arith.h – arithmetic and special functions¶
This module implements arithmetic functions, numbertheoretic and combinatorial special number sequences and polynomials.
Harmonic numbers¶
Stirling numbers¶

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 \(x_{(n)} = x(x1)(x2) \dotsm (xn+1)\) is a falling factorial and \(x^{(n)} = x(x+1)(x+2) \dotsm (x+n1)\) 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_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,klen1)
.To compute a full row, this function can be called with
klen = n+1
. It is assumed thatklen
is at most \(n + 1\).

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 numbersS(n1,0), S(n1,1), S(n1,2), ..., S(n1,klen1)
, computes and stores in the row argumentS(n,0), S(n,1), S(n,2), ..., S(n,klen1)
.If
klen
is greater thann
, the output ends withS(n,n) = 1
followed byS(n,n+1) = S(n,n+2) = ... = 0
. In this case, the input only needs to have lengthn1
; only the input entries up toS(n1,n2)
are read.The
row
andprev
arguments are permitted to be the same, meaning that the row will be updated inplace.

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 \(m1\) and column \(n1\) 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 floatingpoint arithmetic.The
multi_mod
version computes the result modulo several limbsize primes and reconstructs the integer value using the fast Chinese remainder algorithm. A bound for the number of needed primes is computed usingarith_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 \(B_0, B_1, \ldots, B_{n1}\) inclusive. The
recursive
version uses the \(O(n^3 \log n)\) triangular recurrence, while themulti_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
\[B_n = \sum_{k=0}^n \frac{(nk)^n}{(nk)!} \sum_{j=0}^k \frac{(1)^j}{j!}.\]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 \(B_0, B_1, \ldots, B_{n1}\) 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 \(\sum_{k=0}^{\infty} \frac{B_n}{n!} x^n = \exp(e^x1)\), 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 singlefmpq_t
variable.

void _arith_bernoulli_number_vec(fmpz *num, fmpz *den, slong n)¶
Sets the elements of
num
andden
to the reduced numerators and denominators of the Bernoulli numbers \(B_0, B_1, B_2, \ldots, B_{n1}\) inclusive. This function automatically chooses between therecursive
,zeta
andmulti_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 \(B_0, B_1, B_2, \ldots, B_{n1}\) inclusive. This function is equivalent to_arith_bernoulli_number_vec
apart from the output being a singlefmpq
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 StaudtClausen 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 \(n\), \(B_n(x) = \sum_{k=0}^n \binom{n}{k} B_k x^{nk}\) where \(B_k\) is a Bernoulli number. This function basically callsarith_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
andden
to the reduced numerators and denominators of \(B_0, B_1, B_2, \ldots, B_{n1}\) 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
andden
to the reduced numerators and denominators of \(B_0, B_1, B_2, \ldots, B_{n1}\) inclusive. Uses the generating function\[\frac{x^2}{\cosh(x)1} = \sum_{k=0}^{\infty} \frac{(24k) B_{2k}}{(2k)!} x^{2k}\]which is evaluated modulo several limbsize 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 \(x/(e^x1)\) 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 oddindexed 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_vec(fmpz *res, slong n)¶
Computes the Euler numbers \(E_0, E_1, \dotsc, E_{n1}\) for \(n \geq 0\) and stores the result in
res
, which must be an initialisedfmpz
vector of sufficient size.This function evaluates the evenindex \(E_k\) modulo several limbsize primes using the generating function and
nmod_poly
arithmetic. A tight bound for the number of needed primes is computed usingarith_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 \(\tau(p^{r+1}) = \tau(p) \tau(p^r)  p^{11} \tau(p^{r1})\) for prime powers.
The base values \(\tau(p)\) are obtained using the function
arith_ramanujan_tau_series()
. Thus the speed ofarith_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 \(\tau(0),\tau(1), \dotsc, \tau(n1)\), giving the initial \(n\) terms in the series expansion of \(f(q) = q \prod_{k \geq 1} \bigl(1q^k\bigr)^{24}\).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 bymod
. 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
\[A_k(n) = \sum_{h=0}^{k1} \exp\left(\pi i \left[ s(h,k)  \frac{2hn}{k}\right]\right)\]appearing in the HardyRamanujanRademacher 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
structureprod
represents a product of cosines of rational arguments, multiplied by an algebraic prefactor. It must be preinitialised withtrig_prod_init
.This function assumes that \(24k\) and \(24n\) do not overflow a single limb. If \(n\) is larger, it can be prereduced 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 preinitialised MPFR variable \(x\) to the exact value of \(p(n)\). The value is computed using the HardyRamanujanRademacher 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 HardyRamanujanRademacher 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
\[ \begin{align}\begin{aligned}U(x) = \cosh(x) + \frac{\sinh(x)}{x}, \quad C = \frac{\pi}{6} \sqrt{24n1}\\B_k(n) = \sqrt{\frac{3}{k}} \frac{4}{24n1} A_k(n)\end{aligned}\end{align} \]and where \(A_k(n)\) is a certain exponential sum. The remainder satisfies
\[R(n,N) < \frac{44 \pi^2}{225 \sqrt{3}} N^{1/2} + \frac{\pi \sqrt{2}}{75} \left(\frac{N}{n1}\right)^{1/2} \sinh\left(\pi \sqrt{\frac{2}{3}} \frac{\sqrt{n}}{N} \right).\]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 fullprecision 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 \(i = 0, 1, \ldots, n1\), 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.
\[\vartheta_3^k(q) = \left( \sum_{i=\infty}^{\infty} q^{i^2} \right)^k.\]