Remarks on Ramanujan's inequality concerning the prime counting function

Mehdi Hassani

Communications in Mathematics (2021)

  • Volume: 29, Issue: 3, page 473-482
  • ISSN: 1804-1388

Abstract

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In this paper we investigate Ramanujan’s inequality concerning the prime counting function, asserting that π ( x ) 2 < e x log x π x e for x sufficiently large. First, we study its sharpness by giving full asymptotic expansions of its left and right hand sides expressions. Then, we discuss the structure of Ramanujan’s inequality, by replacing the factor x log x on its right hand side by the factor x log x - h for a given h , and by replacing the numerical factor e by a given positive α . Finally, we introduce and study inequalities analogous to Ramanujan’s inequality.

How to cite

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Hassani, Mehdi. "Remarks on Ramanujan's inequality concerning the prime counting function." Communications in Mathematics 29.3 (2021): 473-482. <http://eudml.org/doc/298115>.

@article{Hassani2021,
abstract = {In this paper we investigate Ramanujan’s inequality concerning the prime counting function, asserting that $\pi (x)^2<\frac\{\mathrm \{e\} \,x\}\{\log x\}\,\pi \left(\frac\{x\}\{\mathrm \{e\} \}\right)$ for $x$ sufficiently large. First, we study its sharpness by giving full asymptotic expansions of its left and right hand sides expressions. Then, we discuss the structure of Ramanujan’s inequality, by replacing the factor $\frac\{x\}\{\log x\}$ on its right hand side by the factor $\frac\{x\}\{\log x-h\}$ for a given $h$, and by replacing the numerical factor $\mathrm \{e\} $ by a given positive $\alpha $. Finally, we introduce and study inequalities analogous to Ramanujan’s inequality.},
author = {Hassani, Mehdi},
journal = {Communications in Mathematics},
keywords = {Prime numbers; Ramanujan's inequality; Riemann hypothesis},
language = {eng},
number = {3},
pages = {473-482},
publisher = {University of Ostrava},
title = {Remarks on Ramanujan's inequality concerning the prime counting function},
url = {http://eudml.org/doc/298115},
volume = {29},
year = {2021},
}

TY - JOUR
AU - Hassani, Mehdi
TI - Remarks on Ramanujan's inequality concerning the prime counting function
JO - Communications in Mathematics
PY - 2021
PB - University of Ostrava
VL - 29
IS - 3
SP - 473
EP - 482
AB - In this paper we investigate Ramanujan’s inequality concerning the prime counting function, asserting that $\pi (x)^2<\frac{\mathrm {e} \,x}{\log x}\,\pi \left(\frac{x}{\mathrm {e} }\right)$ for $x$ sufficiently large. First, we study its sharpness by giving full asymptotic expansions of its left and right hand sides expressions. Then, we discuss the structure of Ramanujan’s inequality, by replacing the factor $\frac{x}{\log x}$ on its right hand side by the factor $\frac{x}{\log x-h}$ for a given $h$, and by replacing the numerical factor $\mathrm {e} $ by a given positive $\alpha $. Finally, we introduce and study inequalities analogous to Ramanujan’s inequality.
LA - eng
KW - Prime numbers; Ramanujan's inequality; Riemann hypothesis
UR - http://eudml.org/doc/298115
ER -

References

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  2. Berndt, B.C., Ramanujan's Notebooks. Part IV, 1994, Springer-Verlag, New York, (1994) 
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  5. Hardy, G.H., Ramanujan. Twelve lectures on subjects suggested by his life and work, 1940, Cambridge University Press, Cambridge, England; Macmillan Company, New York, (1940) 
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  8. Mossinghoff, M.J., Trudgian, T.S., 10.1016/j.jnt.2015.05.010, J. Number Theory, 157, 2015, 329-349, (2015) MR3373245DOI10.1016/j.jnt.2015.05.010
  9. Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, Ch.W., NIST Handbook of Mathematical Functions, 2010, U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge, (2010) MR2723248
  10. Platt, D., Trudgian, T., The error term in the prime number theorem, arXiv:, 1809.03134, 2018, Preprint.. (2018) MR3448979
  11. Ramanujan, S., Notebooks. Vols. 1, 2, 1957, Tata Institute of Fundamental Research, Bombay, (1957) 
  12. Trudgian, T., 10.1007/s11139-014-9656-6, Ramanujan J., 39, 2, 2016, 225-234, (2016) MR3448979DOI10.1007/s11139-014-9656-6

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