What does the Riemann zeta function tell us?
Riemann zeta function, function useful in number theory for investigating properties of prime numbers. Written as ζ(x), it was originally defined as the infinite series ζ(x) = 1 + 2−x + 3−x + 4−x + ⋯. For values of x larger than 1, the series converges to a finite number as successive terms are added.
Is the Riemann zeta function holomorphic?
The Riemann zeta function is defined for other complex values via analytic continuation of the function defined for σ > 1. 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.
Why is Riemann zeta function important?
The Zeta function is a very important function in mathematics. While it was not created by Riemann, it is named after him because he was able to prove an important relationship between its zeros and the distribution of the prime numbers. His result is critical to the proof of the prime number theorem.
What would happen if the Riemann hypothesis was solved?
Considered by many to be the most important unsolved problem in mathematics, the Riemann hypothesis makes precise predictions about the distribution of prime numbers. If proved, it would immediately solve many other open problems in number theory and refine our understanding of the behavior of prime numbers.
Is the Riemann zeta function symmetric?
“Furthermore, the fact that ζ(s)=ζ(s∗)∗ for all complex s ≠ 1 (s∗ indicating complex conjugation) implies that the zeros of the Riemann zeta function are symmetric about the real axis.”
Why 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.