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On an arithmetic function considered by Pillai

Florian Luca, Ravindranathan Thangadurai (2009)

Journal de Théorie des Nombres de Bordeaux

For every positive integer n let p ( n ) be the largest prime number p ≤ n . Given a positive integer n = n 1 , we study the positive integer r = R ( n ) such that if we define recursively n i + 1 = n i - p ( n i ) for i ≥ 1 , then n r is a prime or 1 . We obtain upper bounds for R ( n ) as well as an estimate for the set of n whose R ( n ) takes on a fixed value k .

On an estimate of Walfisz and Saltykov for an error term related to the Euler function

Y.-F. S. Pétermann (1998)

Journal de théorie des nombres de Bordeaux

The technique developed by A. Walfisz in order to prove (in 1962) the estimate H ( x ) ≪ ( log x ) 2 / 3 ( log log x ) 4 / 3 for the error term H ( x ) = ∑ n ≤ x φ ( n ) n - 6 π 2 x 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 H ( x ) ≪ ( log x ) 2 / 3 ( log log x ) 1 + ϵ 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...

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