By Chaohua Jia, Kohji Matsumoto

Comprises numerous survey articles on top numbers, divisor difficulties, and Diophantine equations, in addition to study papers on quite a few features of analytic quantity idea difficulties.

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References To examine K(k2), the following observation is useful. )) = 1. 9) for r = 1). On the other hand, to estimate x(X,u) is the simplest linear sieve problem, and it is easy to prove that [I] J . Briidern, Iterationsmethoden in der addi tiven Zahlentheorie. Thesis, Gijttingen 1988. [2] J . Briidern, A problem in additive number theory. Math. Proc. Cambridge Philos. Soc. 103 (1988), 27-33. [3] J. Briidern, A sieve approach to the Waring-Goldbach problem I: Sums of four cubes. Ann. Scient. EC.

Nd that for every integer n with N 5 n 5 (6/5)N. To facilitate our subsequent description, we denote by N(5) the set of all the odd integers in the interval [N, (6/5)N], and put N(4) = Nl n > for all primes p and integers 1 1. 10). 4, respectively. We next discuss a lower bound for Pl(n, Y). 4, we have PI(n, Y) > n (zp2)-' n (1 - 120p-') n (I - 1 2 0 ~ - ' / ~ ) , 74 ANALYTICNUMBER THEORY Ternary problems in additive prime number theory 75 for n E M(kl ) . 1, then we have We shall estimate E(n) on average over n.

Let n be an even integer satisfying N n < (6/5)N with a sufficiently large number N , k2 be an integer with 3 5 kz 5 5, and let Rd(n) be the number of representations of n in the form 19 << N j + i + f e + &+ Nz+&+ae+~ +Ng+i(logN)-~. This establishes the lemma. with integers x and primes p j satisfying x We close this section by showing an easy lemma, which allows us to ignore the integers divisible by a square of a larger prime, in our application of weighted sieves. 7. Let 6 be an arbitrary fixed positive number, and k be an integer with 3 5 k 5 5 .