Some Results On Primitive Pythagorean Triangle

A primeitive Pythagorean triangle is a triple (a, b, c) of natural numbers with a² + b² = c², a≤b, gcd(a, b, c) = 1. It is one of important problems in number theory to study the numbers of primeitive Pythagorean triangle, and many people studied this problem. Let P_p(N) and P_a(N) denote the numbers of Pythagorean triangles with perimeter and area, respectly. less than N. D. H. Lehmer[6] provedP_p(N) = log2·π^-2N + O(N^1/2logN). J. Duttlinger and W. Schwarz[1] provedP_a(N)=c₀N^1/2-c₁N^1/3+R_a(N)withThe eoponent 1/4 in the error term depends on the none-zero domain ofζ(s), which can't be improved now. But under the assumtion of the Riemann Hypothesis. J. Duttlinger and W. Schwarz[1] provedR_a(N)=O(N^5/22+ε)The exponent 5/22 can be replaced by 137/604,127/560,37/164,269/1238,127/616 as proved respectively byMenzer[7], Muller-Nowak-Menzer[9], Miiller-Nowak[8], Nowak[11], and W.Zhai[16].In this dissertation, We mainly study P_p(N) and prove the followingTheorem 1 There exits a constantδ> 0, such thatP_p(N)=log2·π^-2N+R_p(N),withTheorem 2 If the assumption of the Riemann Hypothesis is true, thenR_p(N)=O(N^110/305+ε). In this dissertation, we study P_a(N) on short interval, and obtainTheorem 3 For anyε> 0, H≥N^435/616+ε, we haveP_a(N + H)- P_a(N) = c₀HN^-1/2+O(HN^-1/2-ε).