三个素数混合幂丢番图逼近

Diophantine approximation with three prime and mixed power

采用Davenport-Heilbronn方法改进不等式|λ_1p_1+λ_2p_2+λ_3pk3+η|<(maxp_j)-σ中σ的值,其中k是大于或等于4的正整数,η是任意给定的实数,λ_1,λ_2,λ_3是非零实数,不全同号,并且λ_1/λ_2是无理数,得出了更好的σ值是:当k=4时,0<σ<1/2(2^(k+1)+1);当k≥5时,0<σ<2^(k-1)/(2^(k+1)+1)k(k+1).

The value ofσin the inequality｜λ1 p 1 ＋λ2 p 2 ＋λ3 p k3 ＋η｜＆lt; （maxp j ）-σ is improved by applying Davenport-Heilbronn method,where k is an integer with k ≥4,η is an any real number,andλ1 ,λ2 ,λ3 are nonzero real numbers,not all of the same sign,withλ1/λ2 being irra-tional.It is proved that the better value ofσ is obtained,where 0＆lt;σ＆lt;1/2（2 k ＋1 ＋1）for k =4 and 0＆lt;σ＆lt;2k -1/（2 k ＋1 ＋1）k （k ＋1）for k ≥5.