IRRATIONALITY MEASURE AND LOWER BOUNDS FOR π(x)

One of the most important functions in number theory is π(x), the number of primes at most x. Many of the proofs of the infinitude of primes fall naturally into one of two categories. First, there are those proofs which provide a lower bound for π(x). A classic example of this is Chebyshev’s proof that there is a constant c such that cx/ log x ≤ π(x). Another method of proof is to deduce a contradiction from assuming there are only finitely many primes. One of the nicest such arguments is due to Furstenberg, who gives a topological proof of the infinitude of primes. As is often the case with arguments along these lines, we obtain no information about how rapidly π(x) grows. Sometimes proofs which at first appear to belong to one category in fact belong to another. For example, Euclid proved there are infinitely many primes by noting the following: if not, and if p1, . . . , pN is a complete enumeration, then either p1 · · · pN + 1 is prime or else it is divisible by a prime not in our list. A little thought shows this proof belongs to the first class, as it yields there are at least k primes at most 2 k , that π(x) ≥ log log(x). For the other direction, we examine a standard ‘special value’ proof; see [MT-B] for proofs of all the claims below. Consider the Riemann zeta function