Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-12-02T23:20:42.854Z Has data issue: false hasContentIssue false
  • English
  • Français

Primes of the form [pc] and related questions

Published online by Cambridge University Press:  18 May 2009

Glyn Harman  and
Joël Rivat
Glyn Harman
Affiliation:
School of Mathematics, UWCC, Senghennydd Road, P.O. Box 926, Cardiff CF2 4YH
Joël Rivat
Affiliation:
Bâtiment 425, Université de Paris Sud, 91405 Orsay, France, [email protected]
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Much interest has been shown in determining the range of values of c for which the sequence [n]c contains infinitely many primes. The result is an elementary deduction from the prime number theorem, of course, of 0<c≤l. In 1953, Piatetski–Shapiro [9] showed that

for 1<c<12/11, where xc(X) stands for the number of primes in the set {[nc]n≤x}.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 37 , Issue 2 , May 1995 , pp. 131 - 141
Copyright
Copyright © Glasgow Mathematical Journal Trust 1995

References

REFERENCES

1Baker, R. C., Harman, G. and Rivat, J.. Primes of the form [nc]. to appear, J. Number Theory.Google Scholar
2Balog, A.. On a variant of the Piatetski–Shapiro Prime Number Theorem. Séminaire de Théorie Analytique des Nombres de Paris, 1986.Google Scholar
3Deshouillers, J.-M.. Nombres premiers de la forme [nc]. C. R. Acad. Sci. Paris Series A–B, 282 (3) (1976), A 131–133.Google Scholar
4Harman, G.. Metric Diophantine approximation with two restricted variables III. j. Number Theory, 29 (1988), 364–375.CrossRefGoogle Scholar
5Harman, G.. Fractional and integral parts of pλ. Acta Arith. 58(2) (1991), 141–152.CrossRefGoogle Scholar
6Harman, G.. Metrical theorems on fractional parts of real sequences II. j.Number Theory 44 (1993), 47–57.CrossRefGoogle Scholar
7Landau, E.. Uber einige Summen, die von den Nullstellen der Riemann'schen Zetafunction abhängen. Acta Math. 35 (1912), 271–294.CrossRefGoogle Scholar
8Leitmann, D. and Wolke, D.. Primzahlen der Gestalt [f(n)]. Math. Z. 145 (1975), 81–92.CrossRefGoogle Scholar
9Piatetski–Shapiro, I.. On the distribution of prime numbers in sequences of the form [f(n)]. Mat. Sb. 33(1) (1953), 559–566.Google Scholar
10Rivat, J.. Autour d'un théoréme de Piatetski–Shapiro (Nombres premiers dans la suite [nc]). Thése de troisiéme cycle, Université de Paris-Sud, 1992.Google Scholar
11Schmidt, W. M.. Metrical theorems on fractional parts of sequences. Trans. Amer. Math. Soc., 110 (1964), 493–518.CrossRefGoogle Scholar
12Sprindzuk, V. G.. Metric theory of Diophantine Approximation (translated by Silverman, R. A.). (Winston/Wiley, New York, 1979).Google Scholar
13Titchmarsh, E. C.. The Theory of the Riemann Zeta–function, revised by Brown, D. R. Heath. (Oxford Science Publications, second edition, 1986).Google Scholar