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Considerable interest has been devoted to the problem of obtaining conditions for arithmetic functions (cf. [[Arithmetic function|Arithmetic function]]), in particular for multiplicative functions (cf. [[Multiplicative arithmetic function|Multiplicative arithmetic function]]), that guarantee the existence of a mean value
 
Considerable interest has been devoted to the problem of obtaining conditions for arithmetic functions (cf. [[Arithmetic function|Arithmetic function]]), in particular for multiplicative functions (cf. [[Multiplicative arithmetic function|Multiplicative arithmetic function]]), that guarantee the existence of a 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/h/h110/h110010/h1100101.png" /></td> </tr></table>
+
$$
 +
M ( f ) = {\lim\limits } _ {x \rightarrow \infty } {
 +
\frac{1}{x}
 +
} \cdot \sum _ {n \leq  x } f ( n ) .
 +
$$
  
A strong motivation for this interest was the famous Erdös–Wintner conjecture (see [[#References|[a2]]]): Any multiplicative function assuming only the values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110010/h1100102.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110010/h1100103.png" /> possesses a mean value.
+
A strong motivation for this interest was the famous Erdös–Wintner conjecture (see [[#References|[a2]]]): Any multiplicative function assuming only the values $  + 1 $
 +
and $  - 1 $
 +
possesses a mean value.
  
 
Around 1961, theorems of H. Delange (see [[Delange theorem|Delange theorem]]) and E. Wirsing (see [[Wirsing theorems|Wirsing theorems]]) gave a satisfactory answer for multiplicative functions with non-zero mean value. However, a general mean value theorem, containing a proof of the prime number theorem (cf. also [[De la Vallée-Poussin theorem|de la Vallée-Poussin theorem]]) and of the Erdös–Wintner conjecture, was only given by Wirsing in 1967 (see [[#References|[a7]]]) and G. Halász in 1968 (see [[#References|[a3]]]).
 
Around 1961, theorems of H. Delange (see [[Delange theorem|Delange theorem]]) and E. Wirsing (see [[Wirsing theorems|Wirsing theorems]]) gave a satisfactory answer for multiplicative functions with non-zero mean value. However, a general mean value theorem, containing a proof of the prime number theorem (cf. also [[De la Vallée-Poussin theorem|de la Vallée-Poussin theorem]]) and of the Erdös–Wintner conjecture, was only given by Wirsing in 1967 (see [[#References|[a7]]]) and G. Halász in 1968 (see [[#References|[a3]]]).
  
A simple form of Halász' theorem reads as follows: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110010/h1100104.png" /> is a multiplicative function, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110010/h1100105.png" />, then there exist constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110010/h1100106.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110010/h1100107.png" />, and a slowly oscillating function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110010/h1100108.png" />, satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110010/h1100109.png" />, such that
+
A simple form of Halász' theorem reads as follows: If $  f : \mathbf N \rightarrow \mathbf C $
 +
is a multiplicative function, $  | f | \leq  1 $,  
 +
then there exist constants $  c \in \mathbf C $,  
 +
$  a \in \mathbf R $,  
 +
and a slowly oscillating function $  L $,  
 +
satisfying $  | L | = 1 $,  
 +
such that
  
<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/h/h110/h110010/h11001010.png" /></td> </tr></table>
+
$$
 +
\sum _ {n \leq  x } f ( n ) = c \cdot x ^ {1 + ia } \cdot L ( { \mathop{\rm log} } x ) + o ( x ) .
 +
$$
  
In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110010/h11001011.png" /> is real-valued, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110010/h11001012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110010/h11001013.png" />, and so <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110010/h11001014.png" /> exists. For the proof, Halász used the classical method of complex integration (cf. [[Contour integration, method of|Contour integration, method of]]) in a very skilful way.
+
In particular, if $  f $
 +
is real-valued, then $  a = 0 $,  
 +
$  L = 1 $,  
 +
and so $  M ( f ) $
 +
exists. For the proof, Halász used the classical method of complex integration (cf. [[Contour integration, method of|Contour integration, method of]]) in a very skilful way.
  
