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A function of a natural argument, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067920/n0679201.png" />, equal to the number of natural divisors of the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067920/n0679202.png" />. This arithmetic function is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067920/n0679203.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067920/n0679204.png" />. The following formula holds:
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{{TEX|done}}{{MSC|11A25}}
  
<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/n/n067/n067920/n0679205.png" /></td> </tr></table>
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''of a natural number $n$''
  
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The number of natural divisors of the number $n$. This arithmetic function is denoted by $\tau(n)$ or $d(n)$. The following formula holds:
 +
$$
 +
\tau(n) = (a_1+1) \cdots (a_k+1)
 +
$$
 
where
 
where
 +
$$
 +
n = p_1^{a_1} \cdots p_k^{a_k}
 +
$$
 +
is the canonical expansion of $n$ into prime power factors. For prime numbers $p$, $\tau(p) = 2$, but there exists an infinite sequence of $n$ for which
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$$
 +
\tau(n) \ge 2^{1-\epsilon} \frac{\log n}{\log\log n}\,,\ \ \epsilon > 0 \ .
 +
$$
  
<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/n/n067/n067920/n0679206.png" /></td> </tr></table>
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On the other hand, for all $\epsilon > 0$,
 +
$$
 +
\tau(n) = O(n^\epsilon)\ .
 +
$$
  
is the canonical expansion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067920/n0679207.png" /> into prime factors. For prime numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067920/n0679208.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067920/n0679209.png" />, but there exists an infinite sequence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067920/n06792010.png" /> for which
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$\tau$ is a [[multiplicative arithmetic function]] and is equal to the number of points with natural coordinates on the hyperbola $xy = n$. The average value of $\tau(n)$ is given by Dirichlet's asymptotic formula (cf. [[Divisor problems]]).  
  
<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/n/n067/n067920/n06792011.png" /></td> </tr></table>
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The function $\tau_k(n)$, which is the number of solutions of the equation $n = x_1\cdots x_k$ in natural numbers $x_1,\ldots,x_k$, is a generalization of the function $\tau$.
 
 
On the other hand, for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067920/n06792012.png" />,
 
 
 
<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/n/n067/n067920/n06792013.png" /></td> </tr></table>
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067920/n06792014.png" /> is a [[Multiplicative arithmetic function|multiplicative arithmetic function]] and is equal to the number of points with natural coordinates on the hyperbola <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067920/n06792015.png" />. The average value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067920/n06792016.png" /> is given by Dirichlet's asymptotic formula (cf. [[Divisor problems|Divisor problems]]). The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067920/n06792017.png" />, which is the number of solutions of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067920/n06792018.png" /> in natural numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067920/n06792019.png" />, is a generalization of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067920/n06792020.png" />.
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.M. Vinogradov,  "Elements of number theory" , Dover, reprint  (1954)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  K. Prachar,  "Primzahlverteilung" , Springer  (1957)</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  I.M. Vinogradov,  "Elements of number theory" , Dover, reprint  (1954)  (Translated from Russian)</TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top">  K. Prachar,  "Primzahlverteilung" , Springer  (1957)</TD></TR>
 +
</table>
  
  
  
 
====Comments====
 
====Comments====
 
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The [[Average order of an arithmetic function|average value]] of the number of divisors was obtained by P. Dirichlet in 1849, in the form
 +
$$
 +
\sum_{n \le x} \tau(n) = x \log x + (2 \gamma - 1)x + O(\sqrt x) \ .
 +
$$
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.H. Hardy,  E.M. Wright,  "An introduction to the theory of numbers" , Oxford Univ. Press  (1979)  pp. Chapt. XVI</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  G.H. Hardy,  E.M. Wright,  "An introduction to the theory of numbers" , Oxford Univ. Press  (1979)  pp. Chapt. XVI</TD></TR>
 +
</table>

Revision as of 21:54, 16 December 2020

2020 Mathematics Subject Classification: Primary: 11A25 [MSN][ZBL]

of a natural number $n$

The number of natural divisors of the number $n$. This arithmetic function is denoted by $\tau(n)$ or $d(n)$. The following formula holds: $$ \tau(n) = (a_1+1) \cdots (a_k+1) $$ where $$ n = p_1^{a_1} \cdots p_k^{a_k} $$ is the canonical expansion of $n$ into prime power factors. For prime numbers $p$, $\tau(p) = 2$, but there exists an infinite sequence of $n$ for which $$ \tau(n) \ge 2^{1-\epsilon} \frac{\log n}{\log\log n}\,,\ \ \epsilon > 0 \ . $$

On the other hand, for all $\epsilon > 0$, $$ \tau(n) = O(n^\epsilon)\ . $$

$\tau$ is a multiplicative arithmetic function and is equal to the number of points with natural coordinates on the hyperbola $xy = n$. The average value of $\tau(n)$ is given by Dirichlet's asymptotic formula (cf. Divisor problems).

The function $\tau_k(n)$, which is the number of solutions of the equation $n = x_1\cdots x_k$ in natural numbers $x_1,\ldots,x_k$, is a generalization of the function $\tau$.

References

[1] I.M. Vinogradov, "Elements of number theory" , Dover, reprint (1954) (Translated from Russian)
[2] K. Prachar, "Primzahlverteilung" , Springer (1957)


Comments

The average value of the number of divisors was obtained by P. Dirichlet in 1849, in the form $$ \sum_{n \le x} \tau(n) = x \log x + (2 \gamma - 1)x + O(\sqrt x) \ . $$

References

[a1] G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Oxford Univ. Press (1979) pp. Chapt. XVI
How to Cite This Entry:
Number of divisors. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Number_of_divisors&oldid=18293
This article was adapted from an original article by N.I. Klimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article