Additive problems for integers with a given number of prime divisors
Asymptotic estimates of sums of the form \[ \sum_{n<N, \omega (n)=k} r\bigl(| b\cdot n-a|\bigr)\tag{1} \] is studied, where \(\omega(n)\) is the number of distinct prime divisors of \(n\), and \(r(n)\) is the number of all divisors. For \(a=N\), \(b=1\), in finding the asymptotics of the sum (1), we obtain the asymptotics of the number of solutions of the equation \(N=n+x\cdot y\), where \(\omega(n)=k\) and \(x,y\) are integers; for \(b=1\), we obtain the asymptotics of the number of solutions of the equation \(a= n-x \cdot y\), where \(n<N\), \(\omega(n)=k\), and \(x,y\) are integers.
scientific article; zbMATH DE number 1262506