In a recent paper K. Ramachandra states some conjectures, and gives consequences in the theory of the Riemann zeta function. In this paper we will present two different disproofs of them. The first will be an elementary application of the Szasz-Münto theorem. The second will depend on a version of the Voronin universality theorem, and is also slightly stronger in the sense that it disproves a weaker conjecture. An elementary (but more complicated) disproof has been given by Rusza-Lazkovich.
For arbitrary integers $k\in\mathbb Z$, we investigate the set $C_k$ of the generalised Carmichael number, i.e. the natural numbers $n< \max\{1, 1-k\}$ such that the equation $a^{n+k}\equiv a \mod n$ holds for all $a\in\mathbb N$. We give a characterization of these generalised Carmichael numbers and discuss several special cases. In particular, we prove that $C_1$ is infinite and that $C_k$ is infinite, whenever $1-k>1$ is square-free. We also discuss generalised Carmichael numbers which have one or two prime factors. Finally, we consider the Jeans numbers, i.e. the set of odd numbers $n$ which satisfy the equation $a^n\equiv a \mod n$ only for $a=2$, and the corresponding generalizations. We give a stochastic argument which supports the conjecture that infinitely many Jeans numbers exist which are squares.
For a good Dirichlet series $F(s)$ (see Definition in \S1) which is a quotient of some products of the translates of the Riemann zeta-function, we prove that there are infinitely many poles $p_1+ip_2$ in $\Im (s)>C$ for every fixed $C>0$. Also, we study the gaps between the ordinates of the consecutive poles of $F(s)$.
For ``good Dirichlet series'' $F(s)$ we prove that there are infinitely many poles $p_1+ip_2$ in $\Im (s)>C$ for every fixed $C>0$. Also we study the gaps between the numbers $p_2$ arranged in the non-decreasing order.