Delange's theorem
The set of Delange ( English Delange's theorem ) is a theorem of the mathematical area of analytic number theory , of a work of French mathematician Hubert Delange back in 1961 and addresses the question of the conditions under which statements about averages number theoretic functions can be made . In 1965 Alfréd Rényi provided a simplified proof of the theorem, which is essentially based on an inequality formulated by Jonas Kubilius and Paul Turán .
Formulation of the sentence
Delange's sentence can be summarized as follows:
-
A multiplicative number-theoretic function is given , which should not be the zero function and which for every natural number with regard to the amount of the function value the inequality
- meet.
- Then:
- I.
-
exists with if and only if both of the following conditions are satisfied:
- (1) The series converges.
- (2) There is at least one natural number with .
- II
-
If both of the above conditions are satisfied, the following applies:
Background: The inequality of Turán and Kubilius
The mentioned Turán-Kubilius inequality ( English Turán-Kubilius inequality ) can be formulated as follows, following the monograph by Wolfgang Schwarz :
-
For a given additive number theoretic function let for
-
and
- set.
-
Then there is an absolute constant that is independent of the number theoretic function such that for always the inequality
- is satisfied.
Explanations
- In relation to a given number theoretic function , the (associated) mean value exists if the following limit value exists in the complex number plane :
- There are other and better versions of the Turán-Kubilius inequality presented above , which on the one hand vary the above deduction element and on the other hand the mentioned constant .
See also
literature
- Jean-Marie De Koninck , Florian Luca : Analytic Number Theory . Exploring the Anatomy of Integers ( Graduate Studies in Mathematics . Volume 134 ). American Mathematical Society , Providence, RI 2012, ISBN 978-0-8218-7577-3 ( MR2919246 ).
- Hubert Delange: Sur les fonctions arithmétiques multiplicatives . In: Annales Scientifiques de l'École Normale Supérieure. Troisième Série . tape 78 , 1961, pp. 273-304 ( MR0169829 ).
- J. Kubilius : Probabilistic Methods in the Theory of Numbers (= Translations of Mathematical Monographs . Volume 11 ). American Mathematical Society , Providence, RI 1964 ( MR0160745 ).
- Alfréd Rényi: A new proof of a theorem of Delange . In: Publicationes Mathematicae Debrecen . tape 12 , 1965, p. 323-329 ( MR0190124 ).
- József Sándor , Dragoslav S. Mitrinović , Borislav Crstici : Handbook of Number Theory. I . Springer Verlag , Dordrecht 2006, ISBN 978-1-4020-4215-7 , XVI.27, p. 584 ff . ( MR2186914 ).
- Wolfgang Schwarz : Introduction to number theory (= mathematics. Introductions to the subject matter and results of its sub-areas and related sciences ). Scientific Book Society, Darmstadt 1975 ( MR0434930 ).
Individual evidence
- ↑ a b Wolfgang Schwarz: Introduction to number theory. 1975, p. 121 ff.
- ^ Jean-Marie De Koninck, Florian Luca: Analytic Number Theory. 2012, p. 87 ff.
- ↑ De Koninck / Luca, op.cit., P. 88
- ↑ denotes the amount function .
- ↑ Schwarz, op.cit., P. 122
- ^ József Sándor et al .: Handbook of Number Theory. I, chapter = XVI.3. 2006, p. 561 ff.