Rocky Mountain Journal of Mathematics

Explicit Estimate on Primes Between Consecutive Cubes

Yuan-You Fu-Rui Cheng

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Article information

Rocky Mountain J. Math. Volume 40, Number 1 (2010), 117-153.

First available in Project Euclid: 15 March 2010

Permanent link to this document

Digital Object Identifier

Zentralblatt MATH identifier

Mathematical Reviews number (MathSciNet)

Primary: 11Y35: Analytic computations 11N05: Distribution of primes

Ingham's theorem primes in short interval explicit estimates density theorem divisors


Cheng, Yuan-You Fu-Rui. Explicit Estimate on Primes Between Consecutive Cubes. Rocky Mountain J. Math. 40 (2010), no. 1, 117--153. doi:10.1216/RMJ-2010-40-1-117.

Export citation


  • L.V. Ahlfors, Complex analysis, 2nd ed., McGraw-Hill Book Company, New York, 1979.
  • R.C. Baker and G. Harman, The difference between consecutive primes, Proc. London Math. Soc. 2 (1996), 261.
  • C. Caldwell and Y. (Fred) Cheng, Determining Mills' constants and a note on Honaker's problem J. Integer Sequences 8 (2005), 1-9.
  • K. Chandrasekharan, Arithmetical functions, Springer-Verlag, New York, 1970.
  • Y. (Fred) Cheng, Explicit estimates involving divisor functions, J. Number Theory, submitted.
  • Y. (Fred) Cheng and Sidney W. Graham, Explicit estimates for the Riemann zeta function, Rocky Mountain J. Math. 34 (2004), 1261-1290.
  • Y. (Fred) Cheng and Barnet Weinstock, Explicit estimates on prime numbers, Rocky Mountain J. Math., accepted.
  • H. Davenport, Multiplicative number theory, Springer-Verlag, New York, 1980.
  • H.M. Edwards, Riemann's zeta-function, Academic Press, New York, 1974.
  • K. Ford, Zero-free regions for the Riemann zeta-function, Proc. Millenial Conference on Number Theory, Urbana, IL, 2000.
  • D.A. Goldston and S.M. Gonek, A note on the number of primes in short intervals, Proc. Amer. Math. Soc. 3 (1990), 613-620.
  • G.H. Hardy and E.M. Wright, An introduction to the theory of numbers, 5th ed., Clarendon Press, Oxford, 1979.
  • G. Hoheisel, Primzahlprobleme in der Analysis, Sitz. Preuss. Akad. Wiss. 33 (1930), 580-588.
  • M.N. Huxley, The distribution of prime numbers, Oxford University Press, Oxford, 1972.
  • A.E. Ingham, On the estimation of $N(\sigma, T)$, Quart. J. Math. 11 (1940), 291-292.
  • K. Ireland and M. Rosen, A classical introduction to modern number theory, 2nd ed., Springer-Verlag, New York, 1990.
  • A. Ivić, The Riemann zeta function, John Wiley & Sons, New York, 1985.
  • H. Iwaniec and J. Pintz, Primes in short intervals, Monat. Math. 98 (1984), 115-143.
  • L. Kaniecki, On differences of primes in short intervals under the Riemann hypothesis, Demonstrat. Math 1 (1998), 121-124.
  • F. Kevin, A new result on the upper bound for the Riemann zeta function, preprint, 2001.
  • H.L. Montgomery, Zeros of $L$-functions, Invent. Math. 8 (1969), 346-354.
  • J.B. Rosser, Explicit bounds for some functions of prime numbers, Amer. J. Math. 63 (1941), 211-232.
  • J.B. Rosser and L. Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64-94.
  • --------, Sharper bounds for Chebyshev functions $\theta(x)$ and $\psi(x)$ I, Math. Comp. 29 (1975), 243-269.
  • L. Schoenfeld, Sharper bounds for Chebyshev functions $\theta(x)$ and $\psi(x)$ II, Math. Comp. 30 (1976), 337-360.
  • Titchmarsh, The theory of the Riemann zeta function, Oxford Science Publications, 1986.
  • J. Van de Lune, H.J.J. te Diele and D.T. Winter, On the zeros of the Riemann zeta function in the critical strip, IV, Math. Comp. 47 (1986), 67-681.
  • C.Y. Yildirim, A survey of results on primes in short intervals, in Number theory and its applications, %(Ankara, 1996), Lecture Notes in Pure and Appl. Math. 204 Dekker, New York, 1999.
  • D. Zinoviev, On Vinogradov's constant in Goldbach's ternary problem, J. Number Theory 65 (1997), 334-358.