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asymptotic estimate∗
Wkbj79†
2013-03-21 20:42:30
characteristic function
An asymptotic estimate is an estimate that involves the use of O, o, or ∼.
These are all defined in the entry Landau notation. Examples of asymptotic
estimates are:
X
µ2 (n)
n≤x
π(x)
√
6
x + O( x)
2
π
x
∼
log x
=
(see convolution method for more details)
(see prime number theorem for more details)
Unless otherwise specified, asymptotic estimates are typically valid for x →
∞. An example of an asymptotic estimate that is different from those above in
this aspect is
x2
cos x = 1 −
+ O(x4 ) for |x| < 1.
2
Note that the above estimate would be undesirable for x → ∞, as the error
would be larger than the estimate. Such is not the case for |x| < 1, though.
Tools that are useful for obtaining asymptotic estimates include:
• the Euler-Maclaurin summation formula
• Abel’s lemma
• the convolution method
• the Dirichlet hyperbola method
If A ⊆ N, then an asymptotic estimate for
X
χA (x), where χA denotes the
n≤x
characteristic function of A, enables one to determine the asymptotic density of
A using the formula
1X
lim
χA (x)
x→∞ x
n≤x
∗ hAsymptoticEstimatei
created: h2013-03-21i by: hWkbj79i version: h38027i Privacy
setting: h1i hDefinitioni h11N37i
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1
provided the limit exists. The upper asymptotic density of A and the lower
asymptotic density of A can be computed in a similar manner using lim sup and
lim inf, respectively. (See asymptotic density for more details.)
For example, µ2 is the characteristic function of the squarefree natural numbers. Using the asymptotic estimate above yields the asymptotic density of the
squarefree natural numbers:
√
1 6
1X 2
x + O( x)
µ (n) = lim
lim
x→∞ x
x→∞ x
π2
n≤x
6
+O
x→∞ π 2
= lim
=
6
π2
2
√ x
x