arith.h – arithmetic and special functions

This module implements arithmetic functions, number-theoretic and combinatorial special number sequences and polynomials.

Harmonic numbers

void _arith_harmonic_number(fmpz_t num, fmpz_t den, slong n)

void arith_harmonic_number(fmpq_t x, slong n)

These are aliases for the functions in the fmpq module.

Stirling numbers

void arith_stirling_number_1u(fmpz_t s, ulong n, ulong k)

void arith_stirling_number_1(fmpz_t s, ulong n, ulong k)

void arith_stirling_number_2(fmpz_t s, ulong n, ulong k)

Sets \(s\) to \(S(n,k)\) where \(S(n,k)\) denotes an unsigned Stirling number of the first kind \(|S_1(n, k)|\), a signed Stirling number of the first kind \(S_1(n, k)\), or a Stirling number of the second kind \(S_2(n, k)\). The Stirling numbers are defined using the generating functions

\[ \begin{align}\begin{aligned}x_{(n)} = \sum_{k=0}^n S_1(n,k) x^k\\x^{(n)} = \sum_{k=0}^n |S_1(n,k)| x^k\\x^n = \sum_{k=0}^n S_2(n,k) x_{(k)}\end{aligned}\end{align} \]

where \(x_{(n)} = x(x-1)(x-2) \dotsm (x-n+1)\) is a falling factorial and \(x^{(n)} = x(x+1)(x+2) \dotsm (x+n-1)\) is a rising factorial. \(S(n,k)\) is taken to be zero if \(n < 0\) or \(k < 0\).

These three functions are useful for computing isolated Stirling numbers efficiently. To compute a range of numbers, the vector or matrix versions should generally be used.

void arith_stirling_number_1u_vec(fmpz *row, ulong n, slong klen)

void arith_stirling_number_1_vec(fmpz *row, ulong n, slong klen)

void arith_stirling_number_2_vec(fmpz *row, ulong n, slong klen)

Computes the row of Stirling numbers S(n,0), S(n,1), S(n,2), ..., S(n,klen-1).

To compute a full row, this function can be called with klen = n+1. It is assumed that klen is at most \(n + 1\).

void arith_stirling_number_1u_vec_next(fmpz *row, const fmpz *prev, slong n, slong klen)

void arith_stirling_number_1_vec_next(fmpz *row, const fmpz *prev, slong n, slong klen)

void arith_stirling_number_2_vec_next(fmpz *row, const fmpz *prev, slong n, slong klen)

Given the vector prev containing a row of Stirling numbers S(n-1,0), S(n-1,1), S(n-1,2), ..., S(n-1,klen-1), computes and stores in the row argument S(n,0), S(n,1), S(n,2), ..., S(n,klen-1).

If klen is greater than n, the output ends with S(n,n) = 1 followed by S(n,n+1) = S(n,n+2) = ... = 0. In this case, the input only needs to have length n-1; only the input entries up to S(n-1,n-2) are read.

The row and prev arguments are permitted to be the same, meaning that the row will be updated in-place.

void arith_stirling_matrix_1u(fmpz_mat_t mat)

void arith_stirling_matrix_1(fmpz_mat_t mat)

void arith_stirling_matrix_2(fmpz_mat_t mat)

For an arbitrary \(m\)-by-\(n\) matrix, writes the truncation of the infinite Stirling number matrix:

row 0   : S(0,0)
row 1   : S(1,0), S(1,1)
row 2   : S(2,0), S(2,1), S(2,2)
row 3   : S(3,0), S(3,1), S(3,2), S(3,3)

up to row \(m-1\) and column \(n-1\) inclusive. The upper triangular part of the matrix is zeroed.

For any \(n\), the \(S_1\) and \(S_2\) matrices thus obtained are inverses of each other.

Bell numbers

void arith_bell_number(fmpz_t b, ulong n)

void arith_bell_number_dobinski(fmpz_t res, ulong n)

void arith_bell_number_multi_mod(fmpz_t res, ulong n)

Sets \(b\) to the Bell number \(B_n\), defined as the number of partitions of a set with \(n\) members. Equivalently, \(B_n = \sum_{k=0}^n S_2(n,k)\) where \(S_2(n,k)\) denotes a Stirling number of the second kind.

