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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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<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/e/e110/e110050/e1100504.png" /></td> </tr></table>
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{{TEX|auto}}
 +
{{TEX|done}}
  
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
+
The [[Delange theorem|Delange theorem]], proved in 1961, gives necessary and sufficient conditions for a [[Multiplicative arithmetic function|multiplicative arithmetic function]] $  f : \mathbf N \rightarrow \mathbf C $,
 +
of modulus  $  | f | \leq  1 $,
 +
to possess a non-zero mean value. The unpleasant condition  $  | f | \leq  1 $
 +
was replaced by P.D.T.A. Elliott, in 1975–1980, by boundedness of a [[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/e/e110/e110050/e1100508.png" /></td> </tr></table>
+
$$
 +
\left \| f \right \| _ {q} = \left ( \limsup _ {x \rightarrow \infty }{
 +
\frac{1}{x}
 +
} \cdot \sum _ {n \leq  x } \left | {f ( n ) } \right |  ^ {q} \right ) ^ { {1 / q } } .
 +
$$
  
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
+
More precisely, Elliott showed (see [[#References|[a4]]], [[#References|[a6]]]) the following result. Assume that  $  q > 1 $
 +
and that  $  f $
 +
is a multiplicative arithmetic function with bounded semi-norm  $  \| f \| _ {q} $.  
 +
Then the mean value
 +
 
 +
$$
 +
M ( f ) = {\lim\limits } _ {x \rightarrow \infty } {
 +
\frac{1}{x}
 +
} \cdot \sum _ {n \leq  x } f ( n )
 +
$$
 +
 
 +
of  $  f $
 +
exists and is non-zero if and only if
  
 
i) the four series
 
i) the four 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/e/e110/e110050/e11005010.png" /></td> </tr></table>
+
$$
 +
S _ {1} ( f ) = \sum _ { p } {
 +
\frac{1}{p}
 +
} \cdot ( f ( p ) - 1 ) ,
 +
$$
  
<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/e/e110/e110050/e11005011.png" /></td> </tr></table>
+
$$
 +
S _ {2}  ^  \prime  ( f ) = \sum _ {\left \{ p : {\left | {f ( p ) } \right | \leq  {5 / 4 } } \right \} } {
 +
\frac{1}{p}
 +
} \cdot \left | {f ( p ) - 1 } \right |  ^ {2} ,
 +
$$
  
<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/e/e110/e110050/e11005012.png" /></td> </tr></table>
+
$$
 +
S _ {2,q }  ^ {\prime \prime } ( f ) = \sum _ {\left \{ p : {\left | {f ( p ) } \right | > {5 / 4 } } \right \} } {
 +
\frac{1}{p}
 +
} \cdot \left | {f ( p ) } \right |  ^ {q} ,
 +
$$
  
<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/e/e110/e110050/e11005013.png" /></td> </tr></table>
+
$$
 +
S _ {3,q }  ( f ) = \sum _ { p } \sum _ {k \geq  2 } {
 +
\frac{1}{p  ^ {k} }
 +
} \cdot \left | {f ( p  ^ {k} ) } \right |  ^ {q}
 +
$$
  
 
are convergent; and
 
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" />.
+
ii) $  \sum _ {k = 0 }  ^  \infty  p ^ {- k } \cdot f ( p  ^ {k} ) \neq 0 $
 +
for every prime $  p $.
  
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" />.
+
H. Daboussi [[#References|[a3]]] gave another proof for this result and extended it [[#References|[a2]]] to multiplicative functions $  f $
 +
having at least one non-zero Fourier coefficient $  {\widehat{f}  } ( \alpha ) = M ( n \mapsto f ( n ) \cdot { \mathop{\rm exp} } \{ 2 \pi i \cdot \alpha n \} ) $;  
 +
the necessary and sufficient conditions for this to happen are the convergence of the series $  S _ {1} ( \chi f ) $,
 +
$  S _ {2}  ^  \prime  ( \chi f ) $,  
 +
$  S _ {2,q }  ^ {\prime \prime } ( f ) $,  
 +
and $  S _ {3,q }  ( f ) $
 +
for some [[Dirichlet character|Dirichlet character]] $  \chi $.
  
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.
+
See also [[#References|[a5]]], [[#References|[a7]]], [[#References|[a8]]], [[#References|[a9]]], [[#References|[a1]]]. In fact, the conditions of the Elliott–Daboussi theorem ensure that $  f $
 +
belongs to the space $  {\mathcal B}  ^ {q} $,  
 +
which is the $  \| \cdot \| _ {q} $-
 +
closure of the vector space of linear combinations of the [[Ramanujan sums]]  $  c _ {r} $,
 +
$  r = 1,2, \dots $.  
 +
For details see [[#References|[a10]]], Chapts. VI, VII.
  
