| Title:
|
Remarks on Ramanujan's inequality concerning the prime counting function (English) |
| Author:
|
Hassani, Mehdi |
| Language:
|
English |
| Journal:
|
Communications in Mathematics |
| ISSN:
|
1804-1388 (print) |
| ISSN:
|
2336-1298 (online) |
| Volume:
|
29 |
| Issue:
|
3 |
| Year:
|
2021 |
| Pages:
|
473-482 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
Prime numbers |
| Keyword:
|
Ramanujan's inequality |
| Keyword:
|
Riemann hypothesis |
| MSC:
|
11A41 |
| idZBL:
|
Zbl 07484381 |
| idMR:
|
MR4355420 |
| . |
| Date available:
|
2022-01-10T10:09:08Z |
| Last updated:
|
2022-04-28 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/149330 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
[8] Mossinghoff, M.J., Trudgian, T.S.: Nonnegative trigonometric polynomials and a zero-free region for the Riemann zeta-function.J. Number Theory, 157, 2015, 329-349, MR 3373245, 10.1016/j.jnt.2015.05.010 |
| Reference:
|
[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, MR 2723248 |
| Reference:
|
[10] Platt, D., Trudgian, T.: The error term in the prime number theorem.arXiv:, 1809.03134, 2018, Preprint.. MR 3448979 |
| Reference:
|
[11] Ramanujan, S.: Notebooks. Vols. 1, 2.1957, Tata Institute of Fundamental Research, Bombay, |
| Reference:
|
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| . |
