It has been proved that its value for all even positive numbers is transcendental, but for odd integers, only its value for 3 has been shown to be irrational by Apéry, and that value is sometimes called Apéry's constant. Ramanujan came up with rapidly converging series for its value at odd integers, and his summation can be
It has been proved that its value for all even positive numbers is transcendental, but for odd integers, only its value for 3 has been shown to be irrational by Apéry, and that value is sometimes called Apéry's constant. Ramanujan came up with rapidly converging series for its value at odd integers, and his summation can be used to show that its value at –1 is -1/12, as well as the fact that it is zero for any negative even number, called the trivial roots, but the conjecture that all of its nontrival roots have real part 1/2 is one of the most famous unsolved problems in number theory. For 10 points, what is this function, the subject of a namesake hypothesis, closely related to the distribution of prime numbers?
It has been proved that its value for all even positive numbers is transcendental, but for odd integers, only its value for 3 has been shown to be irrational by Apéry, and that value is sometimes called Apéry's constant.
{Riemann zeta} function