Glasnik Matematicki, Vol. 53, No. 1 (2018), 1-7.

NEW UPPER BOUNDS FOR RAMANUJAN PRIMES

Anitha Srinivasan and Pablo Ares-Gastesi

Department of Applied Mathematics and Statistics, School of Business and Economics, Universidad CEU San Pablo, Madrid, Spain
e-mail: pablo.aresgastesi@ceu.es

Abstract.   For n≥ 1, the n^th Ramanujan prime is defined as the smallest positive integer R_n such that for all x≥ R_n, the interval (x/2, x] has at least n primes. We show that for every ε>0, there is a positive integer N such that if α=2n(1+(log 2+ε)/(log n+j(n))), then R_n< p_[α] for all n>N, where p_i is the i^th prime and j(n)>0 is any function that satisfies j(n)→ ∞ and nj'(n)→ 0.

2010 Mathematics Subject Classification.   11A41, 11N05.

Key words and phrases.   Ramanujan primes, upper bounds.

DOI: 10.3336/gm.53.1.01

References:

1. C. Axler, Über die Primzahl-Zählfunktion, die n-te Primzahl und verallgemeinerte Ramanujan Primzahlen, Ph.D. thesis, 2013 (in German),
   http://docserv.uni-duesseldorf.de/servlets/DocumentServlet?id=26247

2. C. Axler, On generalized Ramanujan primes, Ramanujan J. 39 (2016), 1-30.
   MathSciNet     CrossRef

3. P. Dusart, Explicit estimates of some functions over primes, Ramanujan J. 45 (2018), 227-251.
   MathSciNet     CrossRef

4. S. Laishram, On a conjecture on Ramanujan primes, Int. J. Number Theory 6 (2010), 1869-1873.
   MathSciNet     CrossRef

5. J. B. Paksoy, Derived Ramanujan primes R_n'.
   http:/arxiv.org/abs/1210.6991

6. V. Shevelev, Ramanujan and Labos primes, their generalizations and classifications of primes, J. Integer Seq. 15 (2012), Article 12.1.1.
   MathSciNet    

7. J. Sondow, Ramanujan primes and Bertrand's postulate, Amer. Math. Monthly 116 (2009), 630-635.
   MathSciNet     CrossRef

8. J. Sondow, J. W. Nicholson, T. D. Noe, Ramanujan primes: bounds, runs, twins, and gaps, J. Integer Seq. 14 (2011), Article 11.6.2.
   MathSciNet    

9. A. Srinivasan, An upper bound for Ramanujan primes, Integers 14 (2014), paper A19.
   MathSciNet    

10. A. Srinivasan and J. Nicholson, An improved upper bound for Ramanujan primes, Integers 15 (2015), paper A52.
    MathSciNet    

11. S. Yang and A. Togbé On the estimates of the upper and lower bounds of Ramanujan primes, Ramanujan J. 40 (2016), 245-255.
    MathSciNet     CrossRef