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 ${\displaystyle f\colon \mathbb {N} \to \mathbb {C} }$ ${\displaystyle n}$

${\displaystyle |f(n)|\leq 1}$

meet.

Then:

I.

${\displaystyle \operatorname {M} (f)}$exists with if and only if both of the following conditions are satisfied: ${\displaystyle \operatorname {M} (f)\neq 0}$ ${\displaystyle f}$

(1) The series converges. ${\displaystyle \sum _{p\;{\text{Primzahl}}}{\frac {1-f(p)}{p}}}$

(2) There is at least one natural number with . ${\displaystyle r}$ ${\displaystyle f(2^{r})\neq -1}$

II

If both of the above conditions are satisfied, the following applies: ${\displaystyle f}$

${\displaystyle \operatorname {M} (f)=\prod _{p\;{\text{Primzahl}}}\left({\bigl (}1-{\frac {1}{p}}{\bigr )}\cdot {\bigl (}1+\sum _{k=1}^{\infty }{\frac {f(p^{k})}{p^{k}}}{\bigr )}\right)}$

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 ${\displaystyle a\colon \mathbb {N} \to \mathbb {C} }$ ${\displaystyle N\in \mathbb {N} }$

${\displaystyle A(N):=\sum _{p\;{\text{Primzahl}}\;,\;p\leq N}{\frac {a(p)}{p}}}$

and

${\displaystyle B(N):=\sum _{p\;{\text{Primzahl}}\;,\;k\in \mathbb {N} \;,\;p^{k}\leq N}{\frac {|a(p^{k})|^{2}}{p^{k}}}}$

set.

Then there is an absolute constant that is independent of the number theoretic function such that for always the inequality ${\displaystyle a}$ ${\displaystyle K>0}$ ${\displaystyle N\in \mathbb {N} }$

${\displaystyle \sum _{n=1}^{N}{|a(n)-A(N)|^{2}}\leq K\cdot N\cdot B(N)}$

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 : ${\displaystyle g\colon \mathbb {N} \to \mathbb {C} }$ ${\displaystyle \operatorname {M} (g)}$ ${\displaystyle \mathbb {C} }$

${\displaystyle \lim _{x\in \mathbb {R} \;,\;x\to \infty }{{\frac {1}{x}}{\sum _{n\in \mathbb {N} \;,\;n\leq x}g(n)}}=:\operatorname {M} (g)}$

• 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 . ${\displaystyle A(N)}$ ${\displaystyle K}$

See also

• Article Turán – Kubilius inequality in the English language Wikipedia

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

1. ↑ ^{a b} Wolfgang Schwarz: Introduction to number theory. 1975, p. 121 ff.
2. ^ Jean-Marie De Koninck, Florian Luca: Analytic Number Theory. 2012, p. 87 ff.
3. ↑ De Koninck / Luca, op.cit., P. 88
4. ↑ denotes the amount function . ${\displaystyle |\cdot |}$
5. ↑ Schwarz, op.cit., P. 122
6. ^ József Sándor et al .: Handbook of Number Theory. I, chapter = XVI.3. 2006, p. 561 ff.