# Tau (mathematics)

In number theory, **tau**, written \( \tau(n) \), is the arithmetic function that counts the positive divisors of a positive integer \( n \). It is also denoted \( d(n) \) and \( \sigma_0(n) \), the zeroth member of the divisor-function family \( \sigma_k(n) = \sum_{d \mid n} d^k \), since setting \( k = 0 \) turns the sum of powers of divisors into a count.<sup>[1](https://encyclopediaofmath.org/wiki/Number_of_divisors)</sup><sup> • </sup><sup>[2](https://mathworld.wolfram.com/DivisorFunction.html)</sup>  The letter comes from the German word *Teiler*, meaning divisor.<sup>[3](https://www.simonjensen.com/pdf/On_an_extended_divisor_product_summatory_function.pdf)</sup>

| Key fact | Detail |
|---|---|
| Definition | \( \tau(n) = \sum_{d \mid n} 1 \), the number of positive divisors of \( n \); also written \( d(n) \) or \( \sigma_0(n) \)<sup>[1](https://encyclopediaofmath.org/wiki/Number_of_divisors)</sup><sup> • </sup><sup>[2](https://mathworld.wolfram.com/DivisorFunction.html)</sup>  |
| Product formula | If \( n = p_1^{a_1} \cdots p_k^{a_k} \), then \( \tau(n) = (a_1+1) \cdots (a_k+1) \)<sup>[1](https://encyclopediaofmath.org/wiki/Number_of_divisors)</sup> |
| Multiplicativity | \( \tau \) is multiplicative: \( \tau(mn) = \tau(m)\tau(n) \) when \( \gcd(m,n) = 1 \)<sup>[1](https://encyclopediaofmath.org/wiki/Number_of_divisors)</sup> |
| Average order | \( \sum_{n \le x} \tau(n) = x \log x + (2\gamma - 1)x + O(\sqrt{x}) \), obtained by Dirichlet in 1849<sup>[1](https://encyclopediaofmath.org/wiki/Number_of_divisors)</sup> |
| Growth bounds | \( \tau(n) = O(n^{\epsilon}) \) for every \( \epsilon > 0 \), yet infinitely many \( n \) satisfy \( \tau(n) \ge 2^{1-\epsilon} \log n / \log\log n \)<sup>[1](https://encyclopediaofmath.org/wiki/Number_of_divisors)</sup> |
| Parity | \( \tau(n) \) is odd exactly when \( n \) is a perfect square<sup>[2](https://mathworld.wolfram.com/DivisorFunction.html)</sup> |
| Generalization | \( \tau_k(n) \) counts the ways to write \( n = x_1 \cdots x_k \) in natural numbers; \( \tau \) is the case \( k = 2 \)<sup>[1](https://encyclopediaofmath.org/wiki/Number_of_divisors)</sup><sup> • </sup><sup>[4](https://dlmf.nist.gov/27.2)</sup>  |

## Definition and basic examples

For \( n > 0 \), \( \tau(n) \) is the number of positive divisors of \( n \), counting 1 and \( n \) itself.<sup>[5](https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Elementary_Number_Theory_%28Clark%29/01%3A_Chapters/13%3A_The_Functions_and)</sup> The number 12, whose positive divisors are 1, 2, 3, 4, 6, and 12, has \( \tau(12) = 6 \).<sup>[5](https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Elementary_Number_Theory_%28Clark%29/01%3A_Chapters/13%3A_The_Functions_and)</sup> A prime \( p \) has only the divisors 1 and \( p \), so \( \tau(p) = 2 \).<sup>[1](https://encyclopediaofmath.org/wiki/Number_of_divisors)</sup> The first values, for \( n = 1, 2, 3, \ldots \), run 1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, and the sequence is cataloged as A000005 in the [On-Line Encyclopedia of Integer Sequences](https://www.edgepedia.org/on-line-encyclopedia-of-integer-sequences).<sup>[6](https://oeis.org/A000005)</sup><sup> • </sup><sup>[2](https://mathworld.wolfram.com/DivisorFunction.html)</sup>

