on
Riemann Zeta Function
The zeta function, which is commonly known as the Riemann zeta function, is defined as follows.
\[\zeta (s) = \sum_{n = 1}^{\infty} \frac{1}{n^s}\]The zeta function’s Dirichlet series will converge when the real part of \(s\) is greater than 1. This is under the assumption that \(s\) is a complex number, which can have both real and imaginary components to it.
While the zeta function itself wasn’t discovered by Riemann, it’s most commonly associated to Riemann due to the Riemann hypothesis.
Euler Product Formula
Leonhard Euler proved the relation of the prime numbers and the zeta function as follows, given that \(p \in P\) where \(P\) is a set containing all prime numbers greater than 1.
\[\zeta (s) = \sum_{n=1}^{\infty} \frac{1}{n^s} = \prod_{p} \frac{1}{1 - p^{-s}}\]Wikipedia has the complete proof here. But here’s how the proof is done.
\[\zeta (s) = 1 + \frac{1}{2^s} + \frac{1}{3^s} + \frac{1}{4^s} + \frac{1}{5^s} + ... \\ \frac{1}{2^s} \zeta (s) = \frac{1}{2^s} + \frac{1}{4^s} + \frac{1}{6^s} + \frac{1}{8^s} + \frac{1}{10^s} + ...\]We can substract \(\frac{1}{2^s} \zeta (s)\) from \(\zeta (s)\).
\[(1 - \frac{1}{2^s}) \zeta (s) = 1 + \frac{1}{3^s} + \frac{1}{5^s} + \frac{1}{7^s} + \frac{1}{9^s} + ...\]With \(\frac{1}{3^s} (1 - \frac{1}{2^s}) \zeta (s)\) we’re going to get this.
\[\frac{1}{3^s} (1 - \frac{1}{2^s}) \zeta (s) = \frac{1}{3^s} + \frac{1}{9^s} + \frac{1}{15^s} + \frac{1}{21^s} + \frac{1}{27^s} + ...\]We can subtract it further from \((1 - \frac{1}{2^s}) \zeta (s)\).
\[(1 - \frac{1}{3^s})(1 - \frac{1}{2^s}) \zeta (s) = 1 + \frac{1}{5^s} + \frac{1}{7^s} + \frac{1}{11^s} + \frac{1}{13^s} + \frac{1}{17^s} + ...\]Notice that we’ve eliminated numbers on the right hand side of the equation that have 2 or 3 as their factors. Continuing the process for the factors of 5, 7, and so on will get us to this where \(p \in P\) and \(P\) is the set of all prime numbers.
\[\zeta (s) \prod_{p} 1 - \frac{1}{p^s} = 1 \\ \zeta (s) = \prod_{p} \frac{1}{1 - \frac{1}{p^s}} \\ \zeta (s) = \prod_{p} \frac{1}{1 - p^{-s}}\]Riemann Hypothesis
Bernhard Riemann extended the zeta function for \(s\) containing complex numbers. I honestly don’t think I’ve understood the concepts related to the Riemann hypothesis properly yet, but it’s an important conjecture regarding the distribution of prime numbers. Ben Riffer-Reinert from the University of Chicago wrote a paper on the relation of the zeta function and the prime number theorem here, taking account of trivial and non-trivial zeros of the zeta function as conjectured by Riemann.
According to Riemann, the trivial zeros occur at all negative even integers \(s \in \{ -2, -4, -6, ... \}\) while the non-trivial zeros are located at some points in the critical strip \(0 < \sigma < 1\) where \(s \equiv \sigma + it\) as quoted from this article. The Riemann hypothesis asserts that all non-trivial zeros in the critical strip is located on the critical line where the real part of \(s\) is \(\sigma = \frac{1}{2}\), which is known to be true for the first \(10^{13}\) non-trivial zeros found in the critical strip.
References
Proof of the Euler product formula for the Riemann zeta function
The Zeta Function and Its Relation to the Prime Number Theorem