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''asymptotics of number-theoretical 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013840/a0138401.png" /> there exists an asymptotic if one has an asymptotic indentity
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013840/a0138402.png" /></td> </tr></table>
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$$f(x)=\phi(x)+R(x),$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013840/a0138403.png" /> is the approximating function, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013840/a0138404.png" /> is the error term, about which one knows in general only that
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where $\phi(x)$ is the approximating function, and $R(x)$ is the error term, about which one knows in general only that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013840/a0138405.png" /></td> </tr></table>
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$$\lim_{x\to\infty}\frac{R(x)}{\phi(x)}=0.$$
  
A short notation is: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013840/a0138406.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013840/a0138407.png" /> (cf. [[Asymptotic formula|Asymptotic formula]]).
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A short notation is: $f(x)=\phi(x)+o(\phi(x))$ or $f(x)\sim\phi(x)$ (cf. [[Asymptotic formula|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013840/a0138408.png" /> its average value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013840/a0138409.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013840/a01384010.png" /> a natural number), then the  "irregularity"  of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013840/a01384011.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013840/a01384012.png" />, whose value at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013840/a01384013.png" /> is the number of divisors of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013840/a01384014.png" />, is equal to
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013840/a01384015.png" /></td> </tr></table>
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$$\frac1n\sum_{m\leq n}\tau(n)\sim\ln n.$$
  
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013840/a01384016.png" /> (cf. [[Analytic number theory|Analytic number theory]]).
+
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|Analytic number theory]]).
  
Asymptotics of arithmetic functions play an important a role in additive problems (cf. [[Additive number theory|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013840/a01384017.png" />.
+
Asymptotics of arithmetic functions play an important a role in additive problems (cf. [[Additive number theory|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 $n$.

Revision as of 10:17, 27 September 2014

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 $n$.

How to Cite This Entry:
Asymptotics of arithmetic functions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Asymptotics_of_arithmetic_functions&oldid=33413
This article was adapted from an original article by B.M. Bredikhin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article