Portion of Primes in Short in Intervals and Landau's Inequality


  • Mehdi Hassani Department of Mathematics, University of Zanjan University Blvd., 45371-38791, Zanjan, Iran. Author




Landau’s inequality, Prime numbers


In this paper, we study Landau’s inequality concerning primes counting function, asserting that $\pi(2x)<2\pi(x)$ for $x$  sufficiently large.  We give some variants of this inequality to study portion of primes in intervals with similar length.


C. Axler, "Estimates for π (x) for large values of x and Ramanujan’s prime counting inequality". Integers. Vol. 18, No. A61, pp. 14. (2018).

B. C. Berndt. "Ramanujan’s Notebooks (Part IV)". Springer-Verlag, (1994). DOI: https://doi.org/10.1007/978-1-4612-0879-2

P. Dusart. "Estimates of the kth prime under the Riemann hypothesis". Ramanujan J. Vol. 47, pp. 141- 154. (2018). DOI: https://doi.org/10.1007/s11139-017-9984-4

G. H. Hardy and J. E. Littlewood. "Some problems of 'partitio numerorum'. III. On the expression of a number as a sum of primes". Acta Math. Vol. 44, pp. 1–70. (1923). DOI: https://doi.org/10.1007/BF02403921

D. Hensley and I. Richards. "Primes in intervals". Acta Arith. Vol. 25, pp. 375–391. (1974). DOI: https://doi.org/10.4064/aa-25-4-375-391

C. Karanikolov. On some properties of the function π (x)". Univ. Beograd Publ. Elektrotehn. Fak. Ser. Mat., No. 357–380, pp. 29–30. (1971).

E. Landau. "Handbuch der Lehre von der Verteilung der Primzahlen". AMS Chelsea Publishing. (1974).

J. E. Littlewood. "Sur la distribution des nombres premiers". C. R. Math. Acad. Sci. Paris. Vol. 158, pp. 1869–1872. (1914).

L. Panaitopol. "Inequalities concerning the function π (x) Applications". Acta Arith. Vol. 94, pp. 373–381. (2000). DOI: https://doi.org/10.4064/aa-94-4-373-381

S. Ramanujan. "Notebooks (2 volumes)". Tata Institute of Fundamental Research, Bombay. (1957).

J. B. Rosser and L. Schoenfeld. "Abstract of scientific communications". Intern. Congr. Math. Moscow, Section 3. Theory of Numbers. (1966).

T. Trudgian. "Updating the error term in the prime number theorem". Ramanujan J. Vol.39, pp. 225–234. (2016). DOI: https://doi.org/10.1007/s11139-014-9656-6



How to Cite

Portion of Primes in Short in Intervals and Landau’s Inequality. (2020). Journal of Zankoy Sulaimani - Part A, 22(2), 345-352. https://doi.org/10.17656/jzs.10834

Most read articles by the same author(s)

1 2 3 4 5 6 7 8 9 10 > >>