Insights The History and Importance of the Riemann Hypothesis

AI Thread Summary
The Riemann Hypothesis, introduced by Bernhard Riemann in 1859, posits that all non-trivial zeros of the Riemann zeta function lie on the critical line, a concept crucial to number theory. Its historical roots trace back to early mentions of prime numbers, with significant developments occurring from Euclid's Elements in the 3rd century BC to Hilbert's problems in 1900. The Extended Riemann Hypothesis (ERH) extends this idea to L-functions, impacting modern cryptography, particularly the RSA encryption algorithm, which relies on the complexity of prime factorization. The relationship between prime numbers and the Riemann Hypothesis highlights the evolution of prime number theory, especially from the 19th century onward. Understanding this hypothesis remains vital for advancements in both mathematics and computer science.
fresh_42
Staff Emeritus
Science Advisor
Homework Helper
Insights Author
2024 Award
Messages
20,671
Reaction score
27,922
The Riemann Hypothesis is one of the most famous and long-standing unsolved problems in mathematics, specifically in the field of number theory. It's named after the German mathematician Bernhard Riemann, who introduced the hypothesis in 1859.

  • RH: All non-trivial zeros of the Riemannian zeta function lie on the critical line.
  • ERH: All zeros of L-functions to complex Dirichlet characters of finite cyclic groups within the critical strip lie on the critical line.
  • Related Article: The Extended Riemann Hypothesis and Ramanujan’s Sum
The history of the Riemann hypothesis may be considered to start with the first mention of prime numbers in the Rhind Mathematical Papyrus around 1550 BC. It certainly began with the first treatise of prime numbers in Euclid's Elements in the 3rd century BC. It came to a - hopefully temporary - end on the 8th of August 1900 on the list of Hilbert's famous problems. And primes are the reason why we are more than ever interested in the question of whether ERH holds or not. E.g. the RSA encryption algorithm (Rivest-Shamir-Adleman, 1977) relies on the complexity of the factorization problem FP, that it is NP-hard. FP is probably neither NP-complete nor in P but we do not know for sure. Early factorization algorithms that ran in a reasonable time had to assume the extended Riemann hypothesis (Lenstra, 1988, [1]). So what do prime numbers have in common with the Riemann hypothesis which is about a function defined as a Dirichlet series?
$$
\zeta(s)=\sum_{n=1}^\infty \dfrac{1}{n^s}
$$
One has to admit that what we call prime number theory today originated in the 19th century when Dirichlet began in 1837 to apply analysis to number theory. There is a large gap between Euclid and Euler who published a new proof for the infinite number of primes in 1737.

Continue reading ...
 
Last edited:
  • Like
  • Informative
Likes Steve4Physics, yucheng, Janosh89 and 6 others
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
Thread 'Imaginary Pythagorus'
I posted this in the Lame Math thread, but it's got me thinking. Is there any validity to this? Or is it really just a mathematical trick? Naively, I see that i2 + plus 12 does equal zero2. But does this have a meaning? I know one can treat the imaginary number line as just another axis like the reals, but does that mean this does represent a triangle in the complex plane with a hypotenuse of length zero? Ibix offered a rendering of the diagram using what I assume is matrix* notation...

Similar threads

Replies
3
Views
2K
Replies
2
Views
2K
Replies
1
Views
2K
Replies
12
Views
3K
Replies
1
Views
3K
Back
Top