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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...

Integer Points Close to a Smooth Curve

Trifonov, Ognian (1998)

Serdica Mathematical Journal

∗ This research is partially supported by the Bulgarian National Science Fund under contract MM-403/9We review the existing estimates for the number of integer points close to a smooth curve and improve on some of these.

Joint distribution for the Selmer ranks of the congruent number curves

Ilija S. Vrećica (2020)

Czechoslovak Mathematical Journal

We determine the distribution over square-free integers n of the pair ( dim 𝔽 2 Sel Φ ( E n / ℚ ) , dim 𝔽 2 Sel Φ ^ ( E n ' / ℚ ) ) , where E n is a curve in the congruent number curve family, E n ' : y 2 = x 3 + 4 n 2 x is the image of isogeny Φ : E n → E n ' , Φ ( x , y ) = ( y 2 / x 2 , y ( n 2 - x 2 ) / x 2 ) , and Φ ^ is the isogeny dual to Φ .

Jumping champions.

Odlyzko, Andrew, Rubinstein, Michael, Wolf, Marek (1999)

Experimental Mathematics

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