## An upper bound for the sum of large differences between prime numbers

HTML articles powered by AMS MathViewer

- by R. J. Cook PDF
- Proc. Amer. Math. Soc.
**81**(1981), 33-40 Request permission

## Abstract:

Let ${p_n}$ denote the $n$th prime number, ${d_n} = {p_{n + 1}} - {p_n}$. We estimate the sum $\Sigma {d_n}$ taken over ${p_n} \leqslant x,{d_n} > {x^\mu }$ where $1/6 < \mu < 5/9$.## References

- K. Chandrasekharan,
*Arithmetical functions*, Die Grundlehren der mathematischen Wissenschaften, Band 167, Springer-Verlag, New York-Berlin, 1970. MR**0277490** - R. J. Cook,
*On the occurrence of large gaps between prime numbers*, Glasgow Math. J.**20**(1979), no. 1, 43–48. MR**523787**, DOI 10.1017/S0017089500003700
H. Cramér, - Harold Davenport,
*Multiplicative number theory*, Lectures in Advanced Mathematics, No. 1, Markham Publishing Co., Chicago, Ill., 1967. Lectures given at the University of Michigan, Winter Term, 1966. MR**0217022** - G. Halász and P. Turán,
*On the distribution of roots of Riemann zeta and allied functions. I*, J. Number Theory**1**(1969), 121–137. MR**236124**, DOI 10.1016/0022-314X(69)90031-6 - D. R. Heath-Brown,
*The differences between consecutive primes*, J. London Math. Soc. (2)**18**(1978), no. 1, 7–13. MR**491554**, DOI 10.1112/jlms/s2-18.1.7 - D. R. Heath-Brown,
*The differences between consecutive primes. II*, J. London Math. Soc. (2)**19**(1979), no. 2, 207–220. MR**533319**, DOI 10.1112/jlms/s2-19.2.207 - D. R. Heath-Brown,
*Zero density estimates for the Riemann zeta-function and Dirichlet $L$-functions*, J. London Math. Soc. (2)**19**(1979), no. 2, 221–232. MR**533320**, DOI 10.1112/jlms/s2-19.2.221 - M. N. Huxley,
*On the difference between consecutive primes*, Invent. Math.**15**(1972), 164–170. MR**292774**, DOI 10.1007/BF01418933 - M. N. Huxley,
*A note on large gaps between prime numbers*, Acta Arith.**38**(1980/81), no. 1, 63–68. MR**574125**, DOI 10.4064/aa-38-1-63-68 - A. E. Ingham,
*On the estimation of $N(\sigma ,T)$*, Quart. J. Math. Oxford Ser.**11**(1940), 291–292. MR**3649** - Aleksandar Ivić,
*On sums of large differences between consecutive primes*, Math. Ann.**241**(1979), no. 1, 1–9. MR**531146**, DOI 10.1007/BF01406704 - Matti Jutila,
*Zero-density estimates for $L$-functions*, Acta Arith.**32**(1977), no. 1, 55–62. MR**429790**, DOI 10.4064/aa-32-1-55-62 - Hugh L. Montgomery,
*Topics in multiplicative number theory*, Lecture Notes in Mathematics, Vol. 227, Springer-Verlag, Berlin-New York, 1971. MR**0337847** - Carlos Julio Moreno,
*The average size of gaps between primes*, Mathematika**21**(1974), 96–100. MR**404169**, DOI 10.1112/S0025579300005829 - Atle Selberg,
*On the normal density of primes in small intervals, and the difference between consecutive primes*, Arch. Math. Naturvid.**47**(1943), no. 6, 87–105. MR**12624** - E. C. Titchmarsh,
*The Theory of the Riemann Zeta-Function*, Oxford, at the Clarendon Press, 1951. MR**0046485** - Richard Warlimont,
*Über die Häufigkeit grosser Differenzen konsekutiver Primzahlen*, Monatsh. Math.**83**(1977), no. 1, 59–63 (German). MR**444588**, DOI 10.1007/BF01303013 - Dieter Wolke,
*Grosse Differenzen zwischen aufeinanderfolgenden Primzahlen*, Math. Ann.**218**(1975), no. 3, 269–271 (German). MR**453670**, DOI 10.1007/BF01349699

*Some theorems concerning prime numbers*, Ark. Mat.

**15**(1920), 1-32. —,

*On the order of magnitude of the difference between consecutive prime numbers*, Acta Arith.

**2**(1937), 23-46.

## Additional Information

- © Copyright 1981 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**81**(1981), 33-40 - MSC: Primary 10H15
- DOI: https://doi.org/10.1090/S0002-9939-1981-0589132-3
- MathSciNet review: 589132