Elliott-Daboussi theorem

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. à spectre non vide" Ann. Inst. Fourier Grenoble , 30 (1980) pp. 141–166
[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 -power, " J. Austral. Math. Soc. Ser. A , 29 (1980) pp. 177–205
[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)