Apery's constant is defined using this function, and expanding this function out as a Laurent series around s equals 1 gives the Stieltjes constants.
Karatsuba proved one conjecture regarding this entity posited by Atle Selberg, and the reciprocal of this function gives a Dirchlet series involving the Mobius function. Apery's constant is defined using this function, and expanding this function out as a Laurent series around s equals 1 gives the Stieltjes constants. The Bernoulli numbers form a useful recurrence to define this function, which is equal to pi squared over 6 at 2. This function is equal to the sum from one to infinity of one over n to the s. The real part of the zeros of this function all appear to lie on a critical line, though this has not been proven. For 10 points, name this function at the center of its namesake's unsolved hypothesis about the distribution of primes.
Karatsuba proved one conjecture regarding this entity posited by Atle Selberg, and the reciprocal of this function gives a Dirchlet series involving the Mobius function.
Riemann zeta function