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

Lemma 1. Let P ≥ 10 and assume the above notation. Then

2. 
   (1 ≤ i ≤ 20).
   

Furthermore, if u ≥ 1, we have

3. 
   .
   

Proof. To prove (2.2), we refer to the theorem in Thanigasalam [13] for the case i = 20 and then apply the iterative method in Vaughan [14] to deduce the remaining cases (see the discussion pertaining to k = 7 on pp. 455–459 in [14]).
We deduce (2.3) from (2.2) by a simple version of the circle method. We write (2.4) N = M(Q,P⁷), n = [0,1] \N, and N(q,a) = M(q,a;Q,P⁷), where Q = P^1/9. For α ∈n, Lemma 2.1 in Kawada and Wooley [7] yields

5. 
   .
   

Combining (2.1), (2.2), and (2.5), we obtain

5. 
   
   

Suppose now that α ∈N(q,a) ⊂N. By Lemmas 6.1 and 6.3 in Vaughan [15] ,
q ^−1/7P(1 + N|α − a/q|)⁻¹, if j = 1,
_λ
g α;
q^−1/7P^λj, if 2 ≤ j ≤ 10,
whence
.
We deduce that
(2.7)
The desired bound follows from (2.4), (2.6), and (2.7).
2.2. Exponential sum estimates.
Lemma 2. Let α be real, let ξ_m be complex numbers with |ξ_m|≤ τ(m)^c, and define
(2.8) ,
with ψ(n,z) given by (1.5). Suppose that M ≤ P^11/20, z ≤ ^p2P/M, and that there exist integers a and q satisfying
(2.9) 1 ≤ q ≤ Q, (a,q) = 1, |qα − a| < QP⁻⁷,
with Q ≤ P. Then
(2.10) ,
where Ψ(α) = q + P⁷|qα − a| and L = logP. In particular, we have
(2.11) .
Proof. (2.10) is the case k = 7 of Lemma√ 5.6 in Kumchev [9]. (2.11) follows from
(2.10) on choosing M = 1 and z = 2P.
Lemma 3. Let 1/192 < ρ < 1/141 and let α be a real number such that no integers a and q satisfy (2.9) with Q = P^1/4. Let S(α) be defined by (2.8) and suppose that
z ≤ P1−128ρ and M ≤ P(7−15ρ)/13.
Then
(2.12) .
Proof. This follows from the results in §3 of [9]. Let Q = P^{(49−14ρ)/13}. By Dirichlet’s theorem on diophantine approximation, there exist integers a and q satisfying
(2.13) 1 ≤ q ≤ Q, (a,q) = 1, |qα − a| < Q⁻¹.
By assumption, a and q must also satisfy
(2.14) q + P⁷|qα − a| > P^1/4.
When z ≤ P^1−130ρ, (2.13) and (2.14) suffice to deduce (2.12) from Lemma 3.3 in [9]. When P^1−130ρ < z ≤ P^1−128ρ, we recall Buchstab’s identity
(2.15) ψ(n,z₂) = ψ(n,z₁) − ^{X X} ψ(k,p) (2 ≤ z₁ < z₂).
z₁≤p₂ n=pk
Applying (2.15) with z₁ = P^1−130ρ and z₂ = z, we obtain
S(α) = S₁(α) − S₂(α),
where
.
We can now use Lemma 3.3 in [9] to bound S₁(α) and Lemma 3.1 in [9] ( with
(m,n) = (mk,p)) to bound S₂(α).
2.3. Estimates for sums over integers free of small prime divisors. In this section, we prepare some asymptotic estimates for exponential sums over numbers free of small primes.
Lemma 4. Let 2 ≤ z ≤ y ≤ z^c, let ψ(n,z) be defined by (1.5), and let ω(u) be the continuous solution of the differential delay equation
⁽(uω(u)) ⁰ = ω(u − 1), if u > 2,
ω(u) = u⁻¹, if 1 < u ≤ 2.
Let β be a real number, with |β|≤ y⁻⁷(logy)^B. Then for any A > 0,
,
the implied constant depending at most on A and B. Here, γ is Euler’s constant and .
Proof. When β = 0, this follows from (1.7) in de Bruijn [2] and de la Vall´ee Poussin’s form of the prime number theorem (see §18 in Davenport [1]). The general case follows from the case β = 0 by partial summation.
Lemma 5. Let 2 ≤ z ≤ y ≤ z^c, let ψ(n,z) be defined by (1.5), and let a and q be integers, with (a,q) = 1 and 1 ≤ q ≤ (logy)^B. Then for any A > 0,
,
the implied constant depending at most on A and B.
Proof. This is a generalization of the Siegel–Walfisz theorem. By a variant of (3) in §20 of Davenport [1], it suffices to show that
(2.16) ,
for every nonprincipal character χ mod q. When z ≥ y^1/2, (2.16) follows by partial summation from (3) in §22 of [1]. When z < y^1/2, we apply Buchstab’s identity in the reverse. By (2.15) ,
(2.17) .
The first sum on the right of (2.17) can be estimated as before and the second sum is bounded above by
,
the maximum being over all pairs M,M⁰ with z ≤ M < M⁰ ≤ min(2M,y). Since this quantity can also be estimated by means of (3) in §22 of [1], the desired result follows.

