The asymptotic formula in Waring's problem for one square and seventeen fifth powers

作者：Bruedern Joerg; Kawada Koichi^*

来源：Monatshefte fur Mathematik, 2011, 162(4): 385-407.

DOI：10.1007/s00605-009-0171-6

• Waring's problem
• Asymptotic formula
• Sums of one square and powers

Let R (k,s) (n) denote the number of solutions of the equation in natural numbers x, y (1), . . . , y (s) . By a straightforward application of the circle method, an asymptotic formula for R (k,s) (n) is obtained when k >= 3 and s >= 2 (k-1) + 2. When k >= 6, work of Heath-Brown and Boklan is applied to establish the asymptotic formula under the milder constraint s >= 7 center dot 2 (k-4) + 3. Although the principal conclusions provided by Heath-Brown and Boklan are not available for smaller values of k, some of the underlying ideas are still applicable for k = 5, and the main objective of this article is to establish an asymptotic formula for R (5,17)(n) by this strategy.

• 出版日期2011-4