no zero in the region $\sigma > 1-c_1$, $|t| \le 2$. (Here $s = \sigma + it$ per ...
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1answer
107 views

Is there any relationship between the Riemann z function and strange attractors?

I have this question in mind since the first time I saw a graphical representation of the zeta function (like in the sample below). Just by looking to them I wondered if there is any relationship ...
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2answers
26 views

WHat is the inner steps here?

I was studying abour Riemann zeta function over here where in (3) it has been written "by Abel's theorem", we have $$\sum\limits_{n\geq 1}\frac{1}{n^s}=\sum\limits_{n\geq ...
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1answer
31 views

On the determination of residue

I need your help on the following. (1)First we are to find the residue of $\frac{x^s}{s}$ at $s=0$. Since $s=0$ is the pole of order 1, so we get Res$(\frac{x^s}{s},s=0)=\frac{1}{2\pi ...
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Simple Zero of the Riemann Zeta Function

Let $s=σ+it$. Assume that $ζ(s)-1/(s-1)$ has an analytic continuation to the half plane $σ>0$. Show that if $s = 1 + it$, with $t≠0$, and $ζ(s) = 0$ then $s$ is at most a simple zero of $ζ$. I ...
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2answers
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Summation with Riemann Zeta Function

So the Riemann zeta function $\zeta(s)$ is commonly defined as $\sum \limits_{n=1}^{\infty} n^{-s}$ Now, suppose that $a_k=\zeta (2k).$ How can I find the value of $$\sum \limits_{k=1}^{\infty ...
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1answer
120 views

Evaluate $\sum \sum 1/n^k $

I wanted to evaluate the sum: $$ \sum_{n \ge 2} \left(\zeta(n) - 1\right) $$ I rewrote this as: $$ \sum_{n\ge 2} \sum_{k\ge 2} \frac{1}{n^k} $$ I tried exploiting the symmetry but that didn't seem ...
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2answers
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How to calculate $\zeta(i)$?

As the title says, I'm interested to know how $\zeta(i)$ is calculated. I know the functional equation for the zeta function, but if I put it in that in there, I must know $\zeta(1-i)$. Is it a good ...
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Provide me notes on Riemann zeta function to boast my knowledge to use in Research on Analytical Number Theory

I need your help. I want to study the Riemann zeta function from the very basic level, its concepts, theorems, solved problems etc. I am assigned one problem from Analytical Number Theory related to ...
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1answer
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How the prove that the Dirichlet series converges?

I am ready to calculate an alternated series of the type of Dirichlet $L$-function. The series, as it is, is $$ L = ...
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1answer
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Proving the Riemann Hypothesis and Impact on Cryptography

I was talking with a friend last night, and she raised the topic of the Clay Millennium Prize problems. I mentioned that my "favorite" problem is the Riemann Hypothesis; I explained what it posits ...
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Hunting for some properties of a series similar to Riemann zeta function.

I happened to meat a alternated zeta function in D. Borwein's paper(see eq. 40 of his paper). The series is $$ L_{-3}(s)=1-\frac{1}{2^s}+\frac{1}{4^s}-\frac{1}{5^s}+\frac{1}{7^s}-\frac{1}{8^s}+\dots ...
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Errors in following paper

In the paper "On the Level Curves of the Xi Function" http://arxiv.org/abs/1002.0352v8, John Breslaw takes a very similar approach to a study of the Riemann hypothesis I did while I was in my first ...
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Does this 'alternating' series with $\Lambda(n)$ converge for all $\Re(s)>0$?

The following equation is well known and valid for $\Re(s)>1$: $$\log\big(\zeta(s)\big)=\sum_{n=2}^\infty \dfrac{\Lambda(n)}{\log(n)\,n^s}$$ where $\Lambda(n)$ is the Von Mangoldt function. Take ...