Geometrically, \( \tau(n) \) equals the number of points with natural coordinates on the hyperbola \( xy = n \): each divisor \( d \) pairs with \( n/d \).<sup>[1](https://encyclopediaofmath.org/wiki/Number_of_divisors)</sup> Equivalently, \( \tau(n) \) is the number of ordered ways to write \( n \) as a product \( x \cdot y \) with \( 1 \le x, y \le n \).<sup>[6](https://oeis.org/A000005)</sup>

## The product formula and multiplicativity

The divisor count follows directly from the prime factorization. Building a divisor of \( n = p_1^{a_1} \cdots p_k^{a_k} \) means choosing, independently for each prime \( p_i \), an exponent from 0 up to \( a_i \), giving \( a_i + 1 \) choices per prime; the choices multiply.<sup>[7](https://interactiveprimer.com/concepts/mathematics/number-theory/arithmetic-functions/the-divisor-functions)</sup> Hence

\[ \tau(n) = (a_1 + 1) \cdots (a_k + 1). \]

<sup>[1](https://encyclopediaofmath.org/wiki/Number_of_divisors)</sup> For a prime power \( p^j \) this gives \( \tau(p^j) = j + 1 \).<sup>[8](https://poincare.matf.bg.ac.rs/~zarkom/Book_Math_TheoryOfNumbers_ABaker.pdf)</sup>

The function is multiplicative: when \( \gcd(m, n) = 1 \), every divisor of \( mn \) splits uniquely as a product of a divisor of \( m \) and a divisor of \( n \), so \( \tau(mn) = \tau(m)\tau(n) \).<sup>[9](https://math.umd.edu/~immortal/MATH406/lecturenotes/ch7-2.pdf)</sup> In Dirichlet-convolution terms, \( \tau = \zeta \star \zeta \), where \( \zeta(n) = 1 \) is the constant function.<sup>[10](https://math.stackexchange.com/questions/3339946/where-does-the-identity-taunm-sum-d-mid-m-n-mud-taun-d-taum)</sup>

Several identities follow. The product of the positive divisors of an integer \( n > 1 \) equals \( n^{\tau(n)/2} \), since divisors pair off around \( \sqrt{n} \).<sup>[11](https://assets.zyrosite.com/A1aw0O5jRBiVrR9D/elementary-number-theory-by-david-m.-burton-A0xlz3P72jcpxxNB.pdf)</sup> And \( \tau(n) \) is odd if and only if \( n \) is a square, because only then does the divisor \( \sqrt{n} \) pair with itself.<sup>[2](https://mathworld.wolfram.com/DivisorFunction.html)</sup>

## Growth and average order

Divisor counts fluctuate widely. For every \( \epsilon > 0 \), \( \tau(n) = O(n^{\epsilon}) \), so the function grows more slowly than any positive power of \( n \).<sup>[1](https://encyclopediaofmath.org/wiki/Number_of_divisors)</sup> A more precise version says \( \tau(n) < c\,n^{\delta} \) for any \( \delta > 0 \), with \( c \) depending only on \( \delta \).<sup>[8](https://poincare.matf.bg.ac.rs/~zarkom/Book_Math_TheoryOfNumbers_ABaker.pdf)</sup> In the other direction, there is an infinite sequence of \( n \) for which

\[ \tau(n) \ge 2^{1-\epsilon} \frac{\log n}{\log\log n}, \]

for every \( \epsilon > 0 \).<sup>[1](https://encyclopediaofmath.org/wiki/Number_of_divisors)</sup> Along highly composite numbers, the integers that set divisor-count records, the maximal size takes the form \( \tau(N) = 2^{(1+o(1)) \log N / \log\log N} \), combining Wigert's upper bound with the lower bound from Ramanujan's work.<sup>[12](https://github.com/plasma-ai/erdos/blob/main/library/divisors/kovac_2025_number_divisors_mersenne_numbers/proposition_2.md)</sup>

