ΒΙΒΛΙΟ: The Riemann Curvature Through History (pdf)

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The Riemann Hypothesis

Σαν σήμερα, 17 Σεπτεμβίου, πριν από 198 χρόνια γεννήθηκε ο Bernhard Riemann (1826 - 1866)

Βιογραφία: Georg Friedrich Bernhard Riemann

Bernhard Riemann: «Τα ολοκληρώματα είναι σαν την ποίηση, μερικές φορές»

ζΓ Relation

Riemann zeta function

The Basel Problem

Theorem 

$\zeta (2) = \sum_{n \mathop = 1}^\infty {\dfrac 1 {n^2} } = \dfrac {\pi^2} 6$

where $\zeta$ denotes the Riemann zeta function.

Proof

By Fourier Series of $x^2$, for $x \in ({-\pi}.. \pi)$: 

$ x^2 = \dfrac {\pi^2} 3 + \sum_{n \mathop = 1}^\infty [{({-1})^n \dfrac 4 {n^2} \cos n x}]$

Letting $x \to \pi$ from the left:

\(\pi^2\) \(=\) \(\dfrac {\pi^2} 3 + \sum_{n \mathop = 1}^\infty [{({-1})^n \dfrac 4 {n^2} \cos \pi x}\)]

\(\pi^2\) \(=\) \(\dfrac {\pi^2} 3 + \sum_{n \mathop = 1}^\infty [{({-1})^n ({-1})^n \dfrac 4 {n^2} }\)] Cosine of Multiple of $\pi$

\(\pi^2\) \(=\) \(\dfrac {\pi^2} 3 + 4 \sum_{n \mathop = 1}^\infty \dfrac 1 {n^2}\)

\(\leadsto \ \ \) \(\dfrac {2 \pi^2} 3\) \(=\) \(4 \sum_{n \mathop = 1}^\infty \dfrac 1 {n^2}\)

\(\leadsto \ \ \) \(\sum_{n \mathop = 1}^\infty \dfrac 1 {n^2}\) \(=\) \(\dfrac {\pi^2} 6\)

$\blacksquare$.

From WikiProof

Riemann’s Novel Lecture that Changed the Course of Geometry

That later set the stage for Einstein’s 

general theory of relativity

Albert Einstein revolutionized the early conceptions of gravity in 1915 when he presented a theory of general relativity which was based primarily on the fact that mass and energy warp the fabric of four-dimensional spacetime.

The credit for its underlying geometric or mathematical formulation goes to a mathematician named Georg Friedrich Bernhard Riemann who constructed a class of geometry (or elliptic geometry) that dealt with higher dimensions and hypersurfaces unlike Euclidean geometry (or flat geometry).

Bernhard Riemann (1826–1866) was a German mathematician who studied and served at the University of Göttingen. At Göttingen, he found a great teacher, Carl Friedrich Gauss (1777–1855), famous for his works in a wide range of areas in mathematics.

Source: cantorsparadise

This Result Keeps Me Up At Night

The series of the Riemann zeta function is equal to the Euler product

Lecture 1 | Introduction to Riemannian geometry, curvature and Ricci flow | John W. Morgan

Η Γεωμετρία του Riemann

Riemann Zeta Function

Riemann hypothesis asserts that all nontrivial zeros of the Riemann zeta function have real part $1/2$.

G. F. Bernhard Riemann: Επηρέασε περισσότερο από κάθε άλλον την πορεία των σύγχρονων Μαθηματικών

Ο Γκεόργκ Φρίντριχ Μπέρναρντ Ρίμαν (Georg Friedrich Bernhard Riemann, 17 Σεπτεμβρίου 1826 – 20 Ιουλίου 1866) ήταν Γερμανός μαθηματικός που συνεισέφερε σημαντικά στη Μαθηματική Ανάλυση, την Τοπολογία, την Αναλυτική Θεωρία των αριθμών και την 

Διαφορική γεωμετρία, προωθώντας τη μη ευκλείδεια γεωμετρία και ανοίγοντας έτσι τον δρόμο μεταξύ άλλων και για τη θεμελίωση αργότερα της Γενικής Θεωρίας της Σχετικότητας. Κατά τον D. Struik «με τον Ρίμαν φτάνουμε στον άνθρωπο που επηρέασε περισσότερο από κάθε άλλον την πορεία των σύγχρονων Μαθηματικών».

Riemann Sum

Riemann Sum

Riemann Sum

Riemann sum

Η Γεωμετρία του Riemann και ο παράξενος κόσμος του θείου Αλβέρτου (2 βίντεο)

THE RIEMANN ZETA FUNCTION ζ(s)

The Riemann zeta function is a device/object/function related to prime numbers.

In general, it is a function of complex variable defined by the next equation:

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The Riemann hypothesis

The Riemann hypothesis is one of the most famous and important unsolved problems in number theory. It states that,all the non-trivial zeros of the Riemann zeta function have real part equal to $1/2$. 

The Riemann zeta function is defined by the given relation for any complex number s that is not equal to $1$. 

It has many implications for the distribution of prime numbers and other arithmetic functions.

The Theorem of the Day: The Riemann Explicit Formula

The number of primes not exceeding a given real number $x$ is given by 

$π(x) = R(x) + \sum R(x^ρ)$

where the sum is over all zeros $ρ$ of $ζ$, the Riemann zeta function, and $R(x)$ is the entire function of $log x$ defined by 

$R(x)=1+ \sum_{n-1}^ \infty  \dfrac{(logx)^n}{nn!ζ(n+1)}$.

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Η Γεωμετρία του Riemann