.. _arith:

**arith.h** -- arithmetic and special functions
================================================================================


Primorials
--------------------------------------------------------------------------------


.. function:: 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
--------------------------------------------------------------------------------


.. function:: 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
--------------------------------------------------------------------------------


.. function:: void arith_stirling_number_1u(fmpz_t s, slong n, slong k)

.. function:: void arith_stirling_number_1(fmpz_t s, slong n, slong k)

.. function:: 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

    .. math ::

        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)}

    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.

.. function:: void arith_stirling_number_1u_vec(fmpz * row, slong n, slong klen)

.. function:: void arith_stirling_number_1_vec(fmpz * row, slong n, slong klen)

.. function:: 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`.

.. function:: void arith_stirling_number_1u_vec_next(fmpz * row, fmpz * prev, slong n, slong klen)

.. function:: void arith_stirling_number_1_vec_next(fmpz * row, fmpz * prev, slong n, slong klen)

.. function:: 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.

.. function:: void arith_stirling_matrix_1u(fmpz_mat_t mat)

.. function:: void arith_stirling_matrix_1(fmpz_mat_t mat)

.. function:: 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
--------------------------------------------------------------------------------


.. function:: 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.

.. function:: 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.

.. function:: 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``.

.. function:: 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`.

.. function:: 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.

.. function:: 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``.

.. function:: 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

    .. math ::

        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`.

.. function:: 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.

.. function:: 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.

.. function:: 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`.

.. function:: 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
--------------------------------------------------------------------------------


.. function:: 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``.

.. function:: 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.

.. function:: 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`.

.. function:: 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.

.. function:: 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.

.. function:: 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}`.

.. function:: 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.

.. function:: 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.

.. function:: 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] http://en.wikipedia.org/w/index.php?
    %    title=Bernoulli_number&oldid=405938876

.. function:: 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``.

.. function:: 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 

    .. math ::

        \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.`

.. function:: 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.

.. function:: 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.

.. function:: 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}`.

.. function:: void arith_euler_polynomial(fmpq_poly_t poly, ulong n)

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

    .. math ::

        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.

.. function:: 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
--------------------------------------------------------------------------------


.. function:: 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.

.. function:: 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.

.. function:: 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.

.. function:: 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


    .. math ::


        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
--------------------------------------------------------------------------------


.. function:: 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
    (http://mathworld.wolfram.com/TrigonometryAngles.html ).

.. function:: 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
--------------------------------------------------------------------------------


.. 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
--------------------------------------------------------------------------------


.. function:: 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
--------------------------------------------------------------------------------


.. function:: 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.

.. function:: 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.

.. function:: void arith_hrr_expsum_factored(trig_prod_t prod, mp_limb_t k, mp_limb_t n)

    Symbolically evaluates the exponential sum

    .. math ::

        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`.

.. function:: 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

    .. math ::

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

    where

    .. math ::

        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)

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

    .. math ::

        |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.

.. function:: 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
--------------------------------------------------------------------------------


.. function:: 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.

.. function:: 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.

    .. math ::

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

