On a New Method in Elementary Number Theory Which Leads to An Elementary Proof of the Prime Number Theorem

ERD6iS2K 21/8(kk)-1/6 ) r where (p is a prime) and w = 6 ifp = 3, w =2 if p> 3;. a runs through the P -21 quadratic residues of p that lie between 0 and p, while A runs through 2 the remaining P2 numbers between 0 and p. Specializing again to the case p = 7 we obtain in the usual notation for hypergeometric series: 2r('/7)r(2/7)Pr(4/7) ~(1/2 F(1/4, 1/4, 1; 1/64) = 47 r(3/7)r(6/7)r(6/7)J 5. Let Gd(s) denote the analytical continuation of the function defined for a > 3/2 by the series E/(X2 + y2 + dz2)S- From a formula similar to (4) it is deduced that THEOREM: There exists a real number Od such that Gd(Od) =0 [d > do] where Od 0 as do, but Od 0 0.