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On generalized square-full numbers in an arithmetic progression

Angkana Sripayap, Pattira Ruengsinsub, Teerapat Srichan (2022)

Czechoslovak Mathematical Journal

Let a and b . Denote by R a , b the set of all integers n > 1 whose canonical prime representation n = p 1 α 1 p 2 α 2 p r α r has all exponents α i ( 1 i r ) ...

On the asymmetric divisor problem with congruence conditions

Manfred Kühleitner (1996)

Commentationes Mathematicae Universitatis Carolinae

A certain generalized divisor function d * ( n ) is studied which counts the number of factorizations of a natural number n into integer powers with prescribed exponents under certain congruence restrictions. An Ω -estimate is established for the remainder term in the asymptotic for its Dirichlet summatory function.

On the distribution of the partial sum of Euler's totient function in residue classes

Youness Lamzouri, M. Tip Phaovibul, Alexandru Zaharescu (2011)

Colloquium Mathematicae

We investigate the distribution of Φ ( n ) = 1 + i = 1 φ ( i ) (which counts the number of Farey fractions of order n) in residue classes. While numerical computations suggest that Φ(n) is equidistributed modulo q if q is odd, and is equidistributed modulo the odd residue classes modulo q when q is even, we prove that the set of integers n such that Φ(n) lies in these residue classes has a positive lower density when q = 3,4. We also provide a simple proof, based on the Selberg-Delange method, of a result of T. Dence and...

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