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Multiplicative arithmetic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110110/w1101101.png" /> are determined by their values at the prime powers (cf. [[Multiplicative arithmetic function|Multiplicative arithmetic function]]). Higher prime powers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110110/w1101102.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110110/w1101103.png" />, are rare, and so the behaviour of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110110/w1101104.png" /> at the primes should strongly influence the behaviour of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110110/w1101105.png" /> in general. This vague idea, which also lies behind the theorems of Delange and Elliott (see [[Delange theorem|Delange theorem]]; [[Elliott–Daboussi theorem|Elliott–Daboussi theorem]]), led E. Wirsing in 1961 [[#References|[a6]]] to the following result, which provides asymptotic formulas for a large class of non-negative multiplicative functions.
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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110110/w1101106.png" /> be a non-negative multiplicative function. Assume that the values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110110/w1101107.png" /> at the primes satisfy, with some positive constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110110/w1101108.png" />,
+
{{TEX|auto}}
 +
{{TEX|done}}
  
<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/w/w110/w110110/w1101109.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
+
Multiplicative arithmetic functions  $  f : \mathbf N \rightarrow \mathbf C $
 +
are determined by their values at the prime powers (cf. [[Multiplicative arithmetic function|Multiplicative arithmetic function]]). Higher prime powers  $  p  ^ {k} $,
 +
$  k \geq  2 $,
 +
are rare, and so the behaviour of  $  f $
 +
at the primes should strongly influence the behaviour of  $  f $
 +
in general. This vague idea, which also lies behind the theorems of Delange and Elliott (see [[Delange theorem|Delange theorem]]; [[Elliott–Daboussi theorem|Elliott–Daboussi theorem]]), led E. Wirsing in 1961 [[#References|[a6]]] to the following result, which provides asymptotic formulas for a large class of non-negative multiplicative functions.
  
and that the values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110110/w11011010.png" /> at higher prime powers are not "too large" :
+
Let  $  f $
 +
be a non-negative multiplicative function. Assume that the values of $  f $
 +
at the primes satisfy, with some positive constant  $ \tau $,
  
<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/w/w110/w110110/w11011011.png" /></td> </tr></table>
+
$$ \tag{a1 }
 +
\sum _ {p \leq  x } f ( p ) \cdot { \mathop{\rm log} } p = ( \tau + o ( 1 ) ) \cdot x  \textrm{ as  }  x \rightarrow \infty,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110110/w11011012.png" />. Then
+
and that the values of  $  f $
 +
at higher prime powers are not  "too large" :
  
<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/w/w110/w110110/w11011013.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
+
$$
 +
f ( p  ^ {k} ) \leq  \gamma _ {1} \cdot \gamma _ {2}  ^ {k}  \textrm{ for  }  k = 2,3 \dots
 +
$$
  
<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/w/w110/w110110/w11011014.png" /></td> </tr></table>
+
where  $  0 \leq  \gamma _ {2} < 2 $.  
 +
Then
  
The proof uses an inversion of the order of summation in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110110/w11011015.png" />, to show that
+
$$ \tag{a2 }
 +
\sum _ {n \leq  x } f ( n ) = ( 1 + o ( 1 ) ) \cdot {
 +
\frac{x}{ { \mathop{\rm log} } x }
 +
} \cdot
 +
$$
  
<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/w/w110/w110110/w11011016.png" /></td> </tr></table>
+
$$
 +
\cdot
 +
{
 +
\frac{e ^ {- {\mathcal C} \tau } }{\Gamma ( \tau ) }
 +
} \cdot \prod _ {p \leq  x } \left ( 1 + {
 +
\frac{f ( p ) }{p}
 +
} + {
 +
\frac{f ( p  ^ {2} ) }{p  ^ {2} }
 +
} + \dots \right ) .
 +
$$
 +
 
