On a Problem of Mahler Concerning the Approximation of Exponentials and Logarithms by Michel WALDSCHMIDT Bon

We first propose two conjectural estimates on Diophantine approximation of logarithms of algebraic numbers. Next we discuss the state of the art and we give further partial results on this topic. §1. Two Conjectures on Diophantine Approximation of Logarithms of Algebraic Numbers In 1953 K. Mahler [7] proved that for any sufficiently large positive integers a and b, the estimates log a ≥ a −40 log log a and e b ≥ b −40b (1) hold; here, · denotes the distance to the nearest integer: for x ∈ R, x = min n∈Z |x − n|. In the same paper [7], he remarks: " The exponent 40 log log a tends to infinity very slowly; the theorem is thus not excessively weak, the more so since one can easily show that | log a − b| < 1 a for an infinite increasing sequence of positive integers a and suitable integers b. " (We have replaced Mahler's notation f and a by a and b respectively for coherence with what follows). In view of this remark we shall dub Mahler's problem the following open question: (?) Does there exist an absolute constant c > 0 such that, for any positive integers a and b, |e b − a| ≥ a −c ?