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On the distribution in the arithmetic progressions of reducible quadratic polynomials in short intervals, II

Giovanni Coppola, Saverio Salerno (2001)

Journal de théorie des nombres de Bordeaux

This paper gives further results about the distribution in the arithmetic progressions (modulo a product of two primes) of reducible quadratic polynomials ( a n + b ) ( c n + d ) in short intervals n [ x , x + x ϑ ] , where now ϑ ( 0 , 1 ] . Here we use the Dispersion Method instead of the Large Sieve to get results beyond the classical level ϑ , reaching 3 ϑ / 2 (thus improving also the level of the previous paper, i.e. 3 ϑ - 3 / 2 ), but our new results are different in structure. Then, we make a graphical comparison of the two methods.

On the distribution of complex-valued multiplicative functions

Antanas Laurinčikas (1996)

Journal de théorie des nombres de Bordeaux

Let g j ( m ) , j = 1 , 2 , be complex-valued multiplicative functions. In the paper the necessary and sufficient conditions are indicated for the convergence in some sense of probability measure 1 n card 0 m n : ( g 1 ( m ) , g 2 ( m ) ) A , A ( 2 ) , as n .

On the distribution of ( k , r ) -integers in Piatetski-Shapiro sequences

Teerapat Srichan (2021)

Czechoslovak Mathematical Journal

A natural number n is said to be a ( k , r ) -integer if n = a k b , where k > r > 1 and b is not divisible by the r th power of any prime. We study the distribution of such ( k , r ) -integers in the Piatetski-Shapiro sequence { n c } with c > 1 . As a corollary, we also obtain similar results for semi- r -free integers.

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.

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