How does the Riemann zeta function work?

How does the Riemann zeta function work?

Riemann zeta function, function useful in number theory for investigating properties of prime numbers. For values of x larger than 1, the series converges to a finite number as successive terms are added. If x is less than 1, the sum is again infinite.

Has someone solved the Riemann Hypothesis?

The Riemann hypothesis is one of seven math problems that can win you $1 million from the Clay Mathematics Institute if you can solve it. British mathematician Sir Michael Atiyah claimed on Monday that he solved the 160-year-old problem. Atiyah has already won the the Fields Medal and the Abel Prize in his career.

Is the zeta function meromorphic?

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 the 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 does zeta function tell you?

The Riemann-Zeta function is a complex function that tells us many things about the theory of numbers. Its mystery is increased by the fact it has no closed form – i.e. it can’t be expressed a single formula that contains other standard (elementary) functions. by extending it into the complex plane.

Who proved Riemann zeta function?

Leonhard Euler proved the Euler product formula for the Riemann zeta function in his thesis Variae observationes circa series infinitas (Various Observations about Infinite Series), published by St Petersburg Academy in 1737.

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.”

Is zeta function continuous?

The Riemann zeta function is the infinite sum of terms 1/ns, n ≥ 1. For each n, the 1/ns is a continuous function of s, i.e. 1 ns = 1 ns0 , for all s0 ∈ C, and is differentiable, i.e.

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.

What if Riemann Hypothesis is false?

The Riemann hypothesis implies a bound on the error term in the prime number theorem. Specifically, it implies that π(x)=xlogx+O(√xlogx). If the Riemann hypothesis is shown not to be true, then we will not know that this result is true.