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

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3
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1answer
85 views

On $\sum (1/\sqrt{n})\cos (t_0 \log n)$ and $\sum (1/\sqrt{n})\sin (t_0 \log n)$, from a zero of $\zeta (s)$ of the form $s_0=(1/2)+it_0$

I assume (as hypothesis, for questions too) that $s_0$ a fixed (nontrivial) zero of the Riemann zeta function $\zeta(s)=\sum_{n=1}^\infty 1/n^s$ has the form $(1/2)+it_0$, Thus for a positive real $...
1
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0answers
51 views

How to produce Riemann zeta zero spectrum with the Fourier transform in Mathematica?

All: I post a question generating Riemann Zeta zero spectrum using Mathematica on board of mathematica.stackexchange.com: http://mathematica.stackexchange.com/questions/95294/can-anyone-re-produce-...
1
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0answers
38 views

Analytic continuation vs. series convergence near convergence boundary

Citing Wikipedia, the Riemann zeta function is the analytic continuation of $$ \zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} $$ The series itself is only convergent in the right half complex plane ...
3
votes
1answer
57 views

Are any zeros of Riemann zeta function and the zeros of the derivatives of Riemann zeta function same?

All: Are any zeros of Riemann zeta function and the zeros of the derivatives of Riemann zeta function same ? They shall be all different, right ? Is there a proof of this statement ? Thank you.
2
votes
1answer
65 views

Dirichlet series expansion for Zeta(s)

Wikipedia lists a series expansion for $\zeta(s)$ here. How is the Dirichlet series below derived? I apologize in advance if this is a very simple question, I don't know much about Dirichlet series. ...
2
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0answers
47 views

Where to find Brun's original combinatoric treatment of Brun Sieve?

I tried to understand Brun's original combinatoric treatment of Brun Sieve. (Unfortunately, I do not understand German), so I could not read Brun's original paper as in following: Viggo Brun (1915). ...
2
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1answer
76 views

What are some fundamental symmetries of Riemann Zeta (Xi) function are important to Riemann Hypothesis?

What are some fundamental symmetries of Riemann Zeta (Xi) function are important to Riemann Hypothesis ? (Beside the obvious symmetry of Riemann Xi function, s <--> 1-s reflection) IMHO, at the ...
0
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0answers
37 views

What are some recursive properties of Merten function or Summatory Liouville function?

Both Merten function and Summatory Liouville function show some kinds of "scale invariance" properties. (Those functions also display some kind of "periodic" behavior.( Just wonder if those "scale ...
0
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0answers
39 views

Two-point correlations of the Riemann zeta function

In http://www.americanscientist.org/issues/pub/the-spectrum-of-riemannium/5, the author mentions that the function $$P(x) = 1-\left(\frac{\sin(\pi x)}{(πx)}\right)^2$$ seems to be, assuming the ...
2
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0answers
100 views

Irrationality of $\zeta(\frac{3}{2})$

Is $$ \zeta\left(\frac{3}{2}\right) = \sum_{n=1}^\infty \frac{1}{n^{3/2}} $$ an irrational number?
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2answers
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What's the value of $\sum_{k=1}^{\infty}(\zeta(2k+1)-\zeta(2k+2))$?

I'm confused about this. I have this expression $$ \frac{1}{2}=\sum_{k=1}^{\infty}(\zeta(2k)-\zeta(2k+1)) $$ Now if I want claculate $\zeta(2)$ I'll do the apropriate manipulations to get $$ \zeta(2)=\...
2
votes
1answer
66 views

Estimating a sum related to a short Euler product

The Question Is $$\sum_{\substack{n>y\\ p\mid n\Rightarrow p\leq y}}\frac{\Lambda(n)}{n^s\log n}=O(1/\log T)$$ where $y=(\log T)^{100}$ and $T$ is large? Background Assume that $$\log\zeta(\...
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0answers
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Integral representations of $\zeta(s)$ using the floor/frac functions. How could this one be derived?

