The results given in these papers continue the theme developed in part I of this series. In Part III we prove M(12)>>k(logH0/qn)k2, where pm/qm is the mth convergent of the continued fraction expansion of k, and n is the unique integer such that qnqn+1≥loglogH0>qnqn−1. Section 4 of part III discusses lower bounds of mean values of Titchmarsh series.
If Δm,n denotes the quotient [um,vn]/(um,vn) of the lcm by the gcd, we obtain in this paper a lower bound for the greatest square-free factor Q[Δm,n] of Δm,n when uh=vh,m>n (and un≠0); this implies a lower bound for logQ[un] of the form C(logm)2(loglogm)−1, thereby improving on an earlier result of C. L. Stewart.
In this paper, we apply a combinatorial lemma to a well-known result concerning the transcendency of at least one of the numbers exp(αiβj)(i=1,2,3;j=1,2), where the complex numbers αi,βj satisfy linear independence conditions and show that for any α≠0 and any transcendental number t, we obtain that at most 12+(4N−4+14)1/2 of the numbers exp(αtn)(n=1,2,…,N) are algebraic. Similar statements are given for values of the Weierstrass ℘-function and some connections to related results in the literature are discussed.
We give some polynomial identities involving the first n+1 primes, and deduce from one of these a formula from which pn+1 can be calculated once p1,…,pn are known.