3. 
   Proof of Theorem 1
   

Let n be a large even integer and set

1. 
   and P_j = P^λj (1 ≤ j ≤ 23),
   

where λ₁,...,λ₂₂ are the exponents defined in §2.1 and λ₂₃ = 1. We aim to prove that the set

contains an integer that can be represented as the sum of 23 seventh powers of primes. If m is a natural number, we denote by r(m) the number of solutions of

subject to P_j < p_j ≤ 2P_j. By Cauchy’s inequality,

2. 
   .
   

The first sum on the right side of (3.2) is equal to the total number of 23- tuples p₁,...,p₂₃ with P_j < p_j ≤ 2P_j, while the second sum is bounded above by the integral on the left side of (2.3). Hence, appeals to the prime number theorem and
Lemma 1 yield
,
and the desired conclusion follows from our estimate for E₂₃(n).

3. 
   Proof of Theorem 2
   
   1. 
      Preliminaries. For the sake of simplicity, we present the case s = 23 in detail and then sketch the changes required in the proof of (1.4). Let N be a large parameter. We set and define P₁,...,P₂₂ by (3.1). We also write z = P^15/79, L = logP, and X = PP₁ ···P₂₂N⁻¹.
      

Recalling (2.15), we obtain
(4.1) )
, say.
When P < m ≤ 2P, the left side of (4.1) is equal to the indicator function of the primes, so the number of representations of n as the sum of 23 seventh powers of prime numbers is bounded below by the quantity
R(n) = ^X w(m),
m,p₁,...,p₂₂:(4.2)
where the summation is over the 23-tuples m,p₁,...,p₂₂ subject to
(m 7 + p71 + ··· + p722 = n,
(4.2)
P < m ≤ 2P, P_j < p_j ≤ 2P_j.
We now introduce some notation needed in the application of the circle method. We write
,
Z
R(n;B) = h(α)f₁(α)···f₂₂(α)e(−αn)dα.
B
Further, let

where ω(u) is the function defined in Lemma 4. We define the singular series S(n) by
S .
S(n) is thoroughly studied in Chapter 8 of Hua [5]. In particular, Theorem 12 in [5] asserts that
(4.3) 1
for all odd n. The singular integral associated with R(n) is J(n) = J(n;∞), where
Z ξ
J(n;ξ) = v₁(β)²v₂(β)···v₂₂(β)e(−βn) dβ.
−ξ
Note that a routine application of the Fourier inversion formula yields
(4.4) .
Because of the presence of the sieve weights w(m), we also have to deal with a variant of J(n;ξ), namely,
Z ξ
J^∗(n;ξ) = v^∗(β;z)v₁(β) ···v₂₂(β)e(−βn) dβ.
−ξ
Finally, we define the sets of major and minor arcs by M = M(P^1/4,N) and m = [0,1] \M, respectively.
4.2.