V.27. Problems and Results in Additive Number Theory 715 integral" of n). Some care is needed in interpreting the integral because log 1 = 0, but one can avoid this prob- lem by integrating from 2 to n instead, which changes the function by just an additive constant. The prime number theorem, proved independently by hadamard [VI.65] and de la vallée poussin [VI.67] in 1896, states that li(n) is indeed a good approxima- tion to (n), in the sense that the ratio of the two functions tends to 1 as n tends to infinity. This result is considered one of the great theorems of all time, but it is by no means the end of the story. The proofs of Hadamard and de la Vallée Poussin used the riemann zeta function [IV.2 §3] (s). The Riemann zeta function is defined to be 1 -s +2 -s +3 -s +· · · when- ever s is a complex number with real part greater than 1; this expression defines a holomorphic function [I.3 §5.6], which can be extended (by analytic continu- V.27 Problems and Results in Additive Number Theory Is every even number greater than 4 the sum of two odd primes? Are there infinitely many primes p such that p + 2 is also a prime? Is every sufficiently large positive integer the sum of four cubes? These three questions are all famous unsolved problems in num- ber theory: the first is called the Goldbach conjecture, the second is the twin prime conjecture (discussed in some detail in analytic number theory [IV.2]), and the third is a special case of Waring's problem, which we shall discuss later. These three problems belong to an area of mathemat- ics known as additive number theory. In order to say in general terms what this area is, it is useful to make