The average behavior was measured by Peter Gustav Lejeune Dirichlet, the nineteenth-century German mathematician, in 1849:<sup>[1](https://encyclopediaofmath.org/wiki/Number_of_divisors)</sup>

\[ \sum_{n \le x} \tau(n) = x \log x + (2\gamma - 1)x + O(\sqrt{x}), \]

where \( \gamma \approx 0.57721 \) is the [Euler–Mascheroni constant](https://www.edgepedia.org/euler-mascheroni-constant).<sup>[1](https://encyclopediaofmath.org/wiki/Number_of_divisors)</sup><sup> • </sup><sup>[13](https://www.sciencedirect.com/science/article/abs/pii/S0022314X25002550)</sup>  The average order of \( \tau(n) \) is therefore \( \log n \). This average is misleading as a description of typical values: it is not true that almost all numbers have about \( \log n \) divisors. Almost all numbers have about \( (\log n)^{\log 2} \) divisors, in the sense that for any \( \epsilon > 0 \) the ratio \( \tau(n) / (\log n)^{\log 2} \) eventually lies between \( (\log n)^{\epsilon} \) and \( (\log n)^{-\epsilon} \) for almost all \( n \).<sup>[8](https://poincare.matf.bg.ac.rs/~zarkom/Book_Math_TheoryOfNumbers_ABaker.pdf)</sup>

## The Dirichlet divisor problem

The error term \( \Delta(x) \) in the summatory formula above is the subject of the [Dirichlet divisor problem](https://www.edgepedia.org/dirichlet-divisor-problem), one of the oldest open problems in analytic number theory.<sup>[13](https://www.sciencedirect.com/science/article/abs/pii/S0022314X25002550)</sup> Dirichlet's hyperbola method yields the main terms and the estimate \( \Delta(x) = O(x^{1/2}) \).<sup>[13](https://www.sciencedirect.com/science/article/abs/pii/S0022314X25002550)</sup><sup> • </sup><sup>[14](https://mathworld.wolfram.com/DirichletDivisorProblem.html)</sup>  Hardy and Landau showed in 1916 that the exponent cannot be pushed below \( 1/4 \).<sup>[14](https://mathworld.wolfram.com/DirichletDivisorProblem.html)</sup> Successive improvements have lowered the known upper exponent from Dirichlet's \( 1/2 \) through Voronoi and Sierpiński's \( 1/3 \) to \( 131/416 \approx 0.31490 \), achieved by M. N. Huxley in 2003.<sup>[14](https://mathworld.wolfram.com/DirichletDivisorProblem.html)</sup> The generalized (Piltz) divisor problem asks the same question for \( \tau_k \) with \( k > 2 \), and is closely connected with the behavior of the [Riemann zeta function](https://www.edgepedia.org/riemann-zeta-function) in the critical strip.<sup>[15](https://encyclopediaofmath.org/index.php?title=Divisor_problems)</sup><sup> • </sup><sup>[16](https://link.springer.com/article/10.1007/s11139-026-01422-5)</sup>

## Generalizations and appearances

The function \( \tau_k(n) \), the number of solutions of \( n = x_1 \cdots x_k \) in natural numbers, generalizes \( \tau \), which is the case \( k = 2 \).<sup>[1](https://encyclopediaofmath.org/wiki/Number_of_divisors)</sup> In the DLMF's notation, \( d(n) \) is the special case \( k = 2 \) of \( d_k(n) \), which counts factorizations into \( k \) factors with order taken into account.<sup>[4](https://dlmf.nist.gov/27.2)</sup> The summatory functions \( D_k(x) = \sum_{n \le x} \tau_k(n) \) are the objects of the divisor problems.<sup>[15](https://encyclopediaofmath.org/index.php?title=Divisor_problems)</sup>

The count appears in unexpected places. The number of factors of the polynomial \( x^n - 1 \) over the integers equals \( \tau(n) \), and the same holds for \( a^n - b^n \).<sup>[6](https://oeis.org/A000005)</sup><sup> • </sup><sup>[2](https://mathworld.wolfram.com/DivisorFunction.html)</sup>  For odd \( n \), \( \tau(n) \) is also the number of partitions of \( n \) into consecutive integers.<sup>[6](https://oeis.org/A000005)</sup> The integers with incrementally largest divisor counts are the highly composite numbers, studied by [Srinivasa Ramanujan](https://www.edgepedia.org/srinivasa-ramanujan), the Indian mathematician known for his work in number theory; \( n \) is highly composite when \( \tau(n) \) exceeds \( \tau(m) \) for every positive \( m < n \).<sup>[2](https://mathworld.wolfram.com/DivisorFunction.html)</sup>