The default version automatically selects between table lookup, Dobinski’s formula, and the multimodular algorithm.

The dobinski version evaluates a precise truncation of the series \(B_n = e^{-1} \sum_{k=0}^{\infty} \frac{k^n}{k!}\) (Dobinski’s formula). In fact, we compute \(P = N! \sum_{k=0}^N \frac{k^n}{k!}\) and \(Q = N! \sum_{k=0}^N \frac{1}{k!} \approx N! e\) and evaluate \(B_n = \lceil P / Q \rceil\), avoiding the use of floating-point arithmetic.

The multi_mod version computes the result modulo several limb-size primes and reconstructs the integer value using the fast Chinese remainder algorithm. A bound for the number of needed primes is computed using arith_bell_number_size.

void arith_bell_number_vec(fmpz *b, slong n)

void arith_bell_number_vec_recursive(fmpz *b, slong n)

void arith_bell_number_vec_multi_mod(fmpz *b, slong n)

Sets \(b\) to the vector of Bell numbers \(B_0, B_1, \ldots, B_{n-1}\) inclusive. The recursive version uses the \(O(n^3 \log n)\) triangular recurrence, while the multi_mod version implements multimodular evaluation of the exponential generating function, running in time \(O(n^2 \log^{O(1)} n)\). The default version chooses an algorithm automatically.

ulong arith_bell_number_nmod(ulong n, nmod_t mod)

Computes the Bell number \(B_n\) modulo an integer given by mod.

After handling special cases, we use the formula

\[B_n = \sum_{k=0}^n \frac{(n-k)^n}{(n-k)!} \sum_{j=0}^k \frac{(-1)^j}{j!}.\]

We arrange the operations in such a way that we only have to multiply (and not divide) in the main loop. As a further optimisation, we use sieving to reduce the number of powers that need to be evaluated. This results in \(O(n)\) memory usage.

If the divisions by factorials are impossible, we fall back to calling arith_bell_number_nmod_vec and reading the last coefficient.

void arith_bell_number_nmod_vec(nn_ptr b, slong n, nmod_t mod)

void arith_bell_number_nmod_vec_recursive(nn_ptr b, slong n, nmod_t mod)

void arith_bell_number_nmod_vec_ogf(nn_ptr b, slong n, nmod_t mod)

int arith_bell_number_nmod_vec_series(nn_ptr b, slong n, nmod_t mod)

Sets \(b\) to the vector of Bell numbers \(B_0, B_1, \ldots, B_{n-1}\) inclusive modulo an integer given by mod.

The recursive version uses the \(O(n^2)\) triangular recurrence. The ogf version expands the ordinary generating function using binary splitting, which is \(O(n \log^2 n)\).

The series version uses the exponential generating function \(\sum_{k=0}^{\infty} \frac{B_n}{n!} x^n = \exp(e^x-1)\), running in \(O(n \log n)\). This only works if division by \(n!\) is possible, and the function returns whether it is successful. All other versions support any modulus.

The default version of this function selects an algorithm automatically.

double arith_bell_number_size(ulong n)

Returns \(b\) such that \(B_n < 2^{\lfloor b \rfloor}\). A previous version of this function used the inequality \(B_n < \left(\frac{0.792n}{\log(n+1)}\right)^n\) which is given in [BerTas2010]; we now use a slightly better bound based on an asymptotic expansion.

Bernoulli numbers and polynomials

void _arith_bernoulli_number(fmpz_t num, fmpz_t den, ulong n)

Sets (num, den) to the reduced numerator and denominator of the \(n\)-th Bernoulli number.

void arith_bernoulli_number(fmpq_t x, ulong n)

Sets x to the \(n\)-th Bernoulli number. This function is equivalent to _arith_bernoulli_number apart from the output being a single fmpq_t variable.

void _arith_bernoulli_number_vec(fmpz *num, fmpz *den, slong n)

Sets the elements of num and den to the reduced numerators and denominators of the Bernoulli numbers \(B_0, B_1, B_2, \ldots, B_{n-1}\) inclusive. This function automatically chooses between the recursive, zeta and multi_mod algorithms according to the size of \(n\).

void arith_bernoulli_number_vec(fmpq *x, slong n)

Sets the x to the vector of Bernoulli numbers \(B_0, B_1, B_2, \ldots, B_{n-1}\) inclusive. This function is equivalent to _arith_bernoulli_number_vec apart from the output being a single fmpq vector.

