arb_dirichlet_zeta function

Zeta and Related Functions

Compute the Riemann zeta function, the Hurwitz zeta function, or Lerch's transcendent. Lerch's transcendent $\Phi(z, s, a)$

is defined by [REMOVE_ME]$\sum_{k = 0}^{\infty} \frac{z^{k}}{(k + a)^{s}}sum_{k = 0, 1, \ldots} z^k/(k + a)^s [REMOVE_ME_2]$

for $Mod(z) < 1$ and by analytic continuation elsewhere in the $z$-plane. The Riemann and Hurwitz zeta functions are the special cases $\zeta(s) = \Phi(1, s, 1)$ and $\zeta(s, a) = \Phi(1, s, a)$, respectively. See the references for restrictions on $s$ and $a$.

Description

Compute the Riemann zeta function, the Hurwitz zeta function, or Lerch's transcendent. Lerch's transcendent $\Phi(z, s, a)$

is defined by

for $Mod(z) < 1$ and by analytic continuation elsewhere in the $z$-plane. The Riemann and Hurwitz zeta functions are the special cases $\zeta(s) = \Phi(1, s, 1)$ and $\zeta(s, a) = \Phi(1, s, a)$, respectively. See the references for restrictions on $s$ and $a$.

arb_dirichlet_zeta(s, prec = flintPrec())
acb_dirichlet_zeta(s, prec = flintPrec())

arb_dirichlet_hurwitz(s, a = 1, prec = flintPrec())
acb_dirichlet_hurwitz(s, a = 1, prec = flintPrec())

## arb_dirichlet_lerch_phi(z = 1, s, a = 1, prec = flintPrec())
   acb_dirichlet_lerch_phi(z = 1, s, a = 1, prec = flintPrec())

Arguments

• z, s, a: numeric, complex, arb, or acb vectors.
• prec: a numeric or slong vector indicating the desired precision as a number of bits.

Returns

An arb or acb vector storing function values with error bounds. Its length is the maximum of the lengths of the arguments or zero (zero if any argument has length zero). The arguments are recycled as necessary.

See Also

Classes arb and acb.

References

The FLINT documentation of the underlying

functions: https://flintlib.org/doc/acb_dirichlet.html

NIST Digital Library of Mathematical Functions: https://dlmf.nist.gov/25

Examples

dzet <- acb_dirichlet_zeta
dhur <- acb_dirichlet_hurwitz
dler <- acb_dirichlet_lerch_phi

## Somewhat famous particular values :
debugging <- tolower(Sys.getenv("R_FLINT_CHECK_EXTRA")) == "true"
     s <- .acb(x = c(   -1,    0,      2,       4))
zeta.s <- .acb(x = c(-1/12, -1/2, pi^2/6, pi^4/90))
stopifnot(all.equal(dzet(   s   ), zeta.s),
          all.equal(dhur(   s, 1), zeta.s),
          !debugging ||
          {
          print(cbind(as.complex(dler(1, s, 1)), as.complex(zeta.s)))
          all.equal(dler(1, s, 1), zeta.s) # FLINT bug, report this
          })

set.seed(0xabcdL)
r <- 10L
eps <- 0x1p-4
a    <- flint:::complex.runif(r, modulus = c(    0, 1/eps))
z.l1 <- flint:::complex.runif(r, modulus = c(    0, 1-eps))
z.g1 <- flint:::complex.runif(r, modulus = c(1+eps, 1/eps))
z <- .acb(x = c(z.l1, z.g1))

## A relation with the hypergeometric function from
##   http://dlmf.nist.gov/25.14.E3_3 :
h2f1 <- acb_hypgeom_2f1
stopifnot(all.equal(dler(z.l1, 1, a), h2f1(a, 1, a + 1, z.l1)/a))

## TODO: test values also for z[Mod(z) > 1] ...