On the Primorial Counting Function (Q6551251)

Let \((p_k)_{k\ge 1}\) denote the sequence of the positive prime numbers, and let \((N_k)_{k\ge 1}\) be the sequence of the primorials defined as \(N_k=p_1p_2\cdots p_k\) (\(k\ge 1\)). Let \(\pi(x)\) and \(K(x)\) stand for the functions that count the number of primes and primorials, respectively, not exceeding \(x\). Also, let \(\operatorname{li}(x) =\int_0^x (\log t)^{-1} \operatorname{dt}\) denote the integral logarithm. The author proves the following main theorems.\N\NTheorem 1. The function \(K(x) - \pi(\log x)\) changes sign infinitely many times as \(x\) increases to infinity.\N\NTheorem 2. For every \(x\) satisfying \(\exp(\operatorname{li}^{-1}(4)) < x \le \exp(\operatorname{li}^{-1}(\pi(10^{19})))\), one has \(K(x) < \operatorname{li}(\log x)\).\N\NTheorem 4. The Riemann hypothesis is true if and only if the inequality \(K(x) < \operatorname{li}(\log x)\) holds for every \(x > \exp(\operatorname{li}^{-1}(4))\).