void arith_bernoulli_number_denom(fmpz_t den, ulong n)

Sets den to the reduced denominator of the \(n\)-th Bernoulli number \(B_n\). For even \(n\), the denominator is computed as the product of all primes \(p\) for which \(p - 1\) divides \(n\); this property is a consequence of the von Staudt-Clausen theorem. For odd \(n\), the denominator is trivial (den is set to 1 whenever \(B_n = 0\)). The initial sequence of values smaller than \(2^{32}\) are looked up directly from a table.

double arith_bernoulli_number_size(ulong n)

Returns \(b\) such that \(|B_n| < 2^{\lfloor b \rfloor}\), using the inequality \(|B_n| < \frac{4 n!}{(2\pi)^n}\) and \(n! \le (n+1)^{n+1} e^{-n}\). No special treatment is given to odd \(n\). Accuracy is not guaranteed if \(n > 10^{14}\).

void arith_bernoulli_polynomial(fmpq_poly_t poly, ulong n)

Sets poly to the Bernoulli polynomial of degree \(n\), \(B_n(x) = \sum_{k=0}^n \binom{n}{k} B_k x^{n-k}\) where \(B_k\) is a Bernoulli number. This function basically calls arith_bernoulli_number_vec and then rescales the coefficients efficiently.

void _arith_bernoulli_number_vec_recursive(fmpz *num, fmpz *den, slong n)

Sets the elements of num and den to the reduced numerators and denominators of \(B_0, B_1, B_2, \ldots, B_{n-1}\) inclusive.

The first few entries are computed using arith_bernoulli_number, and then Ramanujan’s recursive formula expressing \(B_m\) as a sum over \(B_k\) for \(k\) congruent to \(m\) modulo 6 is applied repeatedly.

To avoid costly GCDs, the numerators are transformed internally to a common denominator and all operations are performed using integer arithmetic. This makes the algorithm fast for small \(n\), say \(n < 1000\). The common denominator is calculated directly as the primorial of \(n + 1\).

%[1] https://en.wikipedia.org/w/index.php?title=Bernoulli_number&oldid=405938876

void _arith_bernoulli_number_vec_multi_mod(fmpz *num, fmpz *den, slong n)

Sets the elements of num and den to the reduced numerators and denominators of \(B_0, B_1, B_2, \ldots, B_{n-1}\) inclusive. Uses the generating function

\[\frac{x^2}{\cosh(x)-1} = \sum_{k=0}^{\infty} \frac{(2-4k) B_{2k}}{(2k)!} x^{2k}\]

which is evaluated modulo several limb-size primes using nmod_poly arithmetic to yield the numerators of the Bernoulli numbers after multiplication by the denominators and CRT reconstruction. This formula, given (incorrectly) in [BuhlerCrandallSompolski1992], saves about half of the time compared to the usual generating function \(x/(e^x-1)\) since the odd terms vanish.

Euler numbers and polynomials

Euler numbers are the integers \(E_n\) defined by \(\frac{1}{\cosh(t)} = \sum_{n=0}^{\infty} \frac{E_n}{n!} t^n.\) With this convention, the odd-indexed numbers are zero and the even ones alternate signs, viz. \(E_0, E_1, E_2, \ldots = 1, 0, -1, 0, 5, 0, -61, 0, 1385, 0, \ldots\). The corresponding Euler polynomials are defined by \(\frac{2e^{xt}}{e^t+1} = \sum_{n=0}^{\infty} \frac{E_n(x)}{n!} t^n.\)

void arith_euler_number(fmpz_t res, ulong n)

Sets res to the Euler number \(E_n\).

void arith_euler_number_vec(fmpz *res, slong n)

Computes the Euler numbers \(E_0, E_1, \dotsc, E_{n-1}\) for \(n \geq 0\) and stores the result in res, which must be an initialised fmpz vector of sufficient size.