## Distinction from Ramanujan's tau function

The symbol \( \tau(n) \) is also used for Ramanujan's tau function, a different arithmetic function defined by the Fourier expansion \( \Delta(z) = \sum_{n=1}^{\infty} \tau(n) q^n = q \prod_{n=1}^{\infty} (1 - q^n)^{24} \) of the modular discriminant.<sup>[17](https://mathworld.wolfram.com/TauFunction.html)</sup><sup> • </sup><sup>[18](https://www.mathsoc.jp/section/algebra/algsymp_past/algsymp04_files/ono2.pdf)</sup>  The two functions are unrelated in definition, and sources note that the divisor-count notation can be confused with Ramanujan's.<sup>[3](https://www.simonjensen.com/pdf/On_an_extended_divisor_product_summatory_function.pdf)</sup> The lower-case Greek letter \( \tau \) carries many other meanings in mathematics and physics as well, such as the tau particle and torque.<sup>[19](https://mathworld.wolfram.com/Tau.html)</sup>

## References

1. [Number of divisors, Encyclopedia of Mathematics (originator N.I. Klimov)](https://encyclopediaofmath.org/wiki/Number_of_divisors)
2. [Divisor Function, Wolfram MathWorld](https://mathworld.wolfram.com/DivisorFunction.html)
3. [On an extended divisor product summatory function, S. Jensen](https://www.simonjensen.com/pdf/On_an_extended_divisor_product_summatory_function.pdf)
4. [DLMF §27.2 Functions, NIST Digital Library of Mathematical Functions](https://dlmf.nist.gov/27.2)
5. [The Functions σ and τ, Elementary Number Theory (Clark), Mathematics LibreTexts](https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Elementary_Number_Theory_%28Clark%29/01%3A_Chapters/13%3A_The_Functions_and)
6. [A000005: d(n), the number of divisors of n, OEIS](https://oeis.org/A000005)
7. [The divisor functions, Interactive Primer](https://interactiveprimer.com/concepts/mathematics/number-theory/arithmetic-functions/the-divisor-functions)
8. [Arithmetical functions — Average orders, A. Baker](https://poincare.matf.bg.ac.rs/~zarkom/Book_Math_TheoryOfNumbers_ABaker.pdf)
9. [Math 406 Section 7.2: The Sum and Number of Divisors, University of Maryland](https://math.umd.edu/~immortal/MATH406/lecturenotes/ch7-2.pdf)
10. [Where does the identity τ(nm) = Σ μ(d) τ(n/d) τ(m/d) come from?, Mathematics Stack Exchange](https://math.stackexchange.com/questions/3339946/where-does-the-identity-taunm-sum-d-mid-m-n-mud-taun-d-taum)
11. [Elementary Number Theory, D. M. Burton](https://assets.zyrosite.com/A1aw0O5jRBiVrR9D/elementary-number-theory-by-david-m.-burton-A0xlz3P72jcpxxNB.pdf)
12. [Number of divisors of Mersenne numbers, proposition 2, plasma-ai/erdos (GitHub)](https://github.com/plasma-ai/erdos/blob/main/library/divisors/kovac_2025_number_divisors_mersenne_numbers/proposition_2.md)
13. [On certain correlations into the divisor problem, Journal of Number Theory](https://www.sciencedirect.com/science/article/abs/pii/S0022314X25002550)
14. [Dirichlet Divisor Problem, Wolfram MathWorld](https://mathworld.wolfram.com/DirichletDivisorProblem.html)
15. [Divisor problems, Encyclopedia of Mathematics](https://encyclopediaofmath.org/index.php?title=Divisor_problems)
16. [On the generalised divisor problem, The Ramanujan Journal](https://link.springer.com/article/10.1007/s11139-026-01422-5)
17. [Tau Function (Ramanujan), Wolfram MathWorld](https://mathworld.wolfram.com/TauFunction.html)
18. [Ono, Δ(z) and modular forms (Mathematical Society of Japan algebra symposium material)](https://www.mathsoc.jp/section/algebra/algsymp_past/algsymp04_files/ono2.pdf)
19. [Tau, Wolfram MathWorld](https://mathworld.wolfram.com/Tau.html)

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