diff --git a/acb_dirichlet.h b/acb_dirichlet.h
index 0a6e15e7..7e2338dc 100644
--- a/acb_dirichlet.h
+++ b/acb_dirichlet.h
@@ -45,8 +45,6 @@ void acb_dirichlet_zeta(acb_t res, const acb_t s, slong prec);
void acb_dirichlet_zeta_jet_rs(acb_ptr res, const acb_t s, slong len, slong prec);
void acb_dirichlet_zeta_jet(acb_t res, const acb_t s, int deflate, slong len, slong prec);
-void acb_riemann_xi(acb_t res, const acb_t s, slong prec);
-
void acb_dirichlet_hurwitz(acb_t res, const acb_t s, const acb_t a, slong prec);
typedef struct
@@ -75,6 +73,8 @@ void _acb_dirichlet_euler_product_real_ui(arb_t res, ulong s,
void acb_dirichlet_eta(acb_t res, const acb_t s, slong prec);
+void acb_dirichlet_xi(acb_t res, const acb_t s, slong prec);
+
void acb_dirichlet_pairing(acb_t res, const dirichlet_group_t G, ulong m, ulong n, slong prec);
void acb_dirichlet_pairing_char(acb_t res, const dirichlet_group_t G, const dirichlet_char_t a, const dirichlet_char_t b, slong prec);
diff --git a/acb_dirichlet/riemann_xi.c b/acb_dirichlet/riemann_xi.c
deleted file mode 100644
index de3e9245..00000000
--- a/acb_dirichlet/riemann_xi.c
+++ /dev/null
@@ -1,51 +0,0 @@
-/*
- Copyright (C) 2018 D.H.J Polymath
-
- This file is part of Arb.
-
- Arb is free software: you can redistribute it and/or modify it under
- the terms of the GNU Lesser General Public License (LGPL) as published
- by the Free Software Foundation; either version 2.1 of the License, or
- (at your option) any later version. See .
-*/
-
-#include "acb_dirichlet.h"
-
-void acb_riemann_xi(acb_t res, const acb_t s, slong prec)
-{
- acb_t pi, z1, z2, z3, z4;
-
- acb_init(pi);
- acb_const_pi(pi, prec);
-
- /* s*(s-1)/2 */
- acb_init(z1);
- acb_sub_ui(z1, s, 1, prec);
- acb_mul(z1, z1, s, prec);
- acb_mul_2exp_si(z1, z1, -1);
-
- /* pi^(-s/2) */
- acb_init(z2);
- acb_mul_2exp_si(z2, s, -1);
- acb_neg(z2, z2);
- acb_pow(z2, pi, z2, prec);
-
- /* gamma(s/2) */
- acb_init(z3);
- acb_mul_2exp_si(z3, s, -1);
- acb_gamma(z3, z3, prec);
-
- /* zeta(s) */
- acb_init(z4);
- acb_zeta(z4, s, prec);
-
- acb_mul(res, z1, z2, prec);
- acb_mul(res, res, z3, prec);
- acb_mul(res, res, z4, prec);
-
- acb_clear(pi);
- acb_clear(z1);
- acb_clear(z2);
- acb_clear(z3);
- acb_clear(z4);
-}
diff --git a/acb_dirichlet/test/t-riemann_xi.c b/acb_dirichlet/test/t-xi.c
similarity index 75%
rename from acb_dirichlet/test/t-riemann_xi.c
rename to acb_dirichlet/test/t-xi.c
index da2a5fe1..a3819631 100644
--- a/acb_dirichlet/test/t-riemann_xi.c
+++ b/acb_dirichlet/test/t-xi.c
@@ -16,7 +16,7 @@ int main()
slong iter;
flint_rand_t state;
- flint_printf("riemann_xi....");
+ flint_printf("xi....");
fflush(stdout);
flint_randinit(state);
@@ -36,8 +36,8 @@ int main()
arb_randtest_precise(acb_realref(a), state, 1 + n_randint(state, 1000), 3);
arb_randtest_precise(acb_imagref(a), state, 1 + n_randint(state, 1000), 3);
- acb_riemann_xi(b, a, prec1);
- acb_riemann_xi(c, a, prec2);
+ acb_dirichlet_xi(b, a, prec1);
