1 Introduction 2
1.1 Preliminary 3
2 Auxiliary lemmas 4
3 Lemmas 5
3.1 The number of integers that can be written as a square and a square of a
prime in an arithmetic progression 5
3.2 Correlation of r1 and r0 6
3.3 The sum in arithmetic progression 7
4 Upper bound 8
In [2] J. Friedlander and H. Iwaniec established a conditional asymptotic formula for the mean value of A(n)r0(n — 2)r0(n + 2), where A(n) is the von Mangoldt function and A(n) = logn for m = pk with p a prime and is zero otherwise, r0(n) is the number of ways to write n as a sum of two positive squares. This is equivalent to showing a strengthening of Lagrange’s four squares theorem, i.e. finding an asymptotic formula for p = a2+ b2+ c2+ d2subject to hyperbolic condition ad — bc = 1. Here we aim at studying an analogous sum when one of the variables is restricted to prime, i.e. A(n)r1(n — 2)r0(n + 2), where rgn) is the number of ways to represent n as a sum of a square and a square of a prime.
In [2] J. Friedlander and H. Iwaniec established a conditional asymptotic formula for the mean value of A(n)r0(n — 2)r0(n + 2), where A(n) is the von Mangoldt function and A(n) = logn for m = pk with p a prime and is zero otherwise, r0(n) is the number of ways to write n as a sum of two positive squares. This is equivalent to showing a strengthening of Lagrange’s four squares theorem, i.e. finding an asymptotic formula for p = a2+ b2+ c2+ d2subject to hyperbolic condition ad — bc = 1. Here we aim at studying an analogous sum when one of the variables is restricted to prime, i.e. A(n)r1(n — 2)r0(n + 2), where rgn) is the number of ways to represent n as a sum of a square and a square of a prime.
