bernoulli.h

Set the result to $B_n$ in lowest terms with a positive denominator. The value is copied from the cache if it is there; the cache is not extended. Otherwise odd $n$ and $n < 18000$ use _bernoulli_fmpq_ui_zeta, and larger even $n$ use _bernoulli_fmpq_ui_multi_mod with the default tuning.

Set num and den to the numerator and denominator of $B_n$ in lowest terms, by two methods.

Returns $B_n \bmod p$ for a prime $p < 2^{\mathtt{FLINT\_D\_BITS}}$ ($2^{53}$ on 64-bit machines), computed with Harvey's algorithm [Har2010] in time $O(p \log^{O(1)} p)$, independent of $n$. If $p$ divides the denominator of $B_n$ (that is, $n \ge 2$ even with $(p - 1) \mid n$, or $n = 1$ and $p = 2$), returns UWORD_MAX.

Initialises iter so that the first call to bernoulli_rev_next returns $B_n$. An odd $n$ is replaced by $n - 1$.

Sets numer and denom to the numerator and denominator of $B_k$ in lowest terms, where $B_k$ is the next number of the sequence, and advances iter to $B_{k-2}$. Must not be called after $B_0$ has been returned.

Frees the memory used by iter.

Sets res to the num consecutive Bernoulli numbers $B_a, B_{a+1}, \ldots, B_{a + \mathit{num} - 1}$, without reading or writing the cache. To reuse a vector, use bernoulli_cache_compute and bernoulli_fmpq_ui instead. Throws if $a > 2^{31}$ or $\mathit{num} > 10^9$.

Makes sure that $B_0, \ldots, B_{n-1}$ are cached. The cache grows by at least 32 entries (below 128) or 128 entries at a time; new entries are computed with arith_bernoulli_number_vec up to 128 and with bernoulli_fmpq_vec_no_cache beyond.

Returns an integer $b$ with $|B_n| \le 2^b$, using a table for $n < 512$ and otherwise the inequality $|B_n| < 4\, n!/(2\pi)^n < 4 (n+1)^{n+1} e^{-n}/(2\pi)^n$ evaluated in integer arithmetic with a table of logarithms. Returns WORD_MIN if $B_n = 0$. The bound exceeds the true exponent by at most one percent. Throws if $n > 2^{\mathtt{FLINT\_BITS} - 6}$ or the computation would overflow; the function is meant for $n$ small enough that $B_n$ could be computed exactly.

Returns a crude estimate of the number of bits of the denominator of $B_n$: by von Staudt–Clausen the denominator is at most $n^{\sigma_0(n)}$, and the estimate is $\tfrac12 \log_2(n)\, n^{1.066/\log\log n}$, using $\sigma_0(n) < n^{1.066/\log\log n}$ for large $n$.

Returns the working precision of the iterator for $B_{\mathit{nmax}}$: arith_bernoulli_number_size(nmax) + bernoulli_denom_size(nmax). No guard bits are added, as the denominator estimate is pessimistic.

Returns the number $N$ of terms of the zeta series $\sum_{j \le N} j^{-s}$ needed for an error below $2^{-\mathit{prec}}$: about $2^{(\mathit{prec} + 1)/(s - 1)}$, made odd.

Returns the precision for the term $i^{-s_1}$ in a sum computed to absolute precision $\mathit{wp}$: $\mathit{wp} - s_1 \log_2 i$, at least 10.