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

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Non-standard numbers and exponential form of Zeta function

Basic idea For a long time I was looking for a numerical system that would allow to compare infinite sets. In contrast to Cantor's approach that empathizes the possibility of on-to-one correspondence ...
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28 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 ...
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55 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): $$\pi^{-s/2} \Gamma(s/2) \zeta (s)=\frac{1}{2} \int_{0}^{\infty}t^{\frac{s}{2}-1}(v(t)-1)dt$$ ...
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81 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 ...
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177 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 ...
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54 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 ...
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33 views

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 ...
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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 ...
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81 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 ...
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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 ...
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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 ...
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2answers
53 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) = ...
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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 ...
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82 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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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: ...
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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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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 ...
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136 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 ...
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2answers
121 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: $$ ...
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1answer
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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 ...
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2answers
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$\zeta(2n)$ 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 ...
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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 ...
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1answer
48 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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Zeta function, how to solve a finite geomatry summation.

I wanted to solve the zeta function for an undifend 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} ...
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1answer
161 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 = ...
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My problem in the definition of Dirichlet generating function?

In the definition of Dirichlet generating function "for the square-free numbers " is: $$ \frac{\zeta(s)}{\zeta(2s)}=\sum_{n=1}^{\infty} \frac {|\mu(n)|}{n^s} $$ where $\mu$ is Moebius ...
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Extending the Riemann zeta function using Euler's Theorem.

Euler's theorem states that if the real part of a complex number $z$ is larger than 1, then $\zeta(z)=\displaystyle\prod_{n=1}^\infty \frac{1}{1-p_n^{-z}}$, where ...
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1answer
46 views

Estimate for the integral using convexity bound

I'm reading the proof of Hardy and Littlewood's theorem in the book Analytic Number Theory, written by Henryk Iwaniec and Emmanuel Kowalski (p. 547): Theorem (Hardy and Littlewood): Let $N_0(T)$ ...
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272 views

Riemann hypothesis reformulation - again

Yesterday I started to write a paper about the reformulation of the Riemann Hypothesis. My idea was to map the function such that all of the trivial zeros are outside of the unit disk, and the ...
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1answer
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What is the Mobius sum $\sum_{n=1}^\infty \frac{(-1)^{n+1}|\mu(n)|}{n^s}$?

It can be observed that, $$A(s) := \sum_{n=1}^\infty \frac{\mu(n)}{n^s} = \frac{1}{\zeta(s)}$$ $$B(s) := \sum_{n=1}^\infty \frac{|\mu(n)|}{n^s} = \frac{\zeta(s)}{\zeta(2s)}$$ $$C(s) := ...
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Compute $\sum_{b=2}^{\infty}{\left[\sum_{k=1}^{\infty}{\left(\frac{digitsum_b(k)}{k(k+1)}+\left(1+\frac1b\right)\frac{(-1)^k}{kb^k}\right)}\right]}$

Warning: This post contains more than one question and is pretty long. I decided include them all in this post, because they all emerged from the same question. Furthermore, I decided to display all ...
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riemann zeta function : entire and even Laplace transforms

$$\xi(s) = s(s-1)\pi^{-s/2}\Gamma(s/2) \zeta(s)$$ $$\xi(s) = \xi(1-s)$$ thus $\Xi(s) = \xi(1/2+s) = \Xi(-s)$ is even, and furthermore it is an "entire and even Laplace transform" : $$\Xi(s) = ...
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Equating two definitions of Zeta function

The Zeta function $\zeta(s)$ is defined as following $$ \zeta(s)=1+\frac1{2^s}+\frac1{3^s}+\frac1{4^s}+\frac1{5^s}+\cdots=\sum_{n=1}\frac1{n^s} $$ Now it has been shown that $$ \tag1 ...
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36 views

Eulers proof sum of natural numbers

I've to recheck Eulers proof of the sum of the natural numbers, but I dont now exactly what it is? It has something to do with the $\zeta(s)$? Thanks in advance
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A question from Titchmarsh's zeta function book.