Browsing the web, I found quite a few integral representations for $\zeta(s)$ that use the Fractional part {x} or the Floor-function $\lfloor x\rfloor$ e.g.: $$\zeta(s) = \dfrac{s}{s-1} - \frac12+s \...
2
votes
3answers
71 views

Computing the tail of the zeta function $\sum_{n>x}n^{-s}$

I want to compute $$ f_s(x)=\sum_{n>x}n^{-s} $$ for some $s>1$ (in my case, $s=3$). Of course $$ f_s(x)=\zeta(s)-\sum_{n\le x}n^{-s} $$ but for $x$ large this is hard to compute. Are there good ...
8
votes
2answers
145 views

Closed form of $\lim\limits_{n\to\infty}\left(\int_0^{n}\frac{{\rm d}k}{\sqrt{k}}-\sum_{k=1}^n\frac1{\sqrt k}\right)$

Show that $$ L=\lim_{s\rightarrow\infty}\left(\int_0^s\frac{ds'}{\sqrt{s'}}-\sum_{s'=1}^s\frac{1}{\sqrt{s'}}\right) = 1.460\ldots $$ My attempts: To begin, rewriting the limit of the form $$ ...
0
votes
1answer
47 views

Is this infinite product for zeta(2) trivial?

I have crafted an infinite product for zeta(2) shown here. Euler's prime product is the only one I'm aware of. In checking Math World, I don't see any products. Is that because they are trivial?
4
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0answers
207 views

Approximate zeros of a (hypothetical) analog of $\zeta(s)$

[Added numbers 11/13.] Motivation (can skip). When prime powers $p_n$ are used to calculate $$y(x) = \sum_{n=1}^{N}\frac{\sin (x \log p_n)}{p_n},\hspace{5mm}(1)$$ for (say) $N= 30,$ $x>5$, at ...
16
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1answer
269 views

Is there any proof that the Riemann Zeta function is not elementary?

I'm just curious, has anyone ever proved that the Riemann Zeta function is not an elementary function? Here I am using the term "elementary" in the sense of Liouville or as defined in this paper. ...
5
votes
0answers
155 views

Connection between Riemann hypothesis and distribution of primes. [closed]

Honestly, I have to say that I have hardly any experience in number theory. That's maybe one additional reason why the Riemann hypothesis has such a "mystic" appearance for me. You always hear or read ...
2
votes
0answers
211 views

Abel-Plana formula for $\zeta(s)$, is this integral approximation correct?

I wrote a computer program to calculate values for $\zeta(s)$. I was scanning for something that would calculate complex values for $\zeta(s)$. I found the following approximation under the Integral ...
12
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0answers
130 views

Can anyone improve on this work and find a closed form of $\zeta(3)$?

This was something I and another user came across independently, although he decided to post it on reddit. So while its already online, let me reproduce it here with the hope that someone will be able ...
8
votes
3answers
207 views

Closed form for $\sum_{k=1}^\infty(\zeta(4k+1)-1)$

Wikipedia gives $$\sum_{k=2}^\infty(\zeta(k)-1)=1,\quad\sum_{k=1}^\infty(\zeta(2k)-1)=\frac34,\quad\sum_{k=1}^\infty(\zeta(4k)-1)=\frac78-\frac\pi4\left(\frac{e^{2\pi}+1}{e^{2\pi}-1}\right)$$ from ...
9
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0answers
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Riemann zeta function Euler product for primes equivalent to $3$ mod $4$

Question: can $$ \zeta_1(s) = \prod_{p \equiv 3 \pmod{4}} \frac{1}{1 - p^{-s}} $$ be evaluated or written in terms of standard functions? Details: We can write the Riemann zeta function as \begin{...
4
votes
2answers
122 views

Proof of Functional Equation Zeta

$$ \pi^{-s/2}\zeta(s)\Gamma(s/2)=\pi^{-(1-s)/2}\zeta(1-s)\Gamma((1-s)/2) $$ (That's the equation that want to prove) Hello guys, so I'm trying to prove the functional equation of Riemann Zeta, ...
3
votes
0answers
54 views

Is there anything known about the zeros of $\displaystyle \sum_{n=1}^{\infty} \left(\frac{1}{\rho_n^s} +\frac{1}{\overline{\rho_n}^s}\right)$?