This function evaluates the even-index \(E_k\) modulo several limb-size primes using the generating function and nmod_poly arithmetic. A tight bound for the number of needed primes is computed using arith_euler_number_size, and the final integer values are recovered using balanced CRT reconstruction.

double arith_euler_number_size(ulong n)

Returns \(b\) such that \(|E_n| < 2^{\lfloor b \rfloor}\), using the inequality |E_n| < \frac{2^{n+2} n!}{\pi^{n+1}} and \(n! \le (n+1)^{n+1} e^{-n}\). No special treatment is given to odd \(n\). Accuracy is not guaranteed if \(n > 10^{14}\).

void arith_euler_polynomial(fmpq_poly_t poly, ulong n)

Sets poly to the Euler polynomial \(E_n(x)\). Uses the formula

\[E_n(x) = \frac{2}{n+1}\left(B_{n+1}(x) - 2^{n+1}B_{n+1}\left(\frac{x}{2}\right)\right),\]

with the Bernoulli polynomial \(B_{n+1}(x)\) evaluated once using bernoulli_polynomial and then rescaled.

Multiplicative functions

void arith_divisors(fmpz_poly_t res, const fmpz_t n)

Set the coefficients of the polynomial res to the divisors of \(n\), including \(1\) and \(n\) itself, in ascending order.

void arith_ramanujan_tau(fmpz_t res, const fmpz_t n)

Sets res to the Ramanujan tau function \(\tau(n)\) which is the coefficient of \(q^n\) in the series expansion of \(f(q) = q \prod_{k \geq 1} \bigl(1 - q^k\bigr)^{24}\).

We factor \(n\) and use the identity \(\tau(pq) = \tau(p) \tau(q)\) along with the recursion \(\tau(p^{r+1}) = \tau(p) \tau(p^r) - p^{11} \tau(p^{r-1})\) for prime powers.

The base values \(\tau(p)\) are obtained using the function arith_ramanujan_tau_series(). Thus the speed of arith_ramanujan_tau() depends on the largest prime factor of \(n\).

Future improvement: optimise this function for small \(n\), which could be accomplished using a lookup table or by calling arith_ramanujan_tau_series() directly.

void arith_ramanujan_tau_series(fmpz_poly_t res, slong n)

Sets res to the polynomial with coefficients \(\tau(0),\tau(1), \dotsc, \tau(n-1)\), giving the initial \(n\) terms in the series expansion of \(f(q) = q \prod_{k \geq 1} \bigl(1-q^k\bigr)^{24}\).

We use the theta function identity

\[f(q) = q \Biggl( \sum_{k \geq 0} (-1)^k (2k+1) q^{k(k+1)/2} \Biggr)^8\]

which is evaluated using three squarings. The first squaring is done directly since the polynomial is very sparse at this point.

Landau’s function

void arith_landau_function_vec(fmpz *res, slong len)

Computes the first len values of Landau’s function \(g(n)\) starting with \(g(0)\). Landau’s function gives the largest order of an element of the symmetric group \(S_n\).

Implements the “basic algorithm” given in [DelegliseNicolasZimmermann2009]. The running time is \(O(n^{3/2} / \sqrt{\log n})\).

Number of partitions

void arith_number_of_partitions_vec(fmpz *res, slong len)

Computes first len values of the partition function \(p(n)\) starting with \(p(0)\). Uses inversion of Euler’s pentagonal series.

void arith_number_of_partitions_nmod_vec(nn_ptr res, slong len, nmod_t mod)

Computes first len values of the partition function \(p(n)\) starting with \(p(0)\), modulo the modulus defined by mod. Uses inversion of Euler’s pentagonal series.

void trig_prod_init(trig_prod_t prod)

Initializes prod. This is an inline function only.

void arith_hrr_expsum_factored(trig_prod_t prod, ulong k, ulong n)

Symbolically evaluates the exponential sum

\[A_k(n) = \sum_{h=0}^{k-1} \exp\left(\pi i \left[ s(h,k) - \frac{2hn}{k}\right]\right)\]

appearing in the Hardy-Ramanujan-Rademacher formula, where \(s(h,k)\) is a Dedekind sum.

Rather than evaluating the sum naively, we factor \(A_k(n)\) into a product of cosines based on the prime factorisation of \(k\). This process is based on the identities given in [Whiteman1956].

The special trig_prod_t structure prod represents a product of cosines of rational arguments, multiplied by an algebraic prefactor. It must be pre-initialised with trig_prod_init.