 
====References====
 
====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>
+
<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. $B^\lambda$ à spectre non vide"  ''Ann. Inst. Fourier Grenoble'' , '''30'''  (1980)  pp. 141–166. {{ZBL|0446.10040}}</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 $\alpha$-power, $\alpha>1$"  ''J. Austral. Math. Soc. Ser. A'' , '''29'''  (1980)  pp. 177–205. {{ZBL|0426.10050}}</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>

Latest revision as of 10:00, 9 January 2021


The Delange theorem, proved in 1961, gives necessary and sufficient conditions for a multiplicative arithmetic function $ f : \mathbf N \rightarrow \mathbf C $, of modulus $ | f | \leq 1 $, to possess a non-zero mean value. The unpleasant condition $ | f | \leq 1 $ was replaced by P.D.T.A. Elliott, in 1975–1980, by boundedness of a semi-norm

$$ \left \| f \right \| _ {q} = \left ( \limsup _ {x \rightarrow \infty }{ \frac{1}{x} } \cdot \sum _ {n \leq x } \left | {f ( n ) } \right | ^ {q} \right ) ^ { {1 / q } } . $$

More precisely, Elliott showed (see [a4], [a6]) the following result. Assume that $ q > 1 $ and that $ f $ is a multiplicative arithmetic function with bounded semi-norm $ \| f \| _ {q} $. Then the mean value

$$ M ( f ) = {\lim\limits } _ {x \rightarrow \infty } { \frac{1}{x} } \cdot \sum _ {n \leq x } f ( n ) $$

of $ f $ exists and is non-zero if and only if

i) the four series

$$ S _ {1} ( f ) = \sum _ { p } { \frac{1}{p} } \cdot ( f ( p ) - 1 ) , $$

$$ S _ {2} ^ \prime ( f ) = \sum _ {\left \{ p : {\left | {f ( p ) } \right | \leq {5 / 4 } } \right \} } { \frac{1}{p} } \cdot \left | {f ( p ) - 1 } \right | ^ {2} , $$

$$ S _ {2,q } ^ {\prime \prime } ( f ) = \sum _ {\left \{ p : {\left | {f ( p ) } \right | > {5 / 4 } } \right \} } { \frac{1}{p} } \cdot \left | {f ( p ) } \right | ^ {q} , $$

$$ S _ {3,q } ( f ) = \sum _ { p } \sum _ {k \geq 2 } { \frac{1}{p ^ {k} } } \cdot \left | {f ( p ^ {k} ) } \right | ^ {q} $$

are convergent; and

ii) $ \sum _ {k = 0 } ^ \infty p ^ {- k } \cdot f ( p ^ {k} ) \neq 0 $ for every prime $ p $.

H. Daboussi [a3] gave another proof for this result and extended it [a2] to multiplicative functions $ f $ having at least one non-zero Fourier coefficient $ {\widehat{f} } ( \alpha ) = M ( n \mapsto f ( n ) \cdot { \mathop{\rm exp} } \{ 2 \pi i \cdot \alpha n \} ) $; the necessary and sufficient conditions for this to happen are the convergence of the series $ S _ {1} ( \chi f ) $, $ S _ {2} ^ \prime ( \chi f ) $, $ S _ {2,q } ^ {\prime \prime } ( f ) $, and $ S _ {3,q } ( f ) $ for some Dirichlet character $ \chi $.

See also [a5], [a7], [a8], [a9], [a1]. In fact, the conditions of the Elliott–Daboussi theorem ensure that $ f $ belongs to the space $ {\mathcal B} ^ {q} $, which is the $ \| \cdot \| _ {q} $- closure of the vector space of linear combinations of the Ramanujan sums $ c _ {r} $, $ r = 1,2, \dots $. For details see [a10], Chapts. VI, VII.

References

[a1] P. Codecà, M. Nair, "On Elliott's theorem on multiplicative functions" , Proc. Amalfi Conf. Analytic Number Theory , 1989 (1992) pp. 17–34
[a2] H. Daboussi, "Caractérisation des fonctions multiplicatives p.p. $B^\lambda$ à spectre non vide" Ann. Inst. Fourier Grenoble , 30 (1980) pp. 141–166. Zbl 0446.10040
[a3] H. Daboussi, "Sur les fonctions multiplicatives ayant une valeur moyenne non nulle" Bull. Soc. Math. France , 109 (1981) pp. 183–205
[a4] P.D.T.A. Elliott, "A mean-value theorem for multiplicative functions" Proc. London Math. Soc. (3) , 31 (1975) pp. 418–438
[a5] P.D.T.A. Elliott, "Probabilistic number theory" , I–II , Springer (1979–1980)
[a6] P.D.T.A. Elliott, "Mean value theorems for functions bounded in mean $\alpha$-power, $\alpha>1$" J. Austral. Math. Soc. Ser. A , 29 (1980) pp. 177–205. Zbl 0426.10050
[a7] K.-H. Indlekofer, "A mean-value theorem for multiplicative functions" Math. Z. , 172 (1980) pp. 255–271
[a8] W. Schwarz, J. Spilker, "Eine Bemerkung zur Charakterisierung der fastperiodischen multiplikativen zahlentheoretischen Funktionen mit von Null verschiedenem Mittelwert" Analysis , 3 (1983) pp. 205–216
[a9] 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
[a10] W. Schwarz, J. Spilker, "Arithmetical functions" , Cambridge Univ. Press (1994)
How to Cite This Entry:
Elliott-Daboussi theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Elliott-Daboussi_theorem&oldid=22379
This article was adapted from an original article by W. Schwarz (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article