+ acb_dirichlet_xi(c, a, prec2);
if (!acb_overlaps(b, c))
{
@@ -48,10 +48,22 @@ int main()
flint_abort();
}
- /* check riemann_xi(s) = riemann_xi(1-s) */
+ acb_set(c, a);
+ acb_dirichlet_xi(c, c, prec1);
+
+ if (!acb_equal(b, c))
+ {
+ flint_printf("FAIL: aliasing\n\n");
+ flint_printf("a = "); acb_printd(a, 30); flint_printf("\n\n");
+ flint_printf("b = "); acb_printd(b, 30); flint_printf("\n\n");
+ flint_printf("c = "); acb_printd(c, 30); flint_printf("\n\n");
+ flint_abort();
+ }
+
+ /* check xi(s) = xi(1-s) */
acb_sub_ui(c, a, 1, prec1);
acb_neg(c, c);
- acb_riemann_xi(c, c, prec1);
+ acb_dirichlet_xi(c, c, prec1);
if (!acb_overlaps(b, c))
{
diff --git a/acb_dirichlet/xi.c b/acb_dirichlet/xi.c
new file mode 100644
index 00000000..26789be2
--- /dev/null
+++ b/acb_dirichlet/xi.c
@@ -0,0 +1,68 @@
+/*
+ Copyright (C) 2018 D.H.J Polymath
+
+ This file is part of Arb.
+
+ Arb is free software: you can redistribute it and/or modify it under
+ the terms of the GNU Lesser General Public License (LGPL) as published
+ by the Free Software Foundation; either version 2.1 of the License, or
+ (at your option) any later version. See .
+*/
+
+#include "acb_dirichlet.h"
+
+static void
+_acb_dirichlet_xi(acb_t res, const acb_t s, slong prec)
+{
+ acb_t pi, z1, z2, z3;
+
+ acb_init(pi);
+ acb_init(z1);
+ acb_init(z2);
+ acb_init(z3);
+
+ /* (s-1) * pi^(-s/2) * gamma(1 + s/2) * zeta(s) */
+ acb_const_pi(pi, prec);
+ acb_mul_2exp_si(z1, s, -1);
+ acb_neg(z1, z1);
+ acb_pow(z1, pi, z1, prec);
+ acb_mul_2exp_si(z2, s, -1);
+ acb_add_ui(z2, z2, 1, prec);
+ acb_gamma(z2, z2, prec);
+ acb_zeta(z3, s, prec);
+ acb_sub_ui(res, s, 1, prec);
+ acb_mul(res, res, z1, prec);
+ acb_mul(res, res, z2, prec);
+ acb_mul(res, res, z3, prec);
+
+ acb_clear(pi);
+ acb_clear(z1);
+ acb_clear(z2);
+ acb_clear(z3);
+}
+
+/* todo: interval containing 1 */
+void acb_dirichlet_xi(acb_t res, const acb_t s, slong prec)
+{
+ if (!acb_is_finite(s))
+ {
+ acb_indeterminate(res);
+ }
+ else if (acb_is_one(s))
+ {
+ acb_one(res);
+ acb_mul_2exp_si(res, res, -1);
+ }
+ else if (arf_sgn(arb_midref(acb_realref(s))) < 0)
+ {
+ acb_t t;
+ acb_sub_ui(t, s, 1, prec);
+ acb_neg(t, t);
+ _acb_dirichlet_xi(res, t, prec);
+ acb_clear(t);
+ }
+ else
+ {
+ _acb_dirichlet_xi(res, s, prec);
+ }
+}
diff --git a/doc/source/acb_dirichlet.rst b/doc/source/acb_dirichlet.rst
index 24cf9aa3..b5e25916 100644
--- a/doc/source/acb_dirichlet.rst
+++ b/doc/source/acb_dirichlet.rst
@@ -123,6 +123,12 @@ Riemann zeta function
Note that the alternating character `\{1,-1\}` is not itself
a Dirichlet character.
+.. function:: void acb_dirichlet_xi(acb_t res, const acb_t s, slong prec)
+
+ Sets *res* to the Riemann xi function
+ `\xi(s) = \frac{1}{2} s (s-1) \pi^{-s/2} \Gamma(\frac{1}{2} s) \zeta(s)`.
+ The functional equation for xi is `\xi(1-s) = \xi(s)`.
+
Riemann-Siegel formula
-------------------------------------------------------------------------------