Let

The asymptotic behaviours of ƒ(x) and g(x), as x→+0, were first given by G. H. Hardy in (4), (5). In his papers {an}; is a monotone decreasing sequence. Further results on the asymptotic behaviours of ƒ(x) and g(x), as x→+0, for monotone coefficients have been given in (9) and (1). Recently, the results have been generalized to quasi-monotone coefficients.

This paper is concerned with asymptotic behaviours of ƒ(x) and g(x) for δ-quasi-monotone coefficients.

In what follows, we shall denote by L(x) a slowly varying function in the sense of Karamata (6), i.e.

Throughout, ε is any small positive number, θ any real number, n, nj, k, N some positive integers and p, pj any primes. By ‖θ‖ we mean the distance from θ to the nearest integer. Write C(ε), C(ε, k) for positive constants which may depend on the quantities indicated inside the parentheses.

Let λj (1 ≦ j ≦ 4) be any nonzero real numbers which are not all of the same sign and not all in rational ratio and let pj be polynomials of degree one or two with integer coefficients and positive leading coefficients. The author proves that if exactly two pj are of degree two then for any real n there are infinitely many solutions in primes pj of the inequality . where 0 <β < (√(21)–1)∖5760.

In this present paper we shall prove the following. Suppose that λ1, λ2, λ3 are any non-zero real numbers not all of the same sign and that λ1/λ2 is irrational. If η is any real number and, 0 < α < 1/9, then there are infinitely many prime triples (p1, p2, p3) for which

1. Introduction. In 1948, H. Heibronn (6) improved Vinogradov's result (11) and obtained that for any ε > 0 there exists some positive constant C(ε) which depends on ε only such that, for any real number θ and any integer N ≥ 1, there is an integer x satisfying

Here ∥t∥ means the distance from t to the nearest integer.

1. In 1948 Heilbronn [4] proved the following theorem.

THEOREM H. For every real θ and every positive integer N, there is an integer n satisfying

(1.1)

whereis an arbitrarily small number,depends only on, and ‖t‖ means the distance from t to the nearest integer.

The interest of the result (1.1) is that the inequality is uniform in θ, and is therefore analogous to the classical inequality of Dirichlet for the fractional part of θn. In this paper we shall prove the following theorem.

THEOREM. For every real θ and every positive integer N, there is an integer n satisfying

(1.2)

where A is an absolute constant and.