Arithmetical problems with numbers of special type
Let \(E_0\) and \(E_1\) denote the sets of positive integers such that the number of ones in their binary representation is even and odd, respectively. The authors state seven theorems, without any proofs, concerning functions like \(\pi_k(X,m,\ell)=\# \{n<X: n\in E_k\) and \(n\equiv \ell\pmod m\}\) and \(\pi_k(X) = \# \{n<X: n\in E_k\}= \pi_k(X,1,0)\), where \(k=0\) or \(k=1\) and \(m, \ell\) are positive integers. We quote here Theorem 1. Suppose that \(m\) is a positive integer such that \(2\leq m \leq \sqrt{X}\). Then the following asymptotic formula holds: \[ \pi_k(X,m,\ell)= \frac 1{2m} X (1+O(X m^{-2})^{-\lambda_4}), \] where \(\lambda_4 = 1-\log 3/\log 4 = 0.207518\dots\) .
scientific article; zbMATH DE number 1429641