Volume 6 - 1983


1. Mean-value of the Riemann zeta-function and other remarks III

K Ramachandra.
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+1loglogH0>qnqn1. Section 4 of part III discusses lower bounds of mean values of Titchmarsh series.

2. The greatest square free factor of a binary recursive sequence

Tarlok Nath Shorey.
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 un0); 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.

3. A note to a paper by Ramachandra on transctndental numbers

K Ramachandra ; S Srinivasan.
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 α0 and any transcendental number t, we obtain that at most 12+(4N4+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.

4. Primes between pn+1 and p2n+11.

A. Venugopalan.
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.