Assuming the RH and $\rho_n =\frac12 + \gamma_n i$ being the n-th non-trivial zero of $\zeta(s)$, then numerical evidence suggests that: $$f(s) :=\displaystyle \sum_{n=1}^{\infty} \left(\frac{1}{\...
2
votes
1answer
68 views

Trivial zeroes of Zeta are simple

The trivial zeroes of the Riemann Zeta function are located on $-2\mathbb N^*$ and they are simple. It is not difficult to see that, but the proof I have in mind is using the fact that $\xi(-2k)=\xi(1+...
5
votes
1answer
64 views

How to manipulate this functions to an identity involving the Riemann zeta function

The identity I want to prove is the following (from Stein's book, an introduction to Fourier analysis): $$\pi^{-s/2} \Gamma(s/2) \zeta (s)=\frac{1}{2} \int_{0}^{\infty}t^{\frac{s}{2}-1}(v(t)-1)dt$$ ...
1
vote
0answers
104 views

Riemann zeta function, functional equation, what completes this analogy?

What completes this analogy? This: $$\zeta(s) = 2^s\pi^{s-1}\ \sin\left(\frac{\pi s}{2}\right)\ \Gamma(1-s)\ \zeta(1-s)\;\;\;\;\;\;\;\;\;\;(1)$$ is to: $$\chi(s)=\pi ^{-\frac{s}{2}} \Gamma \left(\...
11
votes
1answer
244 views

Fibonacci numbers and the nontrivial zeros of the Riemann zeta function

Is this a mathematical coincidence? For $n=1,\dots,7$: $$ \left\lfloor \prod_{k=1}^n \arg\left(\rho_k\right)\right\rfloor = F_{n+1}, $$ where $\arg$ is the complex argument, $\rho_n$ is the $n$th ...
3
votes
1answer
90 views

Is the functional equation for $\zeta (s) \left(1-\frac{1}{3^{s-1}}\right)$ known?

It says in wikipedia that Hardy gave a simple proof of the functional equation for: $$\eta(s)=\zeta (s) \left(1-\frac{1}{2^{s-1}}\right)$$ and that it is: $$\eta(-s) = 2 \frac{1-2^{-s-1}}{1-2^{-s}}...
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0answers
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Can this relation be made into a functional equation?

I am trying to find the functional equation for this: $$\zeta(s)\left(1 - \frac{1}{n^{(s - 1)}}\right)$$ Therefore I let: $$x_1=\left(1-\frac{1}{n^{s-1}}\right)$$ which I substitute with $s\...
2
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0answers
69 views

Please help me understand Analytic Density $\lim_{\sigma \to 1^+}\frac{1}{\zeta(\sigma)}\sum_{n \in A} \frac{1}{n^{\sigma}}$

$d (A) = \lim_{\sigma \to 1^+}\frac{1}{\zeta(\sigma)}\sum_{n \in B} \frac{1}{n^{\sigma}}$ for $B \subset \Bbb{N}$. So clearly this limit is $0$ for reciprocally summable (convergent) $B$. My goal ...
3
votes
4answers
254 views

regularization of sum $n \ln(n)$

I was testing out a few summation using my previous descriped methodes when i found an error in my reasoning. I'm really hoping someone could help me out. The function which i was evaluating was $\...
8
votes
2answers
148 views

Series with $\zeta$

How do I calculate the following series: $$ \zeta(2)+\zeta(3)+\zeta(4)+ \dots + \zeta(2013) + \zeta(2014) $$ All I know is that $\zeta(2)=\pi^2/6$ and $\zeta(4)=\pi^4/90$. But this is not enough to ...
4
votes
0answers
71 views

How to prove the Riemann hypothesis holds for the first non-trivial zero? [duplicate]

The Riemann hypothesis states that all non-trivial zeros of the Riemann zeta function $\zeta(z)$ lie on the critical line $\Re(z)=1/2$. The MathWorld page on this topic mentions that the hypothesis ...
2
votes
2answers
85 views

Abel's Summation formula help.

I want to be able to show that, \begin{equation} \sum_{p} \log(p) p^{-s} = s \int_{1}^{\infty} \frac{\theta(t)}{t^{s+1}} dt \end{equation} where $\theta(x) =\sum_{p \le x} \log p$. and $\theta(x) = ...
2
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0answers
41 views

Riemann Zeta continued fraction approximants

In the paper Continued-Fraction Expansions for the Riemann Zeta Function and Polylogarithms by Djurdje Cvijovic and Jacek Klinowski, there is a claim that I cannot reproduce. In the abstract they ...
2
votes
1answer
99 views

A question from Titchmarsh's The Theory of the Riemann Zeta-Function.

