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

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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 $-\frac 1 {12}$. ...
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1answer
23 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?
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1answer
36 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 $\zeta(s) = a$ ...
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0answers
37 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 $n\to \infty$?
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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 ...
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1answer
51 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 ...
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1answer
31 views

relation of degree to number of zeros and Riemann hypothesis. [closed]

Assume I have a single variable polynomial with degree n, is there a proof that there are n roots? If there is why hasn't this been applied to the Riemann hypothesis?
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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
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3answers
212 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 ...
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3answers
54 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} ...
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0answers
34 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} ...
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1answer
48 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?
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0answers
16 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 ...
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1answer
108 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 ...
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2answers
102 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
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1answer
870 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 ...
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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 ...
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0answers
37 views

How does the Riemann Hypothesis state that $|\pi(x)-li(x)|<K_\epsilon x^{0.5+\epsilon}$?

In the textbook I'm reading (A Course in Computation Number Theory by Bressoud and Wagon), it says that an equivalent statement to the Riemann Hypothesis being true is that ...
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0answers
43 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 ...
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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 $\Sigma_{p\in X} \frac{1}{p^s}$, ...
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2answers
100 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 !
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1answer
49 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
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1answer
115 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
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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? ...
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2answers
43 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
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2answers
76 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) + ...
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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 ...
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1answer
39 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).
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2answers
124 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 ...
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0answers
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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 ...
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1answer
34 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 ...
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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. ...
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1answer
64 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
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0answers
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$\zeta(2)$ 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 ...
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0answers
41 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 ...
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0answers
32 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
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1answer
317 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 ...
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1answer
100 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 ...
6
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2answers
223 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 ...
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2answers
146 views

Finding the closed form for $\sum_{n=1}^{\infty }\frac{\zeta (4n)}{\beta^{4n-1}}$ [closed]

Finding the closed-form $$\sum_{n=1}^{\infty }\frac{\zeta (4n)}{\beta^{4n-1}}$$ for $\beta\in(1,+\infty)$. I learned from this site many many important things but I till need more, so I need ...
6
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2answers
116 views

The Riemann zeta function $\zeta(s)$ has no zeros for $\Re(s)>1$

I write $\zeta(s)$ for $\Re(s)>1$ as: $\zeta(s) = \prod_{p} (1-p^{-s})^{-1}$ Using this I can show that the Riemann zeta function has no zero for $\Re(s)>1$. I'm however not sure about the ...
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0answers
15 views

seeking upper/lower bounds of a function $F(m)$ related to Jacobi theta function

I am looking for the upper/lower bounds of function $F(m)$ defined and plotted above. The function is related to Jacobi theta function $\theta(x)$ and its derivative values at $x=1$: ...
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Are these identities Newton series?

Newton series is the following expansion of a function: $$f(x)=\sum_{k=0}^\infty \binom{x}k \Delta^k [f]\left (0\right)=\sum_{n=0}^{\infty} {x\choose n} \sum_{k=0}^n{n\choose k}(-1)^{k-n}f(k)$$ Now ...
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60 views

Riemann's explicit formula for $\pi(x)$

Riemann's explicit formula $J(x)=\mathrm{Li}(x)-\sum_{\Im\varrho>0}\left(\mathrm{Li}(x^\varrho)+\mathrm{Li}(x^{1-\varrho})\right)+\int_x^\infty\frac{\mathrm{d}t}{t(t^2-1)\log t}-\log2,$ where ...
2
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1answer
54 views

Summation with Zeta function

I'm currently studiyng Zeta function and I don't understand this identity: $$\sum_{n=1}^\infty x \sum_{p=0}^\infty \frac{x^{2p}}{(n\pi)^{2p+2}} = \sum_{p=1}^\infty \pi^{-2p}\zeta(2p)x^{2p-1} $$ I ...
3
votes
1answer
120 views

Cramer and Riemann Conjecture Implication

Cramer's conjecture gives $$p_{n+1}-p_n= O(\log^2 p_n)$$ while Riemann Hypothesis yields just $$p_{n+1}-p_n= O(\sqrt p_n\log^2 p_n).$$ Does Cramer conjecture on prime gaps imply Riemann Hypothesis ...
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0answers
60 views

Closed form for generating function of Riemann Xi function

What is the closed form for $$f(x)=\ \sum_{k=1}^\infty \frac{\xi(k)x^k}{k!}$$ or $$g(x)=\frac12 \sum_{k=1}^\infty \frac{\xi(k+1/2)x^k}{k!}$$ or $$w(x)=\frac12 \sum_{k=1}^\infty ...
1
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1answer
73 views

How to prove that Riemann zeta function is zero for negative even numbers?

Can anyone please explain to me how to prove that Riemann zeta function is 0 for all negative even numbers. In many references , they have just given the statement without any proof. Any explanation ...
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0answers
48 views

Analytic continuation of the Dirichlet $\eta(s)$ series to $\Re(s) \gt -1$. Why does this work?

Take the known Dirichlet $\eta(s)$ series, $$\displaystyle \eta(s) = \sum _{n=1}^{\infty } \left( {\frac {1}{(2\,n-1)^{s}}} - \frac{1}{(2\,n)^s}\right), \qquad \Re(s)>0$$ and add $\displaystyle ...
4
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1answer
109 views

Conway Complex Analysis Book Exercise 8 in the Riemann Zeta Function Chapter

I am studying the book of John B. Conway Functions of One Complex Variable(1978), and in the section of Riemann Zeta Function chapter 7 I couldn't solve the last exercise. Here it is: Let $\zeta (z)$ ...