An L-function free proof of Hua's Theorem on sums of five prime squares

We provide a new proof of Hua's result that every sufficiently large integer N ≡ 5 (mod 24) can be written as the sum of the five prime squares. Hua's original proof relies on the circle method and uses results from the theory of L-functions. Here, we present a proof based on the transference principle first introduced in[5]. Using a sieve theoretic approach similar to ([10]), we do not require any results related to the distributions of zeros of L- functions. The main technical difficulty of our approach lies in proving the pseudo-randomness of the majorant of the characteristic function of the W-tricked primes which requires a precise evaluation of the occurring Gaussian sums and Jacobi symbols.

Calculation of Gaussian Quadrature with High Accuracy Mathematics Package

We are checking here the dependence of numerical integration accuracy on the quantity of integration points and the accuracy of machine representation of numbers. For this purpose, the package HAHMath is applied. This package allows one to carry out calculations with arbitrary long machine decimal numbers, presented as vectors of integers. Integrals are substituted for Gaussian sums where Hermite polynomials zeros and corresponding weights, computed by the same package are used. It is shown that the chosen case accuracy of final calculations depends on the used machine numbers’ length more strictly than on the quantity of the integration points.