thumb|upright=1.5|Plot of the absolute values of the zeta function, showing some of its features|alt=Plot of the absolute values of the zeta function One of the most famous unsolved questions in mathematics, dating from 1859, and one of the Millennium Prize Problems, is the Riemann hypothesis, which asks where the zeros of the Riemann zeta function are located. This function is an analytic function on the complex numbers. For complex numbers with real part greater than one it equals both an infinite sum over all integers, and an infinite product over the prime numbers, This equality between a sum and a product, discovered by Euler, is called an Euler product. The Euler product can be derived from the fundamental theorem of arithmetic, and shows the close connection between the zeta function and the prime numbers. It leads to another proof that there are infinitely many primes: if there were only finitely many, then the sum-product equality would also be valid at , but the sum would diverge (it is the harmonic series ) while the product would be finite, a contradiction., pp. 191–193.
thumb|upright=1.5|Plot of the absolute values of the zeta function, showing some of its features|alt=Plot of the absolute values of the zeta function One of the most famous unsolved questions in mathematics, dating from 1859, and one of the Millennium Prize Problems, is the Riemann hypothesis, which asks where the zeros of the Riemann zeta function are located. This function is an analytic function on the complex numbers. For complex numbers with real part greater than one it equals both an infinite sum over all integers and an infinite product over the prime numbers, This equality between a sum and a product, discovered by Euler, is called an Euler product. The Euler product can be derived from the fundamental theorem of arithmetic, and shows the close connection between the zeta function and the prime numbers. It leads to another proof that there are infinitely many primes: if there were only finitely many, then the sum-product equality would also be valid at , but the sum would diverge (it is the harmonic series ) while the product would be finite, a contradiction., pp. 191–193.