On pages 35-36 here, we have that the integral $$\frac{1}{2i\sqrt{y}}\int_{1/2-i\infty}^{1/2+i\infty}\phi(s-1/2)\phi(1/2-s)(s-1)\Gamma(1+s/2)\pi^{-s/2}\zeta(s)y^sds$$ equals for $\phi(s)=1$ to: $$\...
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0answers
32 views

Increasing sequences and $\zeta$-type functions

The Riemann zeta function is defined as the sum $\zeta(s) = \sum_{n \geqslant 0} n^s$. The question is whether it globally characterizes the sequence of all natural numbers, in the following sense: ...
2
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0answers
47 views

Which mathematical objects generate the zeroes of $L$-functions?

I've studied analytic and algebraic number theory for years and years, and I encountered a hard question about Riemann zeta function and other kinds of $L$-functions - which might be one of the most ...
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0answers
57 views

how to prove an approximate to Riemann Xi function having only real zeros

I am searching for approximates to Riemann Xi function. Riemann $\Xi(z)$ function is related to Riemann $\zeta(s)$ function via $$\Xi(z)=\xi(1/2+iz)=\xi(s)=(1/2)s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)$$ ...
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An inequality for $|\zeta (s,a)|$, a detailed proof

In page 272 of [1], Apostol leaves as a reader's assigment to complete a proof of a related statement with Hurwitz zeta function, defined initially for $\sigma >1$ by the series $\sum_{n=1}^{\infty}...
5
votes
1answer
154 views

Behaviour of $\zeta(s)$ near $s=1$

I would appreciate if somebody could run this over and see if it works out? any suggestions or pointers would be appreciated. I denote the standard eta function $\eta$ by $\zeta^{*}$. I have not used ...
4
votes
2answers
125 views

Weird thing about $\sum_{k=2}^{\infty}(-1)^k\zeta{(k)}$

Consider the sum $S=\sum_{k=2}^{\infty}(-1)^k\zeta{(k)}$. By a simple manipulation, we can show: $$ S=\sum_{k=2}^{\infty}{(-1)^k\sum_{r=1}^{\infty}\frac{1}{r^k}}=\sum_{r=1}^{\infty}{\sum_{k=2}^{\infty}...
2
votes
1answer
63 views

Upper bound for $|\zeta'(s)|$ near the line $\sigma=1$, a detailed proof

In page 285 Apostol leaves as a reader's asigment the proof that $|\zeta'(s)|=O(\log^{2}t)$, this is for every $T>0$ there exists a positive constant $K$ (depending on T) such that $$|\zeta'(s)|\...
2
votes
2answers
197 views

$\zeta(2n)$ proof [duplicate]

Can anybody pass me on a good source to see the steps in proving, \begin{equation} \zeta(2n) = \frac{(-1)^{k-1}B_2k (2 \pi)^{2k}}{2(2k)!} \end{equation} I know how we start by looking at the product ...
1
vote
0answers
63 views

Looking for a way to apply the Taylor Series expansion to find derivatives for a function.

This post references the Riemann-Siegel formula found at here and at here. I am writing a Java program which implements this formula. I am having trouble with the remainder terms. The Riemann-Siegel ...
0
votes
1answer
53 views

Why is the Bernoulli Number $B_1$ sometimes $+ \frac{1}{2} $?

By using the recursive formula, \begin{equation} \sum_{i=0}^{n} \binom{n+1}{i} B_i = n+1 \end{equation} we find $B_1$ to be $\frac{1}{2}$ and not $- \frac{1}{2}$. Why is this?
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0answers
159 views

Zeta function, how to solve a finite geomatry summation.

I wanted to solve the zeta function for an undefined period "$d$". So for every $d\ge2$. $$\zeta(-s)= \frac{1}{(d^{s+1}-1)}\sum_{m=1}^{\infty} \frac{1}{2^{m+1}}\sum^{m}_{j=1} (-1)^{j+1}(j)^s\dbinom{m}...
12
votes
1answer
345 views

Why is $\zeta(1+it) \neq 0$ equivalent to the prime number theorem?

Reading through Titchmarsh's book on the Riemann zeta function, chapter 3 discusses the Prime Number Theorem. One way to prove this result is to check the zeta function has no zeros on the line $z = ...