A more precise formulation (see [[#References|[a3]]], Satz 2) is as follows. Assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110010/h11001015.png" /> is multiplicative and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110010/h11001016.png" />.
+
A more precise formulation (see [[#References|[a3]]], Satz 2) is as follows. Assume that $  f : \mathbf N \rightarrow \mathbf C $
 +
is multiplicative and that $  | f | \leq  1 $.
  
i) If the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110010/h11001017.png" /> diverges for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110010/h11001018.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110010/h11001019.png" /> exists and is equal to zero.
+
i) If the series $  S _ {f} ( a ) = \sum _ {p} {1 / p } \cdot ( 1 - { \mathop{\rm Re} } ( f ( p ) p ^ {- ia } ) ) $
 +
diverges for all $  a \in \mathbf R $,  
 +
then $  M ( f ) $
 +
exists and is equal to zero.
  
ii) If for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110010/h11001020.png" /> the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110010/h11001021.png" /> is convergent, then, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110010/h11001022.png" />,
+
ii) If for some $  a _ {0} $
 +
the series $  S _ {f} ( a _ {0} ) $
 +
is convergent, then, as $  x \rightarrow \infty $,
  
<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/h/h110/h110010/h11001023.png" /></td> </tr></table>
+
$$
 +
{
 +
\frac{1}{x}
 +
} \cdot \sum _ {n \leq  x } f ( n ) = c x ^ {ia _ {0} } \cdot
 +
$$
  
<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/h/h110/h110010/h11001024.png" /></td> </tr></table>
+
$$
 +
\cdot
 +
{ \mathop{\rm exp} } \left ( - \sum _ {p \leq  x } {
 +
\frac{1}{p}
 +
} \cdot ( 1 - { \mathop{\rm Re} } ( f ( p ) p ^ {- ia _ {0} } ) ) \right )  \cdot
 +
$$
  
<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/h/h110/h110010/h11001025.png" /></td> </tr></table>
+
$$
 +
\cdot
 +
{ \mathop{\rm exp} } ( iA ( x ) ) + o ( 1 ) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110010/h11001026.png" />.
+
where $  A ( x ) = \sum _ {p \leq  x }  {1 / p } \cdot { \mathop{\rm Im} } ( f ( p ) p ^ {- ia _ {0} } ) $.
  
The unpleasant condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110010/h11001027.png" /> in the theorem was (partly) removed, and remainder estimates were given, in, e.g., [[#References|[a4]]], [[#References|[a8]]], [[#References|[a9]]]. K.-H. Indlekofer [[#References|[a6]]] gave a version of the theorem for the class of  "uniformly summable"  multiplicative functions. A uniformly summable multiplicative function is a multiplicative function satisfying
+
The unpleasant condition $  | f | \leq  1 $
 +
in the theorem was (partly) removed, and remainder estimates were given, in, e.g., [[#References|[a4]]], [[#References|[a8]]], [[#References|[a9]]]. K.-H. Indlekofer [[#References|[a6]]] gave a version of the theorem for the class of  "uniformly summable"  multiplicative functions. A uniformly summable multiplicative function is a multiplicative function satisfying
  
<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/h/h110/h110010/h11001028.png" /></td> </tr></table>
+
$$
 +
\left \| f \right \| _ {q} = \left \{ {\lim\limits } _ {x \rightarrow \infty } {
 +
\frac{1}{x}
 +
} \cdot \sum _ {n \leq  x } \left | {f ( n ) } \right |  ^ {q} \right \} ^ { {1 / q } } < \infty,
 +
$$
  
 
and
 
and
  
<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/h/h110/h110010/h11001029.png" /></td> </tr></table>
+
$$
 +
{\lim\limits } _ {K \rightarrow \infty }  \sup  _ {x \geq  1 } {
 +
\frac{1}{x}
 +
} \cdot \sum _ {n \leq  x,  \left | {f ( n ) } \right | \geq  K } \left | {f ( n ) } \right | = 0.
 +
$$
  
 
In 1988, A. Mačiulis gave a version of Halász' theorem with remainder term. Elementary proofs of the Halász theorem were published in [[#References|[a1]]].
 
In 1988, A. Mačiulis gave a version of Halász' theorem with remainder term. Elementary proofs of the Halász theorem were published in [[#References|[a1]]].

Latest revision as of 19:43, 5 June 2020


Considerable interest has been devoted to the problem of obtaining conditions for arithmetic functions (cf. Arithmetic function), in particular for multiplicative functions (cf. Multiplicative arithmetic function), that guarantee the existence of a mean value

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

A strong motivation for this interest was the famous Erdös–Wintner conjecture (see [a2]): Any multiplicative function assuming only the values $ + 1 $ and $ - 1 $ possesses a mean value.

Around 1961, theorems of H. Delange (see Delange theorem) and E. Wirsing (see Wirsing theorems) gave a satisfactory answer for multiplicative functions with non-zero mean value. However, a general mean value theorem, containing a proof of the prime number theorem (cf. also de la Vallée-Poussin theorem) and of the Erdös–Wintner conjecture, was only given by Wirsing in 1967 (see [a7]) and G. Halász in 1968 (see [a3]).

