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

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11
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0answers
27 views

Analogue of Dirichlet $L$-function for $\mathbb{F}_q[T]$.

We consider an analogue of the Dirichlet $L$-function in $\mathbb{F}_q[T]$. Let $g \in \mathbb{F}_q[T]$, $g \neq 0$, let $\chi: (\mathbb{F}_q[T]/(g))^\times \to \mathbb{C}^\times$ be a homomorphism, ...
3
votes
1answer
24 views

Bounding Riemann zeta function by Euler product formula for finite $N$

In a paper concerning a quick calculation of Bernoulli numbers the following inequality is presented (page 3), only referring to it as "not hard to see" that it holds. $$ \sum_{n \leq N}^{} n^{-s} ...
1
vote
4answers
70 views

$\sum\limits _{n=1}^{\infty}\frac{1}{n^s}=50$ Riemann-Zeta

I've to find a value for 's' were the infinit sum gives me the value 50. Is that possible and how do I've to calculate that value. I've no idea how te begin so, help me! Solve s for: $$\sum\limits ...
2
votes
1answer
41 views

Is $f(z)=\sum_{k=1}^{\infty}\frac{1}{(k+\frac{1}{k})^{z}}$ somehow related to Riemann's zeta function?

I was looking at this series $$ f(z)=\sum_{k=1}^{\infty}\frac{1}{(k+\frac{1}{k})^{z}} $$ and wondering if it is somehow realted to the Riemann's zeta function $$ ...
4
votes
0answers
55 views

Is there anything known about the value where the Euler and Hadamard products for $\zeta(s)$ are equal?

Take the Hadamard product for the Riemann $\xi$-function ($\rho$ is a non-trivial zero of $\zeta(s)$): $$\xi(s) =\frac12\, s\,(s-1) \,\pi^{-\frac{s}{2}}\, \Gamma\left(\frac{s}{2}\right)\, \zeta(s) ...
0
votes
3answers
48 views

Integral equal to Riemann Zeta Function

As part of a homework problem in Rudin, I need calculate $$ \int_{1}^{N} \frac{[x]}{x^{s+1}} \,dx$$ where $[x]$ is the floor function. Clearly $[x]$ has derivative $0$ everywhere but the integers. ...
1
vote
1answer
41 views

What are 'Regular Products'?

When looking at the functional equation for the Riemann zeta function, I came across the statement: For $s$ an even positive integer, the product $\sin{(\frac{\pi s}{2})}\Gamma({1-s})$ is regular. ...
1
vote
1answer
51 views

Wrong proof of the functional equation for $ \zeta (s) $ but why is the result correct?

If I introduce the function $ f(x)= |x|^{s-1} $ inside Poisson summatory formula and use the fact that $$ \sum_{n=-\infty}^{\infty}|n|^{s-1}=2\zeta (1-s) $$ If I combine this expression in the ...
3
votes
1answer
49 views

How could I get access to more than the first 2 mln non-trivial zeros of $\zeta(s)$?

I would like to test whether or not the following product (or its complement) $$\displaystyle \displaystyle \prod_{n=1}^\infty \left(1- \frac{s}{\frac12+ (-1)^n\, \gamma_n \, i} \right)$$ converges ...
1
vote
2answers
42 views

Express the Riemann zeta function as the series with the divisor function $d(n)$ [closed]

How do I show this? $$\zeta(s)^2 = \sum_{n=1}^\infty \frac{d(n)}{n^s}$$ Here, $\zeta(s)$ is the Riemann zeta function, and $d(n)$ is the number of positive divisors of $n$. This looks like the ...
0
votes
0answers
24 views

Bounds for the imaginary part of the non-trivial zeros of the Riemann zeta function.

