`arith_ramanujan_tau`

Synopsis#include <flint/arith.h>

void arith_ramanujan_tau(fmpz_t res, const fmpz_t n)source

Sets res to $\tau(n)$ , the coefficient of $q^n$ in $\Delta(q) = q \prod_{k \ge 1} (1 - q^k)^{24}$ , or to 0 for $n \le 0$ .

The function factors $n$ , computes $\tau(p)$ for the prime factors with arith_ramanujan_tau_series, and uses the multiplicativity of $\tau$ and the recurrence $\tau(p^{r+1}) = \tau(p) \tau(p^r) - p^{11} \tau(p^{r-1})$ . The cost is dominated by the series of length $P + 1$ , where $P$ is the largest prime factor of $n$ , which must fit in a word.

Cite this page

The FLINT team, “arith_ramanujan_tau (arith.h),” FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev, 2026. https://flintlib.org/doc/arith/arith_ramanujan_tau.html (generated 2026-11-02).

@misc{flint-doc-arith-arith-ramanujan-tau,
  author       = {{The FLINT team}},
  title        = {{arith\_ramanujan\_tau (arith.h)}},
  howpublished = {FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev},
  year         = {2026},
  url          = {https://flintlib.org/doc/arith/arith_ramanujan_tau.html},
  note         = {Generated 2026-11-02}
}

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