A simple form of Halász' theorem reads as follows: If $ f : \mathbf N \rightarrow \mathbf C $ is a multiplicative function, $ | f | \leq 1 $, then there exist constants $ c \in \mathbf C $, $ a \in \mathbf R $, and a slowly oscillating function $ L $, satisfying $ | L | = 1 $, such that

$$ \sum _ {n \leq x } f ( n ) = c \cdot x ^ {1 + ia } \cdot L ( { \mathop{\rm log} } x ) + o ( x ) . $$

In particular, if $ f $ is real-valued, then $ a = 0 $, $ L = 1 $, and so $ M ( f ) $ exists. For the proof, Halász used the classical method of complex integration (cf. Contour integration, method of) in a very skilful way.

A more precise formulation (see [a3], Satz 2) is as follows. Assume that $ f : \mathbf N \rightarrow \mathbf C $ is multiplicative and that $ | f | \leq 1 $.

i) If the series $ S _ {f} ( a ) = \sum _ {p} {1 / p } \cdot ( 1 - { \mathop{\rm Re} } ( f ( p ) p ^ {- ia } ) ) $ diverges for all $ a \in \mathbf R $, then $ M ( f ) $ exists and is equal to zero.

ii) If for some $ a _ {0} $ the series $ S _ {f} ( a _ {0} ) $ is convergent, then, as $ x \rightarrow \infty $,

$$ { \frac{1}{x} } \cdot \sum _ {n \leq x } f ( n ) = c x ^ {ia _ {0} } \cdot $$

$$ \cdot { \mathop{\rm exp} } \left ( - \sum _ {p \leq x } { \frac{1}{p} } \cdot ( 1 - { \mathop{\rm Re} } ( f ( p ) p ^ {- ia _ {0} } ) ) \right ) \cdot $$

$$ \cdot { \mathop{\rm exp} } ( iA ( x ) ) + o ( 1 ) , $$

where $ A ( x ) = \sum _ {p \leq x } {1 / p } \cdot { \mathop{\rm Im} } ( f ( p ) p ^ {- ia _ {0} } ) $.

The unpleasant condition $ | f | \leq 1 $ in the theorem was (partly) removed, and remainder estimates were given, in, e.g., [a4], [a8], [a9]. K.-H. Indlekofer [a6] gave a version of the theorem for the class of "uniformly summable" multiplicative functions. A uniformly summable multiplicative function is a multiplicative function satisfying

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

and

$$ {\lim\limits } _ {K \rightarrow \infty } \sup _ {x \geq 1 } { \frac{1}{x} } \cdot \sum _ {n \leq x, \left | {f ( n ) } \right | \geq K } \left | {f ( n ) } \right | = 0. $$

In 1988, A. Mačiulis gave a version of Halász' theorem with remainder term. Elementary proofs of the Halász theorem were published in [a1].

References

[a1] H. Daboussi, K.-H. Indlekofer, "Two elementary proofs of Halász's theorem" Math. Z. , 209 (1992) pp. 43–52
[a2] P. Erdös, "Some unsolved problems" Michigan Math. J. , 4 (1957) pp. 291–300
[a3] G. Halász, "Über die Mittelwerte multiplikativer zahlentheoretischer Funktionen" Acta Math. Acad. Sci. Hung. , 19 (1968) pp. 365–403
[a4] G. Halász, "On the distribution of additive and the mean values of multiplicative arithmetic functions" Studia Sci. Math. Hung. , 6 (1971) pp. 211–233
[a5] G. Halász, "Remarks to my paper "On the distribution of additive and the mean values of multiplicative arithmetic functions" " Acta Math. Acad. Sci. Hung. , 23 (1972) pp. 425–432
[a6] K.-H. Indlekofer, "Remark on a theorem of G. Halász" Archiv Math. , 36 (1981) pp. 145–151
[a7] E. Wirsing, "Das asymptotische Verhalten von Summen über multiplikative Funktionen, II" Acta Math. Acad. Sci. Hung. , 18 (1967) pp. 414–467
[a8] A. Parson, J. Tull, "Asymptotic behavior of multiplicative functions" J. Number Th. , 10 (1978) pp. 395–420
[a9] M.I. Tuljaganova, "A generalization of a theorem of Halász" Izv. Akad. Nauk UzSSR , 4 (1978) pp. 35–40; 95 (In Russian)
How to Cite This Entry:
Halász mean value theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hal%C3%A1sz_mean_value_theorem&oldid=13883
This article was adapted from an original article by W. Schwarz (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article