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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.

An initial question is to ask for a function with the same rate of growth as the given function. Given an arithmetic function $f(x)$ we say that $f$ has rate of growth, or is asymptotic to, a given function $\phi$ if $$ \lim_{x\to\infty}\frac{f(x)}{\phi(x)}=1 \ . $$ A short notation is: $f(x)\sim\phi(x)$ (cf. Asymptotic formula).

Examples:

• The Prime number theorem: $\pi(x)$, the number of primes less than $x$ is asymptotic to $\frac{x}{\log x}$.

More precisely, for an arithmetic function $f(x)$ there exists an asymptotic if one has an asymptotic identity $$ f(x)=\phi(x)+R(x)\,, $$ where $\phi(x)$ is the approximating function or main term, 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 $f(x) = (1+o(1))\phi(x)$.

Examples:

• The Prime number theorem in a more precise form: $\pi(x) = (1+o(1)) \mathop{Li}(x)$ where $\mathop{Li}$ is the logarithmic integral function.

Finding asymptotics of arithmetic functions is one of the most important problems in analytic number theory. This is explained by the fact that most arithmetic functions with interesting arithmetical properties display 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 $\tau(n)$, giving the number of divisors of $n$, is equal to $$ \frac1n\sum_{m\leq n}\tau(m)\sim\ln n. $$ (cf. Divisor_problems#Dirichlet's_divisor_problem). 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 $n$.

The average order and normal order of arithemetic functions refer to some simpler or better-understood function which have comparable asymptotic behaviour. It is conventional to assume that the approximating function $g$ is continuous and monotone.

The average order of an arithmetic function is a function which has the same average.

Let $f$, $g$ be functions on the natural numbers. We say that $f$ has average order $g$ if the asymptotic equality $$ \sum_{n \le x} f(n) \sim \sum_{n \le x} g(n) $$ holds as $x$ tends to infinity.

Examples:

• The average order of $d(n)$, the number of divisors of $n$, is $\log n$;
• The average order of $\sigma(n)$, the sum of divisors of $n$, is $ \frac{\pi^2}{6} n$;
• The average order of $\phi(n)$, the Euler totient function of $n$, is $ \frac{6}{\pi^2} n$;
• The average order of $r(n)$, the number of ways of expressing $n$ as a sum of two squares, is $\pi$;
• The Prime Number Theorem is equivalent to the statement that the von Mangoldt function $\Lambda(n)$ has average order 1.

The normal order of an arithmetic function is a function which "usually" takes the same or closely approximate values.

Let $f$ be a function on the natural numbers. We say that the normal order of $f$ is $g$ if for every $\epsilon > 0$, the inequalities $$ (1-\epsilon) g(n) \le f(n) \le (1+\epsilon) g(n) $$ hold for almost all $n$: that is, the proportion of $n < x$ for which this does not hold tends to 0 as $x$ tends to infinity.

Examples:

• The Hardy–Ramanujan theorem: the normal order of $\omega(n)$, the number of distinct prime factors of $n$, is $\log\log n$;
• The normal order of $\log d(n))$, where $d(n)$ is the number of divisors function of $n$, is $\log 2 \log\log n$.

References

• G.H. Hardy; S. Ramanujan; The normal number of prime factors of a number, Quart. J. Math., 48 (1917), pp. 76–92
• G.H. Hardy; E.M. Wright (2008). An Introduction to the Theory of Numbers (6th ed.). Oxford University Press. ISBN 0-19-921986-5
• Gérald Tenenbaum (1995). Introduction to Analytic and Probabilistic Number Theory. Cambridge studies in advanced mathematics 46. Cambridge University Press. ISBN 0-521-41261-7