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Arithmetic Linear Transformations and Abstract Prime Number Theorems

Published online by Cambridge University Press:  20 November 2018

S. A. Amitsur
S. A. Amitsur*
Affiliation:
Hebrew University
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Shapiro and Forman have presented in (4) an abstract formulation of prime number theorems which includes the various prime number theorems; for primes in arithmetic progressions, for prime ideals in ideal classes etc. The methods of proofs are “elementary” and follow closely Shapiro's proof for the primes in arithmetic progression (for reference see bibliography in (4)).

The author has followed in (1) some ideas of Yamamoto (5) on arithmetic linear transformations to introduce a symbolic calculus in dealing with arithmetic functions. This calculus proved to be very useful in unifying many of the “elementary” proofs in the behaviour of arithmetic functions. In (6) Yamamoto has extended his theory to ideals in algebraic number fields, and with this extension the symbolic calculus of (1) can be extended to cover the abstract case of prime number theorem in countable free abelian groups as discussed in (4).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 13 , 1961 , pp. 83 - 109
Copyright
Copyright © Canadian Mathematical Society 1961

References

1. Amitsur, S. A., On arithmetic junctions, J. d'Anal. Math., 5 (1956-7), 273–317.Google Scholar
2. Amitsur, S. A., Some results on arithmetic functions, J. Math. Soc. Japan 11 (1959), 275–290.Google Scholar
3. Beurling, A., Analyse de la loi asymptotique de la distribution des nombre premiers generalises, Acta. Math., 68 (1937), 255–291.Google Scholar
4. Shapiro, H. and Forman, W., Abstract prime number theorems, Comm. Pure and Applied Math., 7 (1954), 587–619.Google Scholar
5. Yamamoto, K., Theory of arithmetic linear transformations and its applications to an elementary proof of Dirichlet's theorem, J. Math. Soc. Japan, 7 (1955), 424–434.Google Scholar
6. Yamamoto, K., Arithmetic linear transformations in an algebraic number field, Mem. Fac. Sci., Kyushu Univ. Sci. A. 12 (1958), 41–66.Google Scholar
7. Hardy, and Wright, , The theory of numbers (Oxford, 1954).Google Scholar