Back to Search
Start Over
Error estimates for a class of continuous Bonse-type inequalities.
- Source :
-
Mathematics of Computation . Sep2022, Vol. 91 Issue 337, p2335-2345. 11p. - Publication Year :
- 2022
-
Abstract
- Let p_n be the nth prime number. In 2000, Papaitopol proved that the inequality p_1\cdots p_n>p_{n+1}^{n-\pi (n)} holds, for all n\geq 2, where \pi (x) is the prime counting function. In 2021, Yang and Liao tried to sharpen this inequality by replacing n-\pi (n) by n-\pi (n)+\pi (n)/\pi (\log n)-2\pi (\pi (n)), however there is a small mistake in their argument. In this paper, we exploit properties of the logarithm error term in inequalities of the type p_1\cdots p_n>p_{n+1}^{k(n,x)}, where k(n,x)=n-\pi (n)+\pi (n)/\pi (\log n)-x\pi (\pi (n)). In particular, we improve Yang and Liao estimate, by showing that the previous inequality at x=1.4 holds for all n\geq 21. [ABSTRACT FROM AUTHOR]
- Subjects :
- *MATHEMATICAL equivalence
Subjects
Details
- Language :
- English
- ISSN :
- 00255718
- Volume :
- 91
- Issue :
- 337
- Database :
- Academic Search Index
- Journal :
- Mathematics of Computation
- Publication Type :
- Academic Journal
- Accession number :
- 157929629
- Full Text :
- https://doi.org/10.1090/mcom/3741