This function assumes that \(24k\) and \(24n\) do not overflow a single limb. If \(n\) is larger, it can be pre-reduced modulo \(k\), since \(A_k(n)\) only depends on the value of \(n \bmod k\).

void arith_number_of_partitions_mpfr(mpfr_t x, ulong n)

Sets the pre-initialised MPFR variable \(x\) to the exact value of \(p(n)\). The value is computed using the Hardy-Ramanujan-Rademacher formula.

The precision of \(x\) will be changed to allow \(p(n)\) to be represented exactly. The interface of this function may be updated in the future to allow computing an approximation of \(p(n)\) to smaller precision.

The Hardy-Ramanujan-Rademacher formula is given with error bounds in [Rademacher1937]. We evaluate it in the form

\[p(n) = \sum_{k=1}^N B_k(n) U(C/k) + R(n,N)\]

where

\[ \begin{align}\begin{aligned}U(x) = \cosh(x) + \frac{\sinh(x)}{x}, \quad C = \frac{\pi}{6} \sqrt{24n-1}\\B_k(n) = \sqrt{\frac{3}{k}} \frac{4}{24n-1} A_k(n)\end{aligned}\end{align} \]

and where \(A_k(n)\) is a certain exponential sum. The remainder satisfies

\[|R(n,N)| < \frac{44 \pi^2}{225 \sqrt{3}} N^{-1/2} + \frac{\pi \sqrt{2}}{75} \left(\frac{N}{n-1}\right)^{1/2} \sinh\left(\pi \sqrt{\frac{2}{3}} \frac{\sqrt{n}}{N} \right).\]

We choose \(N\) such that \(|R(n,N)| < 0.25\), and a working precision at term \(k\) such that the absolute error of the term is expected to be less than \(0.25 / N\). We also use a summation variable with increased precision, essentially making additions exact. Thus the sum of errors adds up to less than 0.5, giving the correct value of \(p(n)\) when rounding to the nearest integer.

The remainder estimate at step \(k\) provides an upper bound for the size of the \(k\)-th term. We add \(\log_2 N\) bits to get low bits in the terms below \(0.25 / N\) in magnitude.

Using arith_hrr_expsum_factored, each \(B_k(n)\) evaluation is broken down to a product of cosines of exact rational multiples of \(\pi\). We transform all angles to \((0, \pi/4)\) for optimal accuracy.

Since the evaluation of each term involves only \(O(\log k)\) multiplications and evaluations of trigonometric functions of small angles, the relative rounding error is at most a few bits. We therefore just add an additional \(\log_2 (C/k)\) bits for the \(U(x)\) when \(x\) is large. The cancellation of terms in \(U(x)\) is of no concern, since Rademacher’s bound allows us to terminate before \(x\) becomes small.

This analysis should be performed in more detail to give a rigorous error bound, but the precision currently implemented is almost certainly sufficient, not least considering that Rademacher’s remainder bound significantly overshoots the actual values.

To improve performance, we switch to doubles when the working precision becomes small enough. We also use a separate accumulator variable which gets added to the main sum periodically, in order to avoid costly updates of the full-precision result when \(n\) is large.

void arith_number_of_partitions(fmpz_t x, ulong n)

Sets \(x\) to \(p(n)\), the number of ways that \(n\) can be written as a sum of positive integers without regard to order.

This function uses a lookup table for \(n < 128\) (where \(p(n) < 2^{32}\)), and otherwise calls arith_number_of_partitions_mpfr.

Sums of squares

void arith_sum_of_squares(fmpz_t r, ulong k, const fmpz_t n)

Sets \(r\) to the number of ways \(r_k(n)\) in which \(n\) can be represented as a sum of \(k\) squares.

If \(k = 2\) or \(k = 4\), we write \(r_k(n)\) as a divisor sum.

Otherwise, we either recurse on \(k\) or compute the theta function expansion up to \(O(x^{n+1})\) and read off the last coefficient. This is generally optimal.

void arith_sum_of_squares_vec(fmpz *r, ulong k, slong n)

For \(i = 0, 1, \ldots, n-1\), sets \(r_i\) to the number of representations of \(i\) a sum of \(k\) squares, \(r_k(i)\). This effectively computes the \(q\)-expansion of \(\vartheta_3(q)\) raised to the \(k\)-th power, i.e.

\[\vartheta_3^k(q) = \left( \sum_{i=-\infty}^{\infty} q^{i^2} \right)^k.\]