A set of generalized numbers showing Beurling's theorem to be sharp
Harold George Diamond · Illinois Journal of Mathematics · 1970
Beurling [1] proved that the prime number theorem holds for generalized (henoeforh e-) numbers if N (z), the number of e-integers not exceeding z, satisfies N (z) cz + 0 (z 1og-z) for c a positive number and , a number greater than {.Fulher, he showed that this result is sharp by giving an example of a "prime measure" and assooiated "integer measure" for which ? { but for which the prime number theorem is false.However, the measures of Beurling's example are continuous and thus differ from the usual (atomic) counting measures of prime number theory.We shall give an example of e-primes and e-integers for which the prime number theorem fails but N (z) cz + 0{z (log z)-8/}.Our oonstruction is based on Beurling's example and a method of approximating measures that we have used in [2].Let r(x) be the number of g-primes not exceeding x, and define II(x) ,:_ln-lr(x/).Set li(x) kf (log t)-ldt(k a constant) and define r (x) by r(x) I1 cos (log ) (log )-g for >_ 1 andr(x) 0forx _ 1, where denotes