Asymptotics of arithmetic functions

asymptotics of number-theoretical functions

Approximate representations of arithmetic functions (functions defined for all natural number values of the argument) by means of comparatively simple expressions with arbitrary small error terms. More precisely, for an arithmetic function $f(x)$ there exists an asymptotic if one has an asymptotic indentity

$$f(x)=\phi(x)+R(x),$$

where $\phi(x)$ is the approximating function, and $R(x)$ is the error term, about which one knows in general only that

$$\lim_{x\to\infty}\frac{R(x)}{\phi(x)}=0.$$

A short notation is: $f(x)=\phi(x)+o(\phi(x))$ or $f(x)\sim\phi(x)$ (cf. Asymptotic formula).

Finding asymptotics of arithmetic functions is one of the most important problems in analytic number theory. This is explained by the fact that almost-all arithmetic functions with interesting arithmetical properties are characterized by extreme irregularity in their changes as the argument increases. If one considers instead of an arithmetic function $f(x)$ its average value $(\sum_{n\leq x}f(n))/x$ ($n$ a natural number), then the "irregularity" of $f(x)$ is smoothed out. Hence, a typical problem for an arithmetic function is that of obtaining an asymptotic for its average value function. For example, the average value of the function $\tau(n)$, whose value at $n$ is the number of divisors of $n$, is equal to

$$\frac1n\sum_{m\leq n}\tau(n)\sim\ln n.$$

The problem that arises here, of the best possible bound for the error term in the asymptotic identity, is still unsolved (1984) for many functions, in particular for the function $\tau(x)$ (cf. Analytic number theory).

Asymptotics of arithmetic functions play an important a role in additive problems (cf. Additive number theory). For many of them there is no known direct proof of the existence of decompositions of a number into terms of a given form. However, as soon as one has an asymptotic for the number of decompositions of the type one is looking for, one can deduce that the required decomposition exists for all sufficiently large numbers $n$.