# Volume 6 - 1983

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

The results given in these papers continue the theme developed in part I of this series. In Part III we prove $M(\frac{1}{2})>\!\!\!>_k (\log H_0/q_n)^{k^2}$, where $p_m/q_m$ is the $m$th convergent of the continued fraction expansion of $k$, and $n$ is the unique integer such that $q_nq_{n+1}\geq \log\log H_0 > q_nq_{n-1}$. 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

If $\Delta_{m,n}$ denotes the quotient $[u_m,v_n]/(u_m,v_n)$ of the lcm by the gcd, we obtain in this paper a lower bound for the greatest square-free factor $Q[\Delta_{m,n}]$ of $\Delta_{m,n}$ when $u_h=v_h, m>n$ (and $u_n\neq0$); this implies a lower bound for $\log Q[u_n]$ of the form $C(\log m)^2(\log\log m)^{-1}$, thereby improving on an earlier result of C. L. Stewart.

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

In this paper, we apply a combinatorial lemma to a well-known result concerning the transcendency of at least one of the numbers $\exp(\alpha_i\beta_j) (i=1, 2, 3; j=1, 2)$, where the complex numbers $\alpha_i,\beta_j$ satisfy linear independence conditions and show that for any $\alpha\neq0$ and any transcendental number $t$, we obtain that at most $\frac{1}{2}+(4N-4+\frac{1}{4})^{1/2}$ of the numbers $\exp(\alpha t^n)~(n=1,2,\ldots,N)$ are algebraic. Similar statements are given for values of the Weierstrass $\wp$-function and some connections to related results in the literature are discussed.

### 4. Primes between $p_n+1$ and $p_{n+1}^2-1.$

We give some polynomial identities involving the first $n+1$ primes, and deduce from one of these a formula from which $p_{n+1}$ can be calculated once $p_1,\ldots,p_n$ are known.