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Inequalities concerning the function π(x): Applications

Laurenţiu Panaitopol — 2000

Acta Arithmetica

Introduction. In this note we use the following standard notations: π(x) is the number of primes not exceeding x, while θ ( x ) = p 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 and Schoenfeld...

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