 +
The proof uses an inversion of the order of summation in  $  \sum _ {n \leq  x }  f ( n ) \cdot { \mathop{\rm log} } n = \sum _ {n \leq  x }  f ( n ) \cdot \sum _ { {{p  ^ {k}  } \mid  n } } { \mathop{\rm log} } p  ^ {k} $,
 +
to show that
 +
 
 +
$$
 +
\sum _ {n \leq  x } f ( n ) \sim \tau \cdot {
 +
\frac{x}{ { \mathop{\rm log} } x }
 +
} \cdot \sum _ {n \leq  x } {
 +
\frac{1}{n}
 +
} \cdot f ( n ) .
 +
$$
  
 
The last sum may be dealt with by elementary arguments or by a skilful application of the Hardy–Littlewood–Karamata Tauberian theorem (cf. [[Tauberian theorems|Tauberian theorems]]).
 
The last sum may be dealt with by elementary arguments or by a skilful application of the Hardy–Littlewood–Karamata Tauberian theorem (cf. [[Tauberian theorems|Tauberian theorems]]).
  
In 1967, B.V. Levin and A.S. Fainleib [[#References|[a5]]] also gave asymptotic evaluations of sums <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110110/w11011017.png" /> for multiplicative functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110110/w11011018.png" />, by reducing the problem to the study of the asymptotic behaviour of solutions of integral equations of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110110/w11011019.png" />.
+
In 1967, B.V. Levin and A.S. Fainleib [[#References|[a5]]] also gave asymptotic evaluations of sums $  \sum _ {n \leq  x }  f ( n ) $
 +
for multiplicative functions $  f $,  
 +
by reducing the problem to the study of the asymptotic behaviour of solutions of integral equations of the form $  t \cdot z ( t ) = \int _ {0}  ^ {t} {K ( t - u ) z ( u ) }  {du } $.
  
In [[#References|[a6]]], Wirsing also deduced results for complex-valued multiplicative functions. Unfortunately, these results did not contain the prime number theorem (in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110110/w11011020.png" />; cf. also [[De la Vallée-Poussin theorem|de la Vallée-Poussin theorem]]), and they did not settle the Erdös–Wintner conjecture: Any multiplicative function assuming only the values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110110/w11011021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110110/w11011022.png" /> has a mean value (see [[#References|[a1]]]).
+
In [[#References|[a6]]], Wirsing also deduced results for complex-valued multiplicative functions. Unfortunately, these results did not contain the prime number theorem (in the form $  \sum _ {n \leq  x }  \mu ( n ) = o ( x ) $;  
 +
cf. also [[De la Vallée-Poussin theorem|de la Vallée-Poussin theorem]]), and they did not settle the Erdös–Wintner conjecture: Any multiplicative function assuming only the values $  + 1 $
 +
and $  - 1 $
 +
has a mean value (see [[#References|[a1]]]).
  
But six years later, in 1967 [[#References|[a7]]], Wirsing was able to settle this conjecture. He proved in an elementary, but complicated, way several results on multiplicative functions. For example: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110110/w11011023.png" /> is a real-valued multiplicative function and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110110/w11011024.png" />, then the mean value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110110/w11011025.png" /> exists [[#References|[a7]]], Satz 1.2.2.
+
But six years later, in 1967 [[#References|[a7]]], Wirsing was able to settle this conjecture. He proved in an elementary, but complicated, way several results on multiplicative functions. For example: If $  f $
 +
is a real-valued multiplicative function and if $  | f | \leq  1 $,  
 +
then the mean value $  M ( f ) $
 +
exists [[#References|[a7]]], Satz 1.2.2.
  
 
The asymptotic formula (a2) can now be proved under the condition
 
The asymptotic formula (a2) can now be proved under the condition
  
<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/w/w110/w110110/w11011026.png" /></td> </tr></table>
+
$$
 +
\sum _ { p } {
 +
\frac{f ( p ) }{p}
 +
} { \mathop{\rm log} } p \sim \tau \cdot { \mathop{\rm log} } x,
 +
$$
  
which is much weaker than (a1). However, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110110/w11011027.png" /> and some other restrictions must be assumed. There are also corresponding, complicated, results on complex-valued multiplicative functions, [[#References|[a7]]], Satz 1.2.
+
which is much weaker than (a1). However, 0 \leq  f ( p ) \leq  G $
 +
and some other restrictions must be assumed. There are also corresponding, complicated, results on complex-valued multiplicative functions, [[#References|[a7]]], Satz 1.2.
  
In 1968, G. Halász [[#References|[a2]]] gave a more satisfactory result (see [[Halász mean value theorem|Halász mean value theorem]]). In 1986, A. Hildebrand [[#References|[a4]]] proved a Wirsing-type theorem by elementary means (his result also contains a proof of the Erdös–Wintner conjecture): There exists a universal constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110110/w11011028.png" /> with the property that for any multiplicative function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110110/w11011029.png" /> with values in the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110110/w11011030.png" /> and with divergent series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110110/w11011031.png" />, the estimate
+
In 1968, G. Halász [[#References|[a2]]] gave a more satisfactory result (see [[Halász mean value theorem|Halász mean value theorem]]). In 1986, A. Hildebrand [[#References|[a4]]] proved a Wirsing-type theorem by elementary means (his result also contains a proof of the Erdös–Wintner conjecture): There exists a universal constant $  \gamma > 0 $
 +
with the property that for any multiplicative function $  f $
 +
with values in the interval $  [ - 1, + 1 ] \subset  \mathbf R $
 +
and with divergent series $  \sum _ {p} {1 / p } \cdot ( 1 - f ( p ) ) $,  
 +
the estimate
  
<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/w/w110/w110110/w11011032.png" /></td> </tr></table>
+
$$
 +
\left | { {
 +
\frac{1}{x}
 +
} \cdot \sum _ {n \leq  x } f ( n ) } \right | \leq  \gamma \cdot \left ( 1 + \sum _ {p \leq  x } {
 +
\frac{1}{p}
 +
} \cdot ( 1 - f ( p ) ) \right ) ^ {- {1 / 2 } }
 +
$$
  
holds for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110110/w11011033.png" />. As shown in [[#References|[a3]]], it is possible to deduce stronger estimates by analytical methods.
+
holds for any $  x \geq  2 $.  
 +
As shown in [[#References|[a3]]], it is possible to deduce stronger estimates by analytical methods.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P. Erdös,  "Some unsolved problems"  ''Michigan Math. J.'' , '''4'''  (1957)  pp. 291–300</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G. Halász,  "Über die Mittelwerte multiplikativer zahlentheoretischer Funktionen"  ''Acta Math. Acad. Sci. Hung.'' , '''19'''  (1968)  pp. 365–403</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G. Halász,  "On the distribution of additive and the mean values of multiplicative arithmetic functions"  ''Studia Sci. Math. Hung.'' , '''6'''  (1971)  pp. 211–233</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A. Hildebrand,  "On Wirsing's mean value theorem for multiplicative functions"  ''Bull. London Math. Soc.'' , '''18'''  (1986)  pp. 147–152</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  B.V. Levin,  A.S. Fainleib,  "Application of certain integral equations to questions of the theory of numbers"  ''Uspekhi Mat. Nauk'' , '''22''' :  3 (135)  (1967)  pp. 119–197  (In Russian)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  E. Wirsing,  "Das asymptotische Verhalten von Summen über multiplikative Funktionen"  ''Math. Ann.'' , '''143'''  (1961)  pp. 75–102</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  E. Wirsing,  "Das asymptotische Verhalten von Summen über multiplikative Funktionen, II"  ''Acta Math. Acad. Sci. Hung.'' , '''18'''  (1967)  pp. 411–467</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P. Erdös,  "Some unsolved problems"  ''Michigan Math. J.'' , '''4'''  (1957)  pp. 291–300</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G. Halász,  "Über die Mittelwerte multiplikativer zahlentheoretischer Funktionen"  ''Acta Math. Acad. Sci. Hung.'' , '''19'''  (1968)  pp. 365–403</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G. Halász,  "On the distribution of additive and the mean values of multiplicative arithmetic functions"  ''Studia Sci. Math. Hung.'' , '''6'''  (1971)  pp. 211–233</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A. Hildebrand,  "On Wirsing's mean value theorem for multiplicative functions"  ''Bull. London Math. Soc.'' , '''18'''  (1986)  pp. 147–152</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  B.V. Levin,  A.S. Fainleib,  "Application of certain integral equations to questions of the theory of numbers"  ''Uspekhi Mat. Nauk'' , '''22''' :  3 (135)  (1967)  pp. 119–197  (In Russian)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  E. Wirsing,  "Das asymptotische Verhalten von Summen über multiplikative Funktionen"  ''Math. Ann.'' , '''143'''  (1961)  pp. 75–102</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  E. Wirsing,  "Das asymptotische Verhalten von Summen über multiplikative Funktionen, II"  ''Acta Math. Acad. Sci. Hung.'' , '''18'''  (1967)  pp. 411–467</TD></TR></table>

Latest revision as of 08:29, 6 June 2020


Multiplicative arithmetic functions $ f : \mathbf N \rightarrow \mathbf C $ are determined by their values at the prime powers (cf. Multiplicative arithmetic function). Higher prime powers $ p ^ {k} $, $ k \geq 2 $, are rare, and so the behaviour of $ f $ at the primes should strongly influence the behaviour of $ f $ in general. This vague idea, which also lies behind the theorems of Delange and Elliott (see Delange theorem; Elliott–Daboussi theorem), led E. Wirsing in 1961 [a6] to the following result, which provides asymptotic formulas for a large class of non-negative multiplicative functions.

Let $ f $ be a non-negative multiplicative function. Assume that the values of $ f $ at the primes satisfy, with some positive constant $ \tau $,

$$ \tag{a1 } \sum _ {p \leq x } f ( p ) \cdot { \mathop{\rm log} } p = ( \tau + o ( 1 ) ) \cdot x \textrm{ as } x \rightarrow \infty, $$

and that the values of $ f $ at higher prime powers are not "too large" :

$$ f ( p ^ {k} ) \leq \gamma _ {1} \cdot \gamma _ {2} ^ {k} \textrm{ for } k = 2,3 \dots $$

where $ 0 \leq \gamma _ {2} < 2 $. Then

$$ \tag{a2 } \sum _ {n \leq x } f ( n ) = ( 1 + o ( 1 ) ) \cdot { \frac{x}{ { \mathop{\rm log} } x } } \cdot $$

$$ \cdot { \frac{e ^ {- {\mathcal C} \tau } }{\Gamma ( \tau ) } } \cdot \prod _ {p \leq x } \left ( 1 + { \frac{f ( p ) }{p} } + { \frac{f ( p ^ {2} ) }{p ^ {2} } } + \dots \right ) . $$

The proof uses an inversion of the order of summation in $ \sum _ {n \leq x } f ( n ) \cdot { \mathop{\rm log} } n = \sum _ {n \leq x } f ( n ) \cdot \sum _ { {{p ^ {k} } \mid n } } { \mathop{\rm log} } p ^ {k} $, to show that

$$ \sum _ {n \leq x } f ( n ) \sim \tau \cdot { \frac{x}{ { \mathop{\rm log} } x } } \cdot \sum _ {n \leq x } { \frac{1}{n} } \cdot f ( n ) . $$

The last sum may be dealt with by elementary arguments or by a skilful application of the Hardy–Littlewood–Karamata Tauberian theorem (cf. Tauberian theorems).

In 1967, B.V. Levin and A.S. Fainleib [a5] also gave asymptotic evaluations of sums $ \sum _ {n \leq x } f ( n ) $ for multiplicative functions $ f $, by reducing the problem to the study of the asymptotic behaviour of solutions of integral equations of the form $ t \cdot z ( t ) = \int _ {0} ^ {t} {K ( t - u ) z ( u ) } {du } $.

In [a6], Wirsing also deduced results for complex-valued multiplicative functions. Unfortunately, these results did not contain the prime number theorem (in the form $ \sum _ {n \leq x } \mu ( n ) = o ( x ) $; cf. also de la Vallée-Poussin theorem), and they did not settle the Erdös–Wintner conjecture: Any multiplicative function assuming only the values $ + 1 $ and $ - 1 $ has a mean value (see [a1]).

But six years later, in 1967 [a7], Wirsing was able to settle this conjecture. He proved in an elementary, but complicated, way several results on multiplicative functions. For example: If $ f $ is a real-valued multiplicative function and if $ | f | \leq 1 $, then the mean value $ M ( f ) $ exists [a7], Satz 1.2.2.

The asymptotic formula (a2) can now be proved under the condition

$$ \sum _ { p } { \frac{f ( p ) }{p} } { \mathop{\rm log} } p \sim \tau \cdot { \mathop{\rm log} } x, $$

which is much weaker than (a1). However, $ 0 \leq f ( p ) \leq G $ and some other restrictions must be assumed. There are also corresponding, complicated, results on complex-valued multiplicative functions, [a7], Satz 1.2.

In 1968, G. Halász [a2] gave a more satisfactory result (see Halász mean value theorem). In 1986, A. Hildebrand [a4] proved a Wirsing-type theorem by elementary means (his result also contains a proof of the Erdös–Wintner conjecture): There exists a universal constant $ \gamma > 0 $ with the property that for any multiplicative function $ f $ with values in the interval $ [ - 1, + 1 ] \subset \mathbf R $ and with divergent series $ \sum _ {p} {1 / p } \cdot ( 1 - f ( p ) ) $, the estimate

$$ \left | { { \frac{1}{x} } \cdot \sum _ {n \leq x } f ( n ) } \right | \leq \gamma \cdot \left ( 1 + \sum _ {p \leq x } { \frac{1}{p} } \cdot ( 1 - f ( p ) ) \right ) ^ {- {1 / 2 } } $$

holds for any $ x \geq 2 $. As shown in [a3], it is possible to deduce stronger estimates by analytical methods.

References

[a1] P. Erdös, "Some unsolved problems" Michigan Math. J. , 4 (1957) pp. 291–300
[a2] G. Halász, "Über die Mittelwerte multiplikativer zahlentheoretischer Funktionen" Acta Math. Acad. Sci. Hung. , 19 (1968) pp. 365–403
[a3] G. Halász, "On the distribution of additive and the mean values of multiplicative arithmetic functions" Studia Sci. Math. Hung. , 6 (1971) pp. 211–233
[a4] A. Hildebrand, "On Wirsing's mean value theorem for multiplicative functions" Bull. London Math. Soc. , 18 (1986) pp. 147–152
[a5] B.V. Levin, A.S. Fainleib, "Application of certain integral equations to questions of the theory of numbers" Uspekhi Mat. Nauk , 22 : 3 (135) (1967) pp. 119–197 (In Russian)
[a6] E. Wirsing, "Das asymptotische Verhalten von Summen über multiplikative Funktionen" Math. Ann. , 143 (1961) pp. 75–102
[a7] E. Wirsing, "Das asymptotische Verhalten von Summen über multiplikative Funktionen, II" Acta Math. Acad. Sci. Hung. , 18 (1967) pp. 411–467
How to Cite This Entry:
Wirsing theorems. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wirsing_theorems&oldid=19040
This article was adapted from an original article by W. Schwarz (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article