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

Theorem 1. Every sufficiently large even integer can be written as the sum of 46 seventh powers of prime numbers.
We deduce Theorem 1 from the following result.
1
Theorem 2. Let 23 ≤ s ≤ 45 and let E_s(x) denote the number of integers n ≤ x, with n ≡ s (mod 2), that cannot be represented as sums of s seventh powers of prime numbers. Then for any A > 0, one has
(1.3) ,
with an implied constant depending at most on A. When s ≥ 24, there exists an absolute constant θ < 1 such that
(1.4) .
The main novelty in Theorem 2 is the estimate for E₂₃(x), which is also the case used in the proof of Theorem 1. A variant of (1.4) was announced earlier as a part of Theorem 3 in Kumchev [8]. It is possible that a more delicate treatment of the major arcs (say, by a variant of the approach of Liu and Zhan [10]) would have allowed us to extend (1.4) to the case s = 23. However, this would entail a significant amount of extra effort that would be spent wiser elsewhere.
As in earlier work on the subject, the proof of Theorem 2 uses the Hardy– Littlewood method. The new ingredients that allow us to bound E₂₃(x) are the exponential sum estimates in Kumchev [9], the main result in Thanigasalam [13] , and the sieve method in Harman [3]. The reader acquainted with Harman’s method will recognize that we apply it in its most primitive form and may wonder whether it is not possible to dispense with the use of sieves altogether. That appears not to be the case. On the other hand, without the result in [13], even the most sophisticated version of the sieve does not seem to yield the desired result.
Notation. Throughout the paper, the letter ε denotes a sufficiently small positive real number. Any statement in which ε occurs holds for each positive ε, and any implied constant in such a statement is allowed to depend on ε. The letter p, with or without subscripts, is reserved for prime numbers; c denotes an absolute constant, not necessarily the same in all occurrences.
As usual in number theory, φ(n) and τ(n) denote Euler’s totient function and the number of divisors function. Also, if z ≥ 2, we define
(
1, if n is divisible by no prime p < z,
(1.5) ψ(n,z) =
0, otherwise,
and we write e(x) = exp(2πix) and (a,b) = gcd(a,b).
Throughout the paper, we use several decompositions of the unit interval into major and minor arcs. If 1 ≤ Y ≤ X, we define the set of major arcs M(Y,X) as the union of the intervals
M
with 0 ≤ a ≤ q ≤ Y and (a,q) = 1.

2. 
   Auxiliary results
   

2.1. Mean-value estimates. Our immediate goal is to describe a set of admissible exponents λ₁,...,λ₂₂ satisfying (2.1) below. Our choice is motivated by the work of Thanigasalam [12, 13] and Vaughan [14]. We set θ₂₂ = 1, θ₂₁ = 12/13, α₂₀ = 37/91 , and then define recursively
,
with j_i given by the following table.

i	13	14	15	16	17	18	19	20
ji	5	5	5	5	4	4	3	3

Table 1. The values of j_i
Further, we choose
if i = 1, and λ_i = θ₁ ···θ_i (1 ≤ i ≤ 22). , if 2 ≤ i ≤ 12,
In particular, a quick calculation reveals that

1. 
   λ₁ + ··· + λ₂₂ > 6.987443 > 7 − 2/159.
   

We now define the generating functions
and .
The following lemma is the main result of this section.