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**On Waring’s problem for cubes.**
*(English)*
Zbl 0574.10046

The author obtains the following results:

(a) The asymptotic formula for the number of representations of integers N as sums of 8 positive cubes. Previously (using Hua’s inequality), the asymptotic formula was established for 9 or more cubes. In the author’s result, the error term differs from the main term by a power of log N.

(b) A lower bound for the number of representations of N as the sum of 7 cubes. This provides a proof of the 7-cubes theorem by the use of the Hardy-Littlewood method.

(c) The number of integers \(\leq N\) that are sums of 3 cubes is \(\gg N^{(19/21)-\epsilon}.\) There are also other related results. Of these, (c) is an improvement of the author’s result [Bull. Lond. Math. Soc. 17, 17-20 (1985; Zbl 0562.10022)], the method for which was derived from the reviewer’s papers [Acta Arith. (to appear; see the following review)] (this being the iterative use of the Hardy-Littlewood method).

The main additional idea required in the present paper lies in a suitable adaptation of an identity of Davenport used in his work on sums of three cubes. The identity is, with \(x_ 2=x_ 1+hm^ 3\), \(4(x^ 3_ 2-x^ 3_ 1)=hm^ 3\{3(2x_ 1+hm^ 3)^ 2+h^ 2m^ 6\}.\) Some ideas of C. Hooley (used in his work on cubes) also are used. The author uses the Hardy-Littlewood method in different ways (especially for estimating the number of solutions of certain equations). The numerous notations have resulted in some misprints.

(a) The asymptotic formula for the number of representations of integers N as sums of 8 positive cubes. Previously (using Hua’s inequality), the asymptotic formula was established for 9 or more cubes. In the author’s result, the error term differs from the main term by a power of log N.

(b) A lower bound for the number of representations of N as the sum of 7 cubes. This provides a proof of the 7-cubes theorem by the use of the Hardy-Littlewood method.

(c) The number of integers \(\leq N\) that are sums of 3 cubes is \(\gg N^{(19/21)-\epsilon}.\) There are also other related results. Of these, (c) is an improvement of the author’s result [Bull. Lond. Math. Soc. 17, 17-20 (1985; Zbl 0562.10022)], the method for which was derived from the reviewer’s papers [Acta Arith. (to appear; see the following review)] (this being the iterative use of the Hardy-Littlewood method).

The main additional idea required in the present paper lies in a suitable adaptation of an identity of Davenport used in his work on sums of three cubes. The identity is, with \(x_ 2=x_ 1+hm^ 3\), \(4(x^ 3_ 2-x^ 3_ 1)=hm^ 3\{3(2x_ 1+hm^ 3)^ 2+h^ 2m^ 6\}.\) Some ideas of C. Hooley (used in his work on cubes) also are used. The author uses the Hardy-Littlewood method in different ways (especially for estimating the number of solutions of certain equations). The numerous notations have resulted in some misprints.

Reviewer: K.Thanigasalam