论文信息 - On divisibility properties of sequences of integers bY

On divisibility properties of sequences of integers bY

In this paper we discuss the results which we obtained on sequences of integers in the last few years and also state some of the problems which we could not settle. First we review the older work on this subject, most of which can be found in the excellent book of Halber-stam and Roth 1131. Let A i {ai<~~c-...] be a sequence of integers. put A(Y) = x 1 nle den&y of A (if it exists) is defined as G;<X ,im A(x)-. x-dd x The logarithmic density is defined as It is obvfous that if the density exists then the logarithmic density exists too, but the converse is not true.

[1] P. Erdös. On the Distribution Function of Additive Functions , 1946 .

[2] E. Szemerédi,et al. On An Extremal Problem Concerning Primitive Sequences , 1967 .

[3] I. Anderson. On Primitive Sequences , 1967 .

[4] P. Turán. On a Theorem of Hardy and Ramanujan , 1934 .

[5] Ralph Alexander. Density and multiplicative structure of sets of integers , 1967 .

[6] F. Behrend,et al. On Sequences of Numbers not Divisible one by another , 1935 .

[7] P. Erdös. Note on Sequences of Integers No One of Which is Divisible By Any Other , 1935 .

[8] E. Szemerédi,et al. On the divisibility properties of sequences of integers (II) , 1968 .