Wirsing theorems

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