arith.h – arithmetic and special functions

Primorials

void arith_primorial(fmpz_t res, slong n)

Sets res to n primorial or \(n \#\), the product of all prime numbers less than or equal to \(n\).

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, slong n, slong k)
void arith_stirling_number_1(fmpz_t s, slong n, slong k)
void arith_stirling_number_2(fmpz_t s, slong n, slong 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, slong n, slong klen)
void arith_stirling_number_1_vec(fmpz * row, slong n, slong klen)
void arith_stirling_number_2_vec(fmpz * row, slong 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, fmpz * prev, slong n, slong klen)
void arith_stirling_number_1_vec_next(fmpz * row, fmpz * prev, slong n, slong klen)
void arith_stirling_number_2_vec_next(fmpz * row, 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)

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.

This function automatically selects between table lookup, binary splitting, and the multimodular algorithm.

void arith_bell_number_bsplit(fmpz_t res, ulong n)

Computes the Bell number \(B_n\) by evaluating a precise truncation of the series \(B_n = e^{-1} \sum_{k=0}^{\infty} \frac{k^n}{k!}\) using binary splitting.

void arith_bell_number_multi_mod(fmpz_t res, ulong n)

Computes the Bell number \(B_n\) using a multimodular algorithm.

This function evaluates the Bell number modulo several limb-size primes using\ arith_bell_number_nmod 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)

Sets \(b\) to the vector of Bell numbers \(B_0, B_1, \ldots, B_{n-1}\) inclusive. Automatically switches between the recursive and multi_mod algorithms depending on the size of \(n\).

void arith_bell_number_vec_recursive(fmpz * b, slong n)

Sets \(b\) to the vector of Bell numbers \(B_0, B_1, \ldots, B_{n-1}\) inclusive. This function uses table lookup if \(B_{n-1}\) fits in a single word, and a standard triangular recurrence otherwise.

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.

This function evaluates the Bell numbers modulo several limb-size primes using\ arith_bell_number_nmod_vec and reconstructs the integer values using the fast Chinese remainder algorithm. A bound for the number of needed primes is computed using arith_bell_number_size.

mp_limb_t arith_bell_number_nmod(ulong n, nmod_t mod)

Computes the Bell number \(B_n\) modulo a prime \(p\) 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.

The divisions by factorials require \(n > p\), so we fall back to calling\ bell_number_nmod_vec_recursive and reading off the last entry when \(p \le n\).

void arith_bell_number_nmod_vec(mp_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 a prime \(p\) given by mod. Automatically switches between the recursive and series algorithms depending on the size of \(n\) and whether \(p\) is large enough for the series algorithm to work.

void arith_bell_number_nmod_vec_recursive(mp_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 a prime \(p\) given by mod. This function uses table lookup if \(B_{n-1}\) fits in a single word, and a standard triangular recurrence otherwise.

void arith_bell_number_nmod_vec_series(mp_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 a prime \(p\) given by mod. This function expands the exponential generating function \sum_{k=0}^{\infty} \frac{B_n}{n!} x^n = \exp(e^x-1). We require that \(p \ge n\).

double arith_bell_number_size(ulong n)

Returns \(b\) such that \(B_n < 2^{\lfloor b \rfloor}\), using the inequality B_n < \left(\frac{0.792n}{\log(n+1)}\right)^n which is given in [BerTas2010].

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. As presently implemented, this function simply calls\ _arith_bernoulli_number_zeta.

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.

Warning: this function does not use proven precision bounds, and could return the wrong results for very large \(n\). It is recommended to use the Bernoulli number functions in Arb instead.

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_zeta(fmpz_t num, fmpz_t den, ulong n)

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

This function first computes the exact denominator and a bound for the size of the numerator. It then computes an approximation of \(|B_n| = 2n! \zeta(n) / (2 \pi)^n\) as a floating-point number and multiplies by the denominator to to obtain a real number that rounds to the exact numerator. For tiny \(n\), the numerator is looked up from a table to avoid unnecessary overhead.

Warning: this function does not use proven precision bounds, and could return the wrong results for very large \(n\). It is recommended to use the Bernoulli number functions in Arb instead.

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_zeta(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 repeated direct calls to\ _arith_bernoulli_number_zeta.

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 \(E_n\). Currently calls _arith_euler_number_zeta.

Warning: this function does not use proven precision bounds, and could return the wrong results for very large \(n\). It is recommended to use the Euler number functions in Arb instead.

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.

void _arith_euler_number_zeta(fmpz_t res, ulong n)

Sets res to the Euler number \(E_n\). For even \(n\), this function uses the relation |E_n| = \frac{2^{n+2} n!}{\pi^{n+1}} L(n+1) where \(L(n+1)\) denotes the Dirichlet \(L\)-function with character \(\chi = \{ 0, 1, 0, -1 \}\).

Warning: this function does not use proven precision bounds, and could return the wrong results for very large \(n\). It is recommended to use the Euler number functions in Arb instead.

Multiplicative functions

void arith_euler_phi(fmpz_t res, const fmpz_t n)
int arith_moebius_mu(const fmpz_t n)
void arith_divisor_sigma(fmpz_t res, const fmpz_t n, ulong k)

These are aliases for the functions in the fmpz module.

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.

Cyclotomic polynomials

void _arith_cos_minpoly(fmpz * coeffs, slong d, ulong n)

For \(n \ge 1\), sets (coeffs, d+1) to the minimal polynomial \(\Psi_n(x)\) of \(\cos(2 \pi / n)\), scaled to have integer coefficients by multiplying by \(2^d\) (\(2^{d-1}\) when \(n\) is a power of two).

The polynomial \(\Psi_n(x)\) is described in [WaktinsZeitlin1993]. As proved in that paper, the roots of \(\Psi_n(x)\) for \(n \ge 3\) are \(\cos(2 \pi k / n)\) where \(0 \le k < d\) and where \(\gcd(k, n) = 1\).

To calculate \(\Psi_n(x)\), we compute the roots numerically with MPFR and use a balanced product tree to form a polynomial with fixed-point coefficients, i.e. an approximation of \(2^p 2^d \Psi_n(x)\).

To determine the precision \(p\), we note that the coefficients in \(\prod_{i=1}^d (x - \alpha)\) can be bounded by the central coefficient in the binomial expansion of \((x+1)^d\).

When \(n\) is an odd prime, we use a direct formula for the coefficients (https://mathworld.wolfram.com/TrigonometryAngles.html ).

void arith_cos_minpoly(fmpz_poly_t poly, ulong n)

Sets poly to the minimal polynomial \(\Psi_n(x)\) of \(\cos(2 \pi / n)\), scaled to have integer coefficients. This polynomial has degree 1 if \(n = 1\) or \(n = 2\), and degree \(\phi(n) / 2\) otherwise.

We allow \(n = 0\) and define \(\Psi_0 = 1\).

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 \(O(n^{3/2} / \sqrt{\log n})\).

Dedekind sums

void arith_dedekind_sum_naive(fmpq_t s, const fmpz_t h, const fmpz_t k)
double arith_dedekind_sum_coprime_d(double h, double k)
void arith_dedekind_sum_coprime_large(fmpq_t s, const fmpz_t h, const fmpz_t k)
void arith_dedekind_sum_coprime(fmpq_t s, const fmpz_t h, const fmpz_t k)
void arith_dedekind_sum(fmpq_t s, const fmpz_t h, const fmpz_t k)

These are aliases for the functions in the fmpq module.

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(mp_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 arith_hrr_expsum_factored(trig_prod_t prod, mp_limb_t k, mp_limb_t 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.

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