Let $\rho_{k}=\beta_{k}+i\gamma_{k}$ the $k-th$ non-trivial zero of the Riemann zeta funcion. We consider only the zeros with $\gamma_{k}>0$ . Then we have$$\gamma_{1}=14.13...$$ ...
2
votes
1answer
69 views

Non-Trivial Zeroes of Riemann Zeta Function

I am in the process of proving the prime number theorem, and I was wondering if there was any easy way of proving that $\sum_{\rho\in R} \rho^{-2}$ converges, where $R$ is the set of all non-trivial ...
0
votes
1answer
46 views

Some clarifications on analytic continuation of Riemann's Zeta function on $\frac 1 2$

Here's my problem: Riemann's Zeta function converge iff $x>1$ so if I want to have a finite value for $\zeta(\frac 1 2)$ I need to use it's analytic continuation but Riemann's hypothesis states ...
0
votes
0answers
51 views

A way to sum supernatural numbers involving Zeta function's analytic continuation

I have this idea on how to sum supernatural numbers assigning them a finite value in a way similar to how we assume that the sum of every natural numbers from 1 to infinity equals $-\frac 1 {12}$. ...
0
votes
1answer
25 views

Inequality $(n+1)^{-s} \leq (2n)^{-s}$ true for all $s\leq1$ and natural $n$?

On the line $S_{2n}-S_n$ I don't understand how the first inequality was established for $s \leq 1$. I see how it works for $0 \leq s \leq 1$ but not s < 0. Any clues?
1
vote
1answer
39 views

What does $O\left(\frac{1}{\log\log T}\right)$ mean?

I've started to work through this paper by Levinson (1975). The abstract says "Let $s = \sigma + it$. For any complex a, all but $O\left(\frac{1}{\log \log T}\right)$ of the roots of $\zeta(s) = a$ ...
0
votes
0answers
42 views

Taylor coefficients for $\zeta(x)\Gamma(x)$

Could you show that in a right neighbourhood of $x=0$ $$\frac{1}{n!}\frac{d^n}{dx^n}\Gamma(x)\zeta(x)$$ behaves like $(-1)^n \zeta (-1)+1$ when $n\to \infty$?
5
votes
1answer
58 views

Why does Titchmarsh say that we can move the derivative under $\frac{2}{\pi}\int_0^\infty \frac{\Xi(t)}{t^2 + \frac{1}{4}} \cosh(\alpha t) \, dt$

If we define the Riemann-Xi function as $$ \Xi(t) = \xi(\frac{1}{2} + it)$$ where $$\xi(s) = \frac{1}{2}s(s-1)\pi^{-\frac{s}{2}}\Gamma(\frac{s}{2})\zeta(s),$$ then according to Titchmarsh in his ...
1
vote
1answer
52 views

Explicit formula for $\zeta$

I can not find anywhere on the internet a proof of this formula : $$\sum_{p\in\mathbb{P}, \; m\geq 1, \; p^m < x} \ln p = x - \sum_\rho \frac{x^\rho}{\rho} - \ln 2\pi - \frac{1}{2}\ln ...
0
votes
0answers
23 views

What is the functional equivalent of this matrix multiplication, if any?

I have a simple short Mathematica program that does the following: First we define a lower triangular matrix $A$ with the defintion: If $k$ divides $n$: $A(n,k) = k^{(1/2 - x)}$ else $A(n,k)=0$ ...
4
votes
3answers
217 views

Zeta function summation

Looking at the question here Surprising identities / equations shows an interesting identity, $$\large (\zeta (2)-1)-(\zeta (3)-1)+(\zeta(4)-1)-\cdots=\frac 12$$ How can we prove that result, or even ...
3
votes
3answers
55 views

Trivial zeros of the zeta function

The trivial zeros of the Riemann zeta function are negative even integers. But I don't understand how that makes sense with the original definition of the function. $\zeta(-2) = \sum_{n=1}^{\infty} ...
1
vote
0answers
35 views

Divergence Dedekind zeta function

Let $K$ be a number field, $\mathcal O_K$ be its ring of integers, T a positive integer and $N$ the norm function. Give an upper bound (in T) for $$\sum_{I\leq \mathcal O_K: N(I)\leq T} ...
0
votes
1answer
51 views

What function satisfies the following equation?

$$f(x)e^{-x}\Gamma(x/\pi)=f(\pi/2-x)e^{x-\pi/2}\Gamma(1/2-x/\pi)$$ I think it should be similar to Zeta function, but what is it exactly?
0
votes
0answers
18 views

the riemann zeta-function and its application

This semester I have à scientific research on zeta function and its application and Riemann Hypothesis . I'd like you to recommend me some links and books and sources what is thé application of zêta ...
8
votes
1answer
113 views

Showing $\sum_{n=1}^{\infty }(-1)^{n-1}\frac{(2n-2)!\zeta (2n)}{\pi ^{2n}}(1-\frac{1}{2^{2n}})(1+\frac{1}{2^{2n-1}})=\frac{\log(2)}{4}$.

