SOME THEOREMS ON DIOPHANTINE APPROXIMATION^) BY CHARLES F. OSGOOD Introduction. The study of the values at rational points of transcendental functions defined by linear differential equations with coefficients in Q[z] (2) can be traced back to Hurwitz [1] who showed that if , . , 1 z 1 z2 Az)= l+-b-lT+WTa)2l + - where « is a positive integer, b is an integer, and b\a is not a negative integer, then for all nonzero z in Q(( - 1)1/2) the number y'(z)jy(z) is not in g(( - 1)1/2). Ratner [2] proved further results. Then Hurwitz [3] generalized his previous results to show that if nZ) ,+g(0) 1! +g(0)-g(l)2! + where f(z) and g(z) are in Q[z\, neither f(z) nor g(z) has a nonnegative integral zero, and degree (/(z)) < degree (g(z)) = r, then for all nonzero z in the imaginary quadratic field Q(( — n)1'2) two of the numbers y(z),y(l)(z),---,yir\z) have a ratio which is not in Q(( - n)112).Perron [4], Popken [5], C. L. Siegel [6], and K. Mahler [7] have obtained important results in this area. In this paper we shall use a generalization of the method which was developed by Mahler [7] to study the approximation of the logarithms of algebraic numbers by rational and algebraic numbers. Definition. Let K denote the field Q(( - n)i/2) for some nonnegative integer «. Definition. For any monic 0(z) in AT[z] of degree k > 0 and such that 6(z) has no positive integral zeros we define the entire function oo d f(z)= £ —--. d^O d n oc«) Presented to the Society, November 23, 1963 under the title A theorem on diophantine approximation; received by the editors November 6, 1964. (t) This paper formed part of the author's doctoral thesis written under the direction of Professor R. S. Lehman at the University of California, Berkeley. p) We denote the rational numbers by Q and the integers by Z. 64 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use