The Riemann zeta function is one of the most important functions in mathematics. It's defined as the infinite sum of one over n raised to the power s, where s is a complex number. For this introduction, we'll consider s as a real number greater than 1, which ensures the series converges. For example, when s equals 2, we get the famous Basel problem sum, which equals pi squared over 6. The function was first introduced by Leonhard Euler in the 18th century and later extensively studied by Bernhard Riemann in connection with the distribution of prime numbers.
The Riemann zeta function's series definition only converges when the real part of s is greater than 1. However, through a process called analytic continuation, mathematicians can extend the function to the entire complex plane, except for a simple pole at s equals 1. This extension satisfies a functional equation that relates zeta of s to zeta of 1 minus s. The extended function has zeros at all negative even integers, called trivial zeros, and infinitely many non-trivial zeros that lie on the critical strip where the real part of s is between 0 and 1. The famous Riemann Hypothesis conjectures that all non-trivial zeros lie exactly on the critical line where the real part equals one-half.
The Riemann Hypothesis, proposed by Bernhard Riemann in 1859, is considered one of the most important unsolved problems in mathematics. It states that all non-trivial zeros of the zeta function have a real part equal to one-half, meaning they all lie on the critical line. Despite extensive computational verification for the first 10 trillion zeros, a complete proof remains elusive. The hypothesis has profound implications for number theory, particularly regarding the distribution of prime numbers. It's one of the seven Millennium Prize Problems, with a million-dollar reward for its solution. The hypothesis connects seemingly unrelated areas of mathematics, from quantum physics to random matrix theory, highlighting the zeta function's fundamental importance.
The Riemann zeta function has a profound connection to the distribution of prime numbers. Euler discovered that the zeta function can be expressed as an infinite product over all prime numbers, known as the Euler product formula. This relationship allows mathematicians to translate properties of the zeta function into statements about prime numbers. The prime number theorem, which describes the asymptotic distribution of primes, is closely related to the fact that the zeta function has no zeros on the line where the real part equals 1. Riemann's explicit formula connects the distribution of primes directly to the zeros of the zeta function. If the Riemann Hypothesis is true, it would provide the best possible error term for the prime number theorem, giving us the most accurate understanding of how prime numbers are distributed.
To summarize what we've learned about the Riemann zeta function: First, it's defined as the infinite sum of 1 over n raised to the power s, which converges when the real part of s is greater than 1. Through analytic continuation, mathematicians extended it to the entire complex plane except for a simple pole at s equals 1. The famous Riemann Hypothesis conjectures that all non-trivial zeros of the function lie on the critical line where the real part equals one-half. The zeta function has deep connections to prime numbers through the Euler product formula and Riemann's explicit formula. Despite over 160 years of effort by the world's greatest mathematicians, proving the Riemann Hypothesis remains one of the most significant unsolved problems in mathematics, with implications across number theory, physics, and beyond.