How is the zeta function related to primes?
The expression states that the sum of the zeta function is equal to the product of the reciprocal of one minus the reciprocal of primes to the power s. This astonishing connection laid the foundation for modern prime number theory, which from this point on used the zeta function ζ(s) as a way of studying primes.
How is the Riemann hypothesis related to primes?
The Riemann hypothesis, formulated by Bernhard Riemann in an 1859 paper, is in some sense a strengthening of the prime number theorem. Whereas the prime number theorem gives an estimate of the number of primes below n for any n, the Riemann hypothesis bounds the error in that estimate: At worst, it grows like √n log n.
Is the Riemann zeta function finite or infinite?
The Riemann zeta function is defined by (1.61) ζ(s) = 1 + 1 2s + 1 3s + 1 4s + ⋯ = ∞ ∑ k = 1 1 ks. The function is finite for all values of s in the complex plane except for the point s = 1. Euler in 1737 proved a remarkable connection between the zeta function and an infinite product containing the prime numbers:
Is the Riemann zeta function holomorphic to the s-plane?
Thus the Riemann zeta function is a meromorphic function on the whole complex s-plane, which is holomorphic everywhere except for a simple pole at s = 1 with residue 1.
What is Riemann’s functional equation for sine?
Riemann’s functional equation. The equation relates values of the Riemann zeta function at the points s and 1 − s, in particular relating even positive integers with odd negative integers. Owing to the zeros of the sine function, the functional equation implies that ζ (s) has a simple zero at each even negative integer s = −2n,…
What is Deligne’s proof of the Riemann hypothesis?
Deligne’s proof of the Riemann hypothesis over finite fields used the zeta functions of product varieties, whose zeros and poles correspond to sums of zeros and poles of the original zeta function, in order to bound the real parts of the zeros of the original zeta function.