On a theorem of Daboussi related to the set of Gaussian integers.
For every positive integer let be the largest prime number . Given a positive integer , we study the positive integer such that if we define recursively for , then is a prime or . We obtain upper bounds for as well as an estimate for the set of whose takes on a fixed value .
The technique developed by A. Walfisz in order to prove (in 1962) the estimate for the error term related to the Euler function is extended. Moreover, the argument is simplified by exploiting works of A.I. Saltykov and of A.A. Karatsuba. It is noted in passing that the proof proposed by Saltykov in 1960 of is erroneous and once corrected “only” yields Walfisz’ result. The generalizations obtained can be applied to error terms related to various classical - and less classical - arithmetical...