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In 1961 H. Delange (see [[#References|[a1]]]) proved that a [[Multiplicative arithmetic function|multiplicative arithmetic function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110120/d1101201.png" /> of modulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110120/d1101202.png" /> possesses a non-zero mean value
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In 1961 H. Delange (see [[#References|[a1]]]) proved that a [[multiplicative arithmetic function]] $f : \mathbf{N} \rightarrow \mathbf{C}$ of modulus $|f| \le 1$ possesses a non-zero mean value
 
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$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110120/d1101203.png" /></td> </tr></table>
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M(f) = \lim_{x\rightarrow\infty} \frac{1}{x} \sum_{n\le x} f(n)
 
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$$
 
if and only if:
 
if and only if:
  
 
i) the Delange series
 
i) the Delange series
 
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$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110120/d1101204.png" /></td> </tr></table>
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S_1 = \sum_p \frac{1}{p}(f(p)-1) \,,
 
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$$
 
extended over the primes, is convergent; and
 
extended over the primes, is convergent; and
  
ii) all the factors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110120/d1101205.png" /> of the [[Euler product|Euler product]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110120/d1101206.png" /> are non-zero.
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ii) all the factors $\sum_{k=0}^\infty f(p^k) p^{-ks}$ of the [[Euler product]] of $\sum_n f(n) n^{-s}$ are non-zero.
  
Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110120/d1101207.png" />, condition ii) is automatically true for every prime <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110120/d1101208.png" />. In [[#References|[a2]]] this theorem was sharpened.
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Since $|f| \le 1$, condition ii) is automatically true for every prime $p>2$. In [[#References|[a2]]] this theorem was sharpened.
  
An elegant proof of the implication  "i) and ii)  Mf exists" , using the Turán–Kubilius inequality, is due to A. Rényi [[#References|[a4]]].
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An elegant proof of the implication  "i) and ii)  $\Rightarrow$ $M(f)$ exists" , using the Turán–Kubilius inequality, is due to A. Rényi [[#References|[a4]]].
  
Using the continuity theorem for characteristic functions, for a real-valued [[Additive arithmetic function|additive arithmetic function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110120/d1101209.png" /> Delange's theorem permits one to deal with the problem of the existence of limit distributions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110120/d11012010.png" />.
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Using the continuity theorem for characteristic functions, for a real-valued [[additive arithmetic function]] $f$ Delange's theorem permits one to deal with the problem of the existence of limit distributions  
 +
$$
 +
\Psi(x) = \lim_{N\rightarrow\infty} \frac{1}{N} \sharp\{ n \le N : f(n) \le x \} \ .
 +
$$
  
Important extensions of Delange's theorem are due to P.D.T.A. Elliott and H. Daboussi; these theorems give necessary and sufficient conditions for multiplicative functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110120/d11012011.png" /> with finite [[Semi-norm|semi-norm]]
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Important extensions of Delange's theorem are due to P.D.T.A. Elliott and H. Daboussi; these theorems give necessary and sufficient conditions for multiplicative functions $f$ with finite [[semi-norm]]
 
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$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110120/d11012012.png" /></td> </tr></table>
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\Vert f \Vert_q = \left({ \limsup_{x\rightarrow\infty} \frac{1}{x} \sum_{n \le x} |f(n)|^q }\right)^{1/q}
 
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$$
to possess a non-zero mean value (respectively, at least one non-zero Fourier coefficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110120/d11012013.png" />). See [[Elliott–Daboussi theorem|Elliott–Daboussi theorem]]. See also [[Wirsing theorems|Wirsing theorems]].
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to possess a non-zero mean value (respectively, at least one non-zero Fourier coefficient $M(n\mapsto f(n) \exp(2\pi i a/r n)$). See [[Elliott–Daboussi theorem]]. See also [[Wirsing theorems]].
  
 
E.V. Novoselov's theory of integration for arithmetic functions (see [[#References|[a3]]]) also leads to many results on mean values of arithmetic functions.
 
