Do sums of 4 biquadrates have a positive density?

Abstract

Since almost one century, the study of sums of s integral s-th powers has attracted the at tention of many mathematicians. Regardless of their efforts, the precise knowledge of their asymptotic behaviour still seems quite out of reach, except in the case of sums of two squares, where Landau [6] proved in 1908 tha t their number up to x is asymptotically equal to Cx(log x) -1/2, for some explicit constant C. On the faith of some computation, P. Barrucand [1] suggested in 1968 that sums of 3 cubes, or sums of 4 fourth powers have a zero asymptotic density, but Ch. Hooley [5] developped in 1986 heuristic arguments in favour of a positive asymptotic density. A third way to tackle the problem has been suggested by ErdSs and Rdnyi [3] in 1960: they built random sequences called pseudo s-th powers, which mimic the behaviour of integral s-th powers and they suggested that the number of representations of an integer as a sum of s pseudo s-th powers should almost surely behave according to a Poisson law, implying the positive asymptotic density of sums of pseudo s-th powers. Besides the fact tha t their proof was not complete (it has been indeed completed by Goguel [4] and Landreau [7]), the drawback of their approach is that it leads to a positive density for sums of 2 pseudo-squares as well. For that reason, in [2], we introduced in the ErdSs-Rdnyi model a refinement which takes into account the arithmetic behaviour of actual s-th powers (which are not well distributed in arithmetic progressions): in our model, sums of 2 pseudo-squares have a zero asymptotic density whereas this density exists and is strictly positive for sums of s pseudo s-th powers, when s > 3. In this paper, we report on our study on sums of 4 biquadrates. In section 2, we explain how to compute the asymptotic density in our probabilistic model. In section 3, we report on the extensive computation we performed on sums of 4 biquadrates, and show a good agreement between the behaviour of the actual biquadrates and that of the pseudo-biquadrates of our probabilistic ErdSs-Rdnyi model.