How do I show that show $$\frac{0!\zeta (2)}{\pi ^2}\left(1-\frac{1}{2^2}\right)\left(1+\frac{1}{2}\right)-\frac{2!\zeta (4)}{\pi ...
8
votes
2answers
106 views

Why is this true ? $(1-s)\zeta(s) = \sum_{k=0}^\infty \frac{\Gamma(k+1-s/2) A_k}{\Gamma(1-s/2) k!}$

On the whole complex plane the following formula is valid : $$(1-s)\zeta(s) = \sum_{k=0}^\infty \frac{\Gamma(k+1-s/2) A_k}{\Gamma(1-s/2) k!}$$ where $A_k = \sum_{j=0}^k (2j-1) \zeta(2j+2) (k,j)$ ...
12
votes
1answer
884 views

What exactly *is* the Riemann zeta function? [duplicate]

I'm doing a little project on the $\zeta$ function, and I am at a complete loss of what it is actually doing. I understand it is way over my head, but when I am plugging say $\zeta(1 + i)$ into ...
0
votes
2answers
60 views

$\zeta (1/2 + i) = 0$, correct?

Plugging $\zeta (1/2 + i)$ into Wolfram Alpha yields me some complex number, but I was under the understanding that $\zeta (1/2 + it) = 0$, for all $t$ we have yet calculated... Is Wolfram Alpha just ...
3
votes
0answers
47 views

Is there an equivalent statement of Riemann Hypothesis in term of Random Matrix or physics theory?

We all know that Riemann Hypothesis has many equivalent statements. After Montgomery’s works on pair-relationship, we now know that ZEROs of Riemann Zeta function has similar properties as ...
1
vote
1answer
36 views

Putting a bound on sum of primes with a certain property

I'm trying to prove that the Dirichlet density of the set $X$ of primes of the form $p = n^2+1$ is zero. The only way I can think to do it is to put a bound on the sum $\Sigma_{p\in X} \frac{1}{p^s}$, ...
1
vote
2answers
103 views

A proof of $\sum{\mu(n)/n}=0$

I am looking for a proof (or references) of the following statement $$\sum_{n=1}^{\infty}{\frac{\mu(n)}{n}}=0$$ where $\mu$ is the Möbius function. Many thanks !
0
votes
1answer
50 views

Riemann Zeta Function and Including Complex Numbers [duplicate]

I'm a high-school senior attempting to make sense of the zeta function. I know Riemann regularized it to include complex numbers. Apparently, from this we could obtain that the sum of natural numbers ...
7
votes
1answer
119 views

Proving $\zeta(2) - \beta(1) + \zeta(4) - \beta(3) + \zeta(6)- \beta(5) + \ldots=1$

Trying to prove $$\zeta(2) - \beta(1) + \zeta(4) - \beta(3) + \zeta(6)- \beta(5) + \ldots=1$$ I found by numerical calculation that (when $k$ goes to infinity) $$\sum_{n=1}^{k}\zeta ...
4
votes
1answer
63 views

What is known about the complex solutions to $\zeta(s)=-1$?

I don't recall ever coming across a discussion of the complex solutions to the equation $$\zeta(s)=-1,$$ where $s\in\mathbb{C}$. How many such solutions exist? Is there any literature on this? ...
0
votes
2answers
44 views

Properties of $\underset{k\geq1}{\sum}\frac{1}{\left(2k-1\right)^{s}}$

Is this function $$\underset{k\geq1}{\sum}\frac{1}{\left(2k-1\right)^{s}},\,Re(s)>1$$ well known? In particular I'm interessed about analytic continuation and its zeros and poles. Have this ...
2
votes
2answers
81 views

Riemann Hypothesis: Proving a relation between $\psi(x)$ and $\pi(x)$

I am trying to prove the following If $\pi(x) = \operatorname{Li}(x) + O(x^{\frac{1}{2}}\log(x))$ then $\psi(x) = x + O(x^{\frac{1}{2}} \log^2(x))$ I have tried using $\psi(x) = \theta(x) + ...
1
vote
1answer
98 views

Curve profile for the logarithm-integral sum term of Riemann explicit formula?

