The History and Importance of the Riemann Hypothesis

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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.

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