On simultaneous diagonal equations and inequalities

J. Brüdern; R. J. Cook

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Introduction. In this note we use the following standard notations: π(x) is the number of primes not exceeding x, while $θ (x) = \sum_{p \leq x} l o g p$ . The best known inequalities involving the function π(x) are the ones obtained in [6] by B. Rosser and L. Schoenfeld: (1) x/(log x - 1/2) < π(x) for x ≥ 67 (2) x/(log x - 3/2) > π(x) for $x > e^{3 / 2}$ . The proof of the above inequalities is not elementary and is based on the first 25 000 zeros of the Riemann function ξ(s) obtained by D. H. Lehmer [4]. Then Rosser, Yohe...