Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-28T14:23:07.064Z Has data issue: false hasContentIssue false
  • English
  • Français

Generalized Euler constants

Published online by Cambridge University Press:  01 July 2008

HAROLD G. DIAMOND  and
KEVIN FORD
HAROLD G. DIAMOND
Affiliation:
Department of Mathematics, University of Illinois, 1409 West Green St., Urbana, IL 61801, USA. e-mail: [email protected], [email protected]
KEVIN FORD
Affiliation:
Department of Mathematics, University of Illinois, 1409 West Green St., Urbana, IL 61801, USA. e-mail: [email protected], [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the distribution of a family of generalized Euler constants arising from integers sieved by finite sets of primes . For , the set of the first r primes, as r → ∞. Calculations suggest that is monotonic in r, but we prove it is not. Also, we show a connection between the distribution of and the Riemann hypothesis.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 145 , Issue 1 , July 2008 , pp. 27 - 41
Copyright
Copyright © Cambridge Philosophical Society 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Bateman, P. T. and Diamond, H. G.. Analytic Number Theory (World Scientific, 2004).CrossRefGoogle Scholar
[2]Bays, C. and Hudson, R. H.. A new bound for the smallest x with π(x) > li(x). Math. Comp. 69 (2000), 1285–1296.CrossRefGoogle Scholar
[3]Davenport, H.. Multiplicative number theory, 3rd ed. Revised and with a preface by H. L. Montgomery. Graduate Texts in Math. 74 (Springer-Verlag, 2000).Google Scholar
[4]Havil, J.. Gamma: Exploring Euler's Constant; with a foreword by Freeman Dyson, (Princeton University Press, 2003).Google Scholar
[5]Lehman, R. S.. On the difference π(x)-li(x). Acta Arith. 11 (1966), 397–410.CrossRefGoogle Scholar
[6]Littlewood, J. E.. Sur la distribution des nombres premiers. Comptes Rendus 158 (1914), 1869–1872.Google Scholar
[7]Pintz, J.. On the remainder term of the prime number formula I. On a problem of Littlewood. Acta Arith. 36 (1980), 341–365.CrossRefGoogle Scholar
[8]Pintz, J.. On the remainder term of the prime number formula II. On a theorem of Ingham. Acta Arith. 37 (1980), 209–220.CrossRefGoogle Scholar
[9]Ramaré, O. and Rumely, R.. Primes in arithmetic progressions. Math. Comp. 65 (1996), 397–425.CrossRefGoogle Scholar
[10]Rosser, J. B. and Schoenfeld, L.. Approximate formulas for some functions of prime numbers. Illinois J. Math. 6 (1962), 64–94.CrossRefGoogle Scholar
[11]Schoenfeld, L.. Sharper bounds for the Chebyshev functions θ(x) and ψ(x). Math. Comp. 30 (1976), 337–360.Google Scholar