E.V. Novoselov's theory of integration for arithmetic functions (see [[#References|[a3]]]) also leads to many results on mean values of arithmetic functions.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Delange,  "Sur les fonctions arithmétiques multiplicatives"  ''Ann. Sci. Ecole Norm. Sup. (3)'' , '''78'''  (1961)  pp. 273–304</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Delange,  "On a class of multiplicative functions"  ''Scripta Math.'' , '''26'''  (1963)  pp. 121–141</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  E.V. Novoselov,  "A new method in probabilistic number theory"  ''Transl. Amer. Math. Soc.'' , '''52'''  (1966)  pp. 217–275  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''28'''  (1964)  pp. 307–364</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A. Rényi,  "A new proof of a theorem of Delange"  ''Publ. Math. Debrecen'' , '''12'''  (1965)  pp. 323–329</TD></TR></table>
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<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Delange,  "Sur les fonctions arithmétiques multiplicatives"  ''Ann. Sci. Ecole Norm. Sup. (3)'' , '''78'''  (1961)  pp. 273–304</TD></TR>
 +
<TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Delange,  "On a class of multiplicative functions"  ''Scripta Math.'' , '''26'''  (1963)  pp. 121–141</TD></TR>
 +
<TR><TD valign="top">[a3]</TD> <TD valign="top">  E.V. Novoselov,  "A new method in probabilistic number theory"  ''Transl. Amer. Math. Soc.'' , '''52'''  (1966)  pp. 217–275  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''28'''  (1964)  pp. 307–364</TD></TR>
 +
<TR><TD valign="top">[a4]</TD> <TD valign="top">  A. Rényi,  "A new proof of a theorem of Delange"  ''Publ. Math. Debrecen'' , '''12'''  (1965)  pp. 323–329</TD></TR>
 +
</table>
 +
 
 +
{{TEX|done}}

Latest revision as of 11:03, 21 October 2017

In 1961 H. Delange (see [a1]) proved that a multiplicative arithmetic function $f : \mathbf{N} \rightarrow \mathbf{C}$ of modulus $|f| \le 1$ possesses a non-zero mean value $$ M(f) = \lim_{x\rightarrow\infty} \frac{1}{x} \sum_{n\le x} f(n) $$ if and only if:

i) the Delange series $$ S_1 = \sum_p \frac{1}{p}(f(p)-1) \,, $$ extended over the primes, is convergent; and

ii) all the factors $\sum_{k=0}^\infty f(p^k) p^{-ks}$ of the Euler product of $\sum_n f(n) n^{-s}$ are non-zero.

Since $|f| \le 1$, condition ii) is automatically true for every prime $p>2$. In [a2] this theorem was sharpened.

An elegant proof of the implication "i) and ii) $\Rightarrow$ $M(f)$ exists" , using the Turán–Kubilius inequality, is due to A. Rényi [a4].

Using the continuity theorem for characteristic functions, for a real-valued additive arithmetic function $f$ Delange's theorem permits one to deal with the problem of the existence of limit distributions $$ \Psi(x) = \lim_{N\rightarrow\infty} \frac{1}{N} \sharp\{ n \le N : f(n) \le x \} \ . $$

Important extensions of Delange's theorem are due to P.D.T.A. Elliott and H. Daboussi; these theorems give necessary and sufficient conditions for multiplicative functions $f$ with finite semi-norm $$ \Vert f \Vert_q = \left({ \limsup_{x\rightarrow\infty} \frac{1}{x} \sum_{n \le x} |f(n)|^q }\right)^{1/q} $$ to possess a non-zero mean value (respectively, at least one non-zero Fourier coefficient $M(n\mapsto f(n) \exp(2\pi i a/r n)$). See Elliott–Daboussi theorem. See also Wirsing theorems.

E.V. Novoselov's theory of integration for arithmetic functions (see [a3]) also leads to many results on mean values of arithmetic functions.

References

[a1] H. Delange, "Sur les fonctions arithmétiques multiplicatives" Ann. Sci. Ecole Norm. Sup. (3) , 78 (1961) pp. 273–304
[a2] H. Delange, "On a class of multiplicative functions" Scripta Math. , 26 (1963) pp. 121–141
[a3] E.V. Novoselov, "A new method in probabilistic number theory" Transl. Amer. Math. Soc. , 52 (1966) pp. 217–275 Izv. Akad. Nauk SSSR Ser. Mat. , 28 (1964) pp. 307–364
[a4] A. Rényi, "A new proof of a theorem of Delange" Publ. Math. Debrecen , 12 (1965) pp. 323–329
How to Cite This Entry:
Delange theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Delange_theorem&oldid=18449
This article was adapted from an original article by W. Schwarz (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article