View source for Elliott-Daboussi theorem

The [[Delange theorem|Delange theorem]], proved in 1961, gives necessary and sufficient conditions for a [[Multiplicative arithmetic function|multiplicative arithmetic function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110050/e1100501.png" />, of modulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110050/e1100502.png" />, to possess a non-zero mean value. The unpleasant condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110050/e1100503.png" /> was replaced by P.D.T.A. Elliott, in 1975–1980, by boundedness of a [[Semi-norm|semi-norm]]

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More precisely, Elliott showed (see [[#References|[a4]]], [[#References|[a6]]]) the following result. Assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110050/e1100505.png" /> and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110050/e1100506.png" /> is a multiplicative arithmetic function with bounded semi-norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110050/e1100507.png" />. Then the mean value

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of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110050/e1100509.png" /> exists and is non-zero if and only if

i) the four series

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are convergent; and

ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110050/e11005014.png" /> for every prime <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110050/e11005015.png" />.

H. Daboussi [[#References|[a3]]] gave another proof for this result and extended it [[#References|[a2]]] to multiplicative functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110050/e11005016.png" /> having at least one non-zero Fourier coefficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110050/e11005017.png" />; the necessary and sufficient conditions for this to happen are the convergence of the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110050/e11005018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110050/e11005019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110050/e11005020.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110050/e11005021.png" /> for some [[Dirichlet character|Dirichlet character]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110050/e11005022.png" />.

See also [[#References|[a5]]], [[#References|[a7]]], [[#References|[a8]]], [[#References|[a9]]], [[#References|[a1]]]. In fact, the conditions of the Elliott–Daboussi theorem ensure that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110050/e11005023.png" /> belongs to the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110050/e11005024.png" />, which is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110050/e11005025.png" />-closure of the vector space of linear combinations of the Ramanujan sums <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110050/e11005026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110050/e11005027.png" />. For details see [[#References|[a10]]], Chapts. VI, VII.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P. Codecà,   M. Nair,   "On Elliott's theorem on multiplicative functions" , ''Proc. Amalfi Conf. Analytic Number Theory'' , '''1989'''  (1992)  pp. 17–34</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Daboussi,   "Caractérisation des fonctions multiplicatives p.p. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110050/e11005028.png" /> à spectre non vide"  ''Ann. Inst. Fourier Grenoble'' , '''30'''  (1980)  pp. 141–166</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H. Daboussi,   "Sur les fonctions multiplicatives ayant une valeur moyenne non nulle"  ''Bull. Soc. Math. France'' , '''109'''  (1981)  pp. 183–205</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  P.D.T.A. Elliott,   "A mean-value theorem for multiplicative functions"  ''Proc. London Math. Soc. (3)'' , '''31'''  (1975)  pp. 418–438</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  P.D.T.A. Elliott,   "Probabilistic number theory" , '''I–II''' , Springer  (1979–1980)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  P.D.T.A. Elliott,   "Mean value theorems for functions bounded in mean <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110050/e11005029.png" />-power, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110050/e11005030.png" />"  ''J. Austral. Math. Soc. Ser. A'' , '''29'''  (1980)  pp. 177–205</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  K.-H. Indlekofer,   "A mean-value theorem for multiplicative functions"  ''Math. Z.'' , '''172'''  (1980)  pp. 255–271</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  W. Schwarz,   J. Spilker,   "Eine Bemerkung zur Charakterisierung der fastperiodischen multiplikativen zahlentheoretischen Funktionen mit von Null verschiedenem Mittelwert"  ''Analysis'' , '''3'''  (1983)  pp. 205–216</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  W. Schwarz,   J. Spilker,   "A variant of proof of Daboussi's theorem on the characterization of multiplicative functions with non-void Fourier–Bohr spectrum"  ''Analysis'' , '''6'''  (1986)  pp. 237–249</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  W. Schwarz,   J. Spilker,   "Arithmetical functions" , Cambridge Univ. Press  (1994)</TD></TR></table>