Chebyshev estimates for Beurling generalized prime numbers
Author:
Harold G. Diamond
Journal:
Proc. Amer. Math. Soc. 39 (1973), 503508
MSC:
Primary 10H20
MathSciNet review:
0314782
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We consider a Beurling generalized number system for which the counting function of the integers satisfies for some positive and . Beurling showed that the prime number theorem must hold if , but it can fail to hold if . It was shown by Hall in the preceding article that the Chebyshev prime counting estimates can fail if . Here we shall prove that these estimates hold for any system satisfying . The proof uses a convolution approximate inverse of the measure .
 [1]
Paul
T. Bateman and Harold
G. Diamond, Asymptotic distribution of Beurling’s generalized
prime numbers, Studies in Number Theory, Math. Assoc. Amer.
(distributed by PrenticeHall, Englewood Cliffs, N.J.), 1969,
pp. 152–210. MR 0242778
(39 #4105)
 [2]
A. Beurling, Analyse de la loi asymptotique de la distribution des nombres premiers généralisés. I, Acta Math. 68 (1937), 255291.
 [3]
T.
M. K. Davison, A Tauberian theorem and analogues of the prime
number theorem, Canad. J. Math. 20 (1968),
362–367. MR 0224569
(37 #168)
 [4]
Harold
G. Diamond, Asymptotic distribution of Beurling’s generalized
integers, Illinois J. Math. 14 (1970), 12–28.
MR
0252334 (40 #5555)
 [5]
R. S. Hall, Theorems about Beurling's generalized primes and the associated zeta function, Ph.D. Thesis, University of Illinois, Urbana, Ill., 1967.
 [6]
S.
L. Segal, A Tauberian theorem for Dirichlet convolutions,
Illinois J. Math. 13 (1969), 316–320. MR 0238779
(39 #143)
 [1]
 P. T. Bateman and H. G. Diamond, Asymptotic distribution of Beurling's generalized prime numbers, Studies in Number Theory, vol. 6, Math. Assoc. Amer., PrenticeHall, Englewood Cliffs, N.J., 1969, pp. 152210. MR 39 #4105. MR 0242778 (39:4105)
 [2]
 A. Beurling, Analyse de la loi asymptotique de la distribution des nombres premiers généralisés. I, Acta Math. 68 (1937), 255291.
 [3]
 T. M. K. Davison, A Tauberian theorem and analogues of the prime number theorem, Canad. J. Math. 20 (1968), 362367. MR 37 #168. MR 0224569 (37:168)
 [4]
 H. G. Diamond, Asymptotic distribution of Beurling's generalized integers, Illinois J. Math. 14 (1970), 1228. MR 40 #5555. MR 0252334 (40:5555)
 [5]
 R. S. Hall, Theorems about Beurling's generalized primes and the associated zeta function, Ph.D. Thesis, University of Illinois, Urbana, Ill., 1967.
 [6]
 S. L. Segal, A Tauberian theorem for Dirichlet convolutions, Illinois J. Math. 13 (1969), 316320. MR 39 #143. MR 0238779 (39:143)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC:
10H20
Retrieve articles in all journals
with MSC:
10H20
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939197303147824
PII:
S 00029939(1973)03147824
Article copyright:
© Copyright 1973
American Mathematical Society