I am considering the following term from the Riemann explicit formula (see here >>>): $$\sum_{\rho(\Im>0)}{\mathrm{li}(x^\rho)}$$ with $\rho$ non-trivial zeros of $\zeta$-function. I have a plot ...
0
votes
1answer
42 views

At what hight is the nth zero Riemann zeta function? [closed]

At what hight is the nth zero Riemann zeta function? It is a mathematical formula but I can not get under it on the Internet (once I found).
4
votes
2answers
128 views

Divisor and Riemann Zeta functions series proofs

Where d(n) is the number of divisors of n, show that $\sum_{n=1}^\infty d(n)z^n = \sum_{k=1}^\infty \frac{z^k}{1-z^k}$ where both sides converge for |z|<1 and show that $\sum_{n=1}^\infty ...
1
vote
0answers
59 views

Has the Riemann Hypothesis been generalized to the Octonions and the Quaternions?

I've noticed that it uses imaginary numbers. I know that sometimes when I have too few dimensions like (-1)^n, dots show where I might expect lines due to imaginary numbers. So perhaps there is a ...
1
vote
1answer
35 views

Understanding a series representation of the logarithm of the zeta function

I am reading through M. Ram Murty's Problems in Analytic Number Theory and have the following question regarding the first step in his proof of Dirichlet's Theorem. Given this definition for the zeta ...
1
vote
0answers
24 views

how to prove $\Phi(t)$ is divergent when $Im(t)=\pi/2$?

The Riemann $\Xi(z)$ function admits a Fourier transform of a even kernel $\Phi(t)=3e^{5t/4}\theta'(e^{t})+2e^{9t/4}\theta''(e^{t})$. Here $\theta(z)$ is the Jacobi theta function. ...
3
votes
1answer
66 views

Moving the integral $Q(x) = -\frac{e^{-1/2x}}{4i}\int_{1/2-i\infty}^{1/2+i\infty} \zeta(s)\Gamma(\frac{s}{2})\pi^{-s/2}e^{xs} ds$ past Re(s) = 1.

Given the integral $$Q(x) = -\frac{e^{-1/2x}}{4i}\int_{1/2-i\infty}^{1/2+i\infty} \zeta(s)\Gamma(\frac{s}{2})\pi^{-s/2}e^{xs} ds,$$ I know that the integrand is holomorphic except for simple poles at ...
5
votes
0answers
49 views

$\zeta(2)$ Euler's proof (Basel problem) [duplicate]

At one point Euler assumes that $$\frac{\sin x}{x} = \prod_{n=1}^{\infty}\left(1-\frac{x}{n\pi} \right)\left(1-\frac{-x}{n\pi} \right)$$ Why does he assume that? If we factor random functions ...
2
votes
0answers
42 views

Comprehensive summary of where the function $\pi^{-\frac x\pi}$ can be encountered

I am studying the special functions, including the Riemann Xi and Zeta, and everywhere a function $\pi^{-\frac x\pi}$ pops up, usually as multiplier to the Gamma function. But yet I am not sure this ...
0
votes
0answers
34 views

Explicit analytic continuation of the zeta function and shift operators

Is there a way to compute the radius of convergence of the expansion of the zeta function, e.g. around $a=2+2i$? We have $\zeta(a)=\sum_{n=1}^\infty n^{-a}\approx 0.867.. - i\,0.275.. $, but I ...
5
votes
1answer
410 views

Double sum and zeta function

This is a personal research that came to an end , since the results were not those which were being anticipated. I was unable to come up with a solution therefore I post the topic here: Prove (it ...
6
votes
1answer
105 views

Integral Representation of the Zeta Function

How does one get from this $$\zeta(s)=\sum_{k=1}^{\infty}\frac1{k^s}$$ to the integral representation $$\zeta(s)=\frac1{\Gamma(s)}\int_{0}^\infty \frac{x^{s-1}}{e^x-1}dx$$ of the Riemann Zeta ...
11
votes
2answers
377 views

Using the rules that prove the sum of all natural numbers is $-\frac{1}{12}$, how can you prove that the harmonic series diverges?

I think I understand intuitively how we can assign a value to the sum of all natural numbers. But of all the proofs that I've seen that show why $\zeta(-1) = -\frac{1}{12}$, none of them use their own ...