An inequality related to the Hardy-Littlewood conjecture (Q2762850)

Let \(\pi(x)\) be the number of primes not exceeding \(x\). Related to the Hardy-Littlewood conjecture: \(\pi(x+y)\leq \pi(x)+\pi(y)\), in this paper there is studied the sign of \(D_\alpha(x,y)=\pi^\alpha(x+y)-\pi^\alpha(x)-\pi^\alpha(y)\) for \(\alpha\neq 1\) and natural numbers \(x\) and \(y\). The following results are proved:NEWLINENEWLINENEWLINE1. For \(\alpha>1\) there exists no natural number \(m\) such that for all \(x,y\geq m\) the function \(D_\alpha(x,y)\) has constant sign.NEWLINENEWLINENEWLINE2. For \(\alpha <1\) there exists \(m=m(\alpha)\) explicitly computable such that for all \(x,y\geq m\) we have \(D_\alpha(x,y)\leq 0\).