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
How to Cite This Entry:
Wirsing theorems. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wirsing_theorems&oldid=49229
This article was adapted from an original article by W. Schwarz (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article