View source for Delange theorem

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

<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>

if and only if:

i) the Delange series

<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>

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.

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.

An elegant proof of the implication  "i) and ii)  Mf 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" />.

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]]

<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>

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]].

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====
<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>