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On the distribution of the Euler function of shifted smooth numbers

Stefanie S. Loiperdinger, Igor E. Shparlinski (2010)

Colloquium Mathematicae

We give asymptotic formulas for some average values of the Euler function on shifted smooth numbers. The result is based on various estimates on the distribution of smooth numbers in arithmetic progressions which are due to A. Granville and É. Fouvry & G. Tenenbaum.

On the divisor function over Piatetski-Shapiro sequences

Hui Wang, Yu Zhang (2023)

Czechoslovak Mathematical Journal

Let [ x ] be an integer part of x and d ( n ) be the number of positive divisor of n . Inspired by some results of M. Jutila (1987), we prove that for 1 < c < 6 5 , ∑ n ≤ x d ( [ n c ] ) = c x log x + ( 2 γ - c ) x + O x log x , where γ is the Euler constant and [ n c ] is the Piatetski-Shapiro sequence. This gives an improvement upon the classical result of this problem.

On the Euler Function on Differences Between the Coordinates of Points on Modular Hyperbolas

Igor E. Shparlinski (2008)

Bulletin of the Polish Academy of Sciences. Mathematics

For a prime p > 2, an integer a with gcd(a,p) = 1 and real 1 ≤ X,Y < p, we consider the set of points on the modular hyperbola a , p ( X , Y ) = ( x , y ) : x y ≡ a ( m o d p ) , 1 ≤ x ≤ X , 1 ≤ y ≤ Y . We give asymptotic formulas for the average values ∑ ( x , y ) ∈ a , p ( X , Y ) x ≠ y * φ ( | x - y | ) / | x - y | and ∑ ( x , y ) ∈ a , p ( X , X ) x ≠ y * φ ( | x - y | ) with the Euler function φ(k) on the differences between the components of points of a , p ( X , Y ) .

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