On page 30, he writes that $\xi(0)=-\zeta(0)=1/2$, but on page 16 he writes that: $\xi(s)=1/2 s(s-1)\pi^{-1/2s}\Gamma(1/2s)\zeta(s)$ in eq.(2.1.12); so if I plug into this equation $s=0$ then I get ...
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1answer
41 views

Bernoulli Numbers generating function and Riemann Zeta function

I've been studying Bernoulli numbers and I came across this summation: $$ \sum_{n=1}^{\infty}\frac{B_n x^n}{n!} = \sum_{n=1}^{\infty}\frac{-n \zeta(1-n) x^n}{n!} = -\sum_{n=1}^{\infty}\frac{\zeta(1-n) ...
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Out of all the proofs of the PNT, which one is the most accessible?

I have been studying the continuation of the Riemann zeta function $\zeta(s)$ for the past while. I can prove that all the zeroes must lie in the critical strip.I am currently in the process of using ...
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How shall I get an estimate of $\int_{1-\frac{c}{log t}-iT}^{1-\frac{c}{log t}+iT}\frac{\zeta(s-1)}{\zeta(s)}\frac{x^s}{s}ds$?

Please help me on the following. I need to estimate $$\int_{1-\frac{c}{\log t}-iT}^{1-\frac{c}{\log t}+iT}\frac{\zeta(s-1)}{\zeta(s)}\frac{x^s}{s}ds$$ where $c$ is a constant, $T>0$. What i ...
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When is $\zeta(s)=0$?

At what real constant does $\zeta(s)=0$? Does that constant have any significance? Thank you very much for any help provided.
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Is this Riemann Zeta function integral formula known about?

I discovered that $$\zeta(s)=\int_0^1\frac{(-\log(1-x))^{s-1}}{x(s-1)!}dx.$$ Is this an obvious result that is not worth much interest or is this new and unique?
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Using The Riemann Zeta Functional Equation

Riemann was able to establish the following link between the Riemann zeta function and the weighted prime counting function $J(x)$. $$\ln(\zeta(s))=s\int_1^\infty J(x)x^{-s-1}dx$$ Using the Mellin ...
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Probability number is divisible by half the square of a prime?

Let $p$ be a prime. What is the probability that a number of the form $\left \lceil \frac{p^2}{2} \right \rceil$ divides a random positive integer $n$. I have a solution that involves the Riemann-Zeta ...
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Prove that these Dirichlet L function are equal to these zeta function products.

Prove or disprove that: $$2 L_{2,1}(s)-2 \zeta (s)+\zeta (s)=\left(1-\frac{1}{2^{s-1}}\right) \zeta (s)$$ $$3 L_{3,1}(s)-3 \zeta (s)+\zeta (s)=\left(1-\frac{1}{3^{s-1}}\right) \zeta (s)$$ Where ...
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59 views

How $\zeta(-1)$ is defined? [duplicate]

I know some proof that $\zeta(-1)$ equals to $-\frac{1}{12}$. here it is: Let $S_1 = (1)+(-1)+(1)+(-1)+...$ Then we have that $2S_1=1$ because if we shift the second $S_1$ for 1 to right we have ...
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1answer
36 views

Identity for $L(s,\chi)L(s,\bar\chi)$

I was told recently that there is an identity roughly of the form $$L(s,\chi)L(s,\bar\chi)=\zeta(s)^2$$ If true, it seems like there should be a short proof of this. Could someone supply a ...
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+100

Weierstrass's M-test example for uniform convergence and switching Sum and Integral.

How would I go about finding $M_n$ in \begin{equation} \sum_{n=1}^{\infty} \int_{0}^\infty x^{\frac{s}{2}-1}e^{-\pi n^{2}x}dx \end{equation} to show that it is uniformly convergent? UPDATE: Just ...
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$\eta(1) = \ln(2)$ proof using Abel's Theorem

Hi I was just wondering how does one justify $\eta(1) = \ln(2)$. Looking at the power series for $\ln(1+x)$ we have \begin{equation} \ln(1+x)= \sum_{n=1}^{\infty} \frac{(-1)^{n+1}x^{n}}{n} ...
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Applying the inversion formula to Hardy's theorem

So, in the proof of Hardy's theorem, who says that $\zeta$ has infinite zeros on the critical line, we have eventually that ...
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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 ...