Erd} os, P.: On a new method in elementary number theory which leads to an elementary proof of the prime number theorem. Proc. Nat. Acad. Sci. U. S. A. 35 (1949), 374-384.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Mathematicians, such as Bohr, Hardy and Ingham then observed how the Prime Number Theorem is, intrinsically, equivalent to the associated analytic problem, and so asserted that no elementary proof was feasible. It thus came as a surprise when, in the late forties, Selberg [Se1] and Erdos [Er1] constructed an elementary proof, though Ingham [In] later showed that it is essentially equivalent to the earlier analytic one of De La Vall ee Poussin. The proofs of Selberg and Erdos, which are based on Selberg s formula (1:1) x)log x X px x p log p = 2xlog x O(x) are ....

....the behaviour of g (x) P px; p2S; ff(p) g log p, as x 1: I) g (x) x=jGj for each g 2 G. II) There exists a subgroup H of G, of rank 2, such that g (x) ae 2x=jGj O(x=log x) if g 62 H ; O(x=log x) if g 2 H . We remark that our proof here bears many similarities to that in [Er1]. It is clear that Theorem 1 follows from Theorem 2, by taking S to be the set of primes that do not divide q, and ff(p) a where a is the residue class that p belongs to, modulo q. We can also prove a result analogous to Corollary 1: With G; S and ff as in Theorem 2, suppose that we are given ....

[Article contains additional citation context not shown here]

ERD OS, P., On a new method in elementary number theory which leads to an elementary proof of the Prime Number Theorem, Proc. Nat. Acad. Sci., 35 (1949), 374--384.

No context found.

Erd} os, P.: On a new method in elementary number theory which leads to an elementary proof of the prime number theorem. Proc. Nat. Acad. Sci. U. S. A. 35 (1949), 374-384.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC