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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.

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|Tauberian theorems]]).

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  $  \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  $  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

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

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, [[#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  $  \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 [[#References|[a3]]], it is possible to deduce stronger estimates by analytical methods.

====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>