On Hypothesis \(K\) for biquadrates (Q1961737)

Let \(r_4(n)\) denote the number of representations of an integer \(n\) as a sum of 4 fourth powers of integers. Hypothesis \(K\), due to Hardy and Littlewood, is that \(r_4(n) = O\bigl( n^\varepsilon \bigr)\), for each \(\varepsilon >0\). The author's result is \[ r_4(n) = O\biggl( {n^{1/2} {(\log\log n)}^8 \over \log n }\biggr) . \] Apart from the factor \({(\log\log n)}^8\), this result improves by the factor \(\log n\) over what would be relatively trivial: an estimate \(r_4(n) = O\bigl( n^{1/2+\varepsilon }\bigr)\) would follow using the fact that the number \(r_{2,4}(n)\) of representations of \(n\) as a sum of two fourth powers is \(O( n^\varepsilon)\), and the estimate \(r_4(n) = O( n^{1/2})\) would follow after an appeal to results of the reviewer [Math. Z. 94, 223-234 (1966; Zbl 0151.03802] or the author [J. Reine Angew. Math. 475, 55-75 (1996; Zbl 0848.11041)] on the sum \(\sum_{n \leq x} r_{2,4}^2(n)\). The factor \(\log n\) in the denominator of the author's estimate is obtained using a sifting procedure based on the fact that the representable numbers \(n\) satisfy an equation \(n-x^4-y^4 = z^4+t^4\) in which the prime factors of the right side are subject to a congruence condition mod 8.