Article in Proceedings of the American Mathematical Society · October 005 doi: 10. 2307/4097907 citations reads 10 72 author

The major arcs. Let B be a real number with B ≥ A + c, where c is the constant appearing in (4.13) below. We may assume that B ≥ 50. Our first order of business is to approximate R(n;M₀), where M₀ = M(L^B,N). When
,
Lemma 5, the prime number theorem, and some partial summation yield
(1 ≤ j ≤ 22)
and
.
Furthermore, by Lemma 4 ,
.
Since the measure of M₀ is O(L^2BN⁻¹), we deduce that
(4.5) .
In the next section, we show that
(4.6) ,
where δ is a positive absolute constant. Since by Lemma 8.5 in Hua [5] we have , (4.4)–(4.6) imply that
(4.7) .
We deal with the contribution from the remainder of the major arcs by means of Lemma 2. Let K = M\M₀ and write
⁽M(q,a;P ^1/4,N) \M(q,a;L^B,N), if q ≤ L^B, K(q,a) =
M(q,a;P^1/4,N), if L^B < q ≤ P^1/4.
We want to bound the cardinality of the set
(4.8) .
By Bessel’s inequality,
(4.9)
For α ∈K(q,a), (2.11) gives
q ^−1/2+εL^cP ₁(1 + N|α − a/q|)^−1/2, when j = 1,
(4.10) ^j q ^−1/2+εL^cP_j, when j = 2,3,4.
Furthermore, by (4.1), h(α) is the difference of two sums of the form (2.8) with M ≤ P^1/2 and z = P^15/79. Applying (2.10) to each of those sums, we find that when α ∈K(q,a) ,
(4.11) .
Combining (4.9)–(4.11) and the trivial bounds for f₅(α),...,f₂₂(α), we get
.
Recalling (4.8), we deduce that
(4.13) .
4.3. The singular integral. In this section, we establish (4.6). By Lemma 6.2 in Vaughan [15] ,
(1 ≤ j ≤ 22),
and a similar argument yields
.
Also, some elementary analysis reveals that
,
where
.
We remark that numerical integration shows that δ₀ > 0.01. Using the above estimates, we obtain

Thus, (4.6) with δ = (79/15)δ₀ follows from the formula (see Ingham [6], p. 24)
.
4.4. The minor arcs. Next we proceed to bound the cardinality of the set
(4.14) .
By Bessel’s inequality,
(4.15)
We can estimate the last integral by means of Lemmas 1 and 3. By Lemma 3 with ρ = 1/158 (note that 15/79 = 1 − 128/158) ,
(4.16) sup h(α) P157/158+ε.
Also, by comparing the underlying diophantine equations, we get
Z 1 Z 1
|f₁(α)···f₂₂(α)^|2 dα ≤ |G₁(α)^|2 dα,
0 0
where G₁(α) is the generating function appearing in Lemma 1. Thus, (2.2) yields
(4.17) .
Recalling (2.1), we conclude from (4.14)–(4.17) that
(4.18) ,
where η = 2/159 − 1/79.
Finally, suppose that n ∈ (N,2N] is an odd integer such that n ∈X∪Y6. Then, by (4.3), (4.4), (4.7), (4.8), and (4.14), we obtain
.
In view of (4.13) and (4.18), this implies that all but O(NL^−A) odd n ∈ (N,2N] are representable as the sum of 23 seventh powers of primes. This establishes the case s = 23 of the theorem.
4.5. The case s ≥ 24. The generating function h(α)f₁(α)···f₂₂(α) that we used above is now replaced by
(4.19) h(α)f₁(α)^s−22f₂(α)···f₂₂(α).
First, we note that when s ≥ 24, we can use (4.10) and (4.11) to estimate directly R(n;K) in a similar fashion to (4.12). Hence, when s ≥ 24, we have X = ∅. Further, by the case k = 7 of Theorem 3 in [9] ,
.