sagemathgh-39391: Add wrapper for FLINT's acb_dirichlet_zeta_zeros

    
New method `ComplexBallField.zeta_zeros` that wraps
`acb_dirichlet_zeta_zeros` for computing zeros of the Rieman ζ function.

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URL: sagemath#39391
Reported by: Pierre Lairez
Reviewer(s): Marc Mezzarobba

src/sage/rings/complex_arb.pyx

@@ -168,6 +168,7 @@ from sage.libs.flint.acb_hypgeom cimport *
 from sage.libs.flint.acb_elliptic cimport *
 from sage.libs.flint.acb_modular cimport *
 from sage.libs.flint.acb_poly cimport *
+from sage.libs.flint.acb_dirichlet cimport *
 from sage.libs.flint.arf cimport arf_init, arf_get_d, arf_get_mpfr, arf_clear, arf_set, arf_is_nan
 from sage.libs.flint.mag cimport (mag_init, mag_clear, mag_set_d,
         MAG_BITS, mag_zero, mag_set_ui_2exp_si,
@@ -1258,6 +1259,70 @@ class ComplexBallField(UniqueRepresentation, sage.rings.abc.ComplexBallField):
 
         return res
 
+    def zeta_zeros(self, count, start=1):
+        r"""
+        Compute consecutive zeros of the Riemann zeta function.
+
+        INPUT:
+
+        - ``count`` -- positive integer; number of zeros to be computed, must fit in a machine integer
+
+        - ``start`` -- positive integer (default: 1); index of the first zero to be computed
+
+        OUTPUT:
+
+        A list of ``count`` consecutive zeros of the Riemann zeta function, starting from the ``start``-th zero.
+        Indexing starts at one, following usual mathematical notations.
+
+        EXAMPLES::
+
+            sage: CBF.zeta_zeros(10)
+            [0.5000000000000000 + [14.134725141734...]*I,
+            0.5000000000000000 + [21.0220396387715...]*I,
+            0.5000000000000000 + [25.010857580145...]*I,
+            0.5000000000000000 + [30.4248761258595...]*I,
+            0.5000000000000000 + [32.935061587739...]*I,
+            0.5000000000000000 + [37.586178158825...]*I,
+            0.5000000000000000 + [40.918719012147...]*I,
+            0.5000000000000000 + [43.32707328091...]*I,
+            0.5000000000000000 + [48.005150881167...]*I,
+            0.5000000000000000 + [49.773832477672...]*I]
+
+            sage: CBF.zeta_zeros(6, start=5)
+            [0.5000000000000000 + [32.935061587739...]*I,
+            0.5000000000000000 + [37.586178158825...]*I,
+            0.5000000000000000 + [40.918719012147...]*I,
+            0.5000000000000000 + [43.32707328091...]*I,
+            0.5000000000000000 + [48.005150881167...]*I,
+            0.5000000000000000 + [49.773832477672...]*I]
+        """
+        cdef fmpz_t _start
+        fmpz_init(_start)
+        fmpz_set_mpz(_start, (<Integer> Integer(start)).value)
+
+        cdef long _count = count
+        if _count < 1:
+            raise ValueError("count must be positive")
+
+        cdef acb_ptr ar = _acb_vec_init(_count)
+
+        sig_on()
+        acb_dirichlet_zeta_zeros(ar, _start, _count, self._prec)
+        sig_off()
+
+        res = []
+        cdef ComplexBall b
+        for i in range(_count):
+            b = ComplexBall.__new__(ComplexBall)
+            b._parent = self
+            acb_swap(b.value, &ar[i])
+            res.append(b)
+
+        _acb_vec_clear(ar, _count)
+        fmpz_clear(_start)
+
+        return res
+
 
 cdef inline bint _do_sig(long prec) noexcept:
     """