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In this paper we investigate about the effective distribution of the numbers coprime with the primes up to p_n . More precisely we prove that these numbers form a periodically monotone sequence . Then we examine some properties of this sequence which, in a certain sense, are transferred to the sequence of primes. Moreover we study the distribution of twin and cousin terms within the above sequence . This study also makes furthermore strongly plausible that the set of twin primes as well as the set of cousin primes is infinite.

n this paper we give a new proof of a known diophantine approximation result, then we apply this to prove convergence of a class of series in a Banach space, whose terms are defined recursively.

In this paper we investigate the uniform distribution in [0,T] (T ∊ R+- Q) of a suitable sequence. Then we give an interesting application to the study of a class of series whose terms are defined recursively.

In this paper we prove some theorems that emphasize properties of the series whose terms are defined recursively. Suitable examples and counter-examples complete the theory.