[x64] Reductions are too eager to promote to 64-bit
```python
In [1]: import jax.numpy as jnp

In [2]: jnp.arange(4, dtype='int8').sum()
Out[2]: DeviceArray(6, dtype=int64)
```

The root cause is this line: https://github.com/google/jax/blob/d849f495193c69e8c133d0c6a424ba80d72938cb/jax/_src/numpy/reductions.py#L81

It's not an issue for floating-point types, because numpy does not promote those in most cases
diff --git a/jax/_src/numpy/reductions.py b/jax/_src/numpy/reductions.py
index 00834edfd..40848eb62 100644
--- a/jax/_src/numpy/reductions.py
+++ b/jax/_src/numpy/reductions.py
@@ -26,7 +26,7 @@ from jax import lax
 from jax._src import api
 from jax._src import dtypes
 from jax._src.numpy.ndarray import ndarray
-from jax._src.numpy.util import _broadcast_to, _check_arraylike, _complex_elem_type, _promote_dtypes_inexact, _where, _wraps
+from jax._src.numpy.util import _broadcast_to, _check_arraylike, _complex_elem_type, _promote_dtypes_inexact, _promote_dtypes_numeric, _where, _wraps
 from jax._src.lax import lax as lax_internal
 from jax._src.util import canonicalize_axis as _canonicalize_axis, maybe_named_axis
 
@@ -62,7 +62,7 @@ def _upcast_f16(dtype):
 def _reduction(a, name, np_fun, op, init_val, has_identity=True,
                preproc=None, bool_op=None, upcast_f16_for_computation=False,
                axis=None, dtype=None, out=None, keepdims=False, initial=None,
-               where_=None, parallel_reduce=None):
+               where_=None, parallel_reduce=None, promote_integers=False):
   bool_op = bool_op or op
   # Note: we must accept out=None as an argument, because numpy reductions delegate to
   # object methods. For example `np.sum(x)` will call `x.sum()` if the `sum()` method
@@ -86,7 +86,18 @@ def _reduction(a, name, np_fun, op, init_val, has_identity=True,
     if not _all(core.greater_equal_dim(shape[d], 1) for d in pos_dims):
       raise ValueError(f"zero-size array to reduction operation {name} which has no identity")
 
-  result_dtype = dtypes.canonicalize_dtype(dtype or dtypes.dtype(np_fun(np.ones((), dtype=dtypes.dtype(a)))))
+  result_dtype = dtypes.canonicalize_dtype(dtype or dtypes.dtype(a))
+
+  # promote_integers=True matches NumPy's behavior for sum() and prod(), which promotes
+  # all int-like inputs to the widest available dtype.
+  if dtype is None and promote_integers:
+    if dtypes.issubdtype(result_dtype, np.bool_):
+      result_dtype = dtypes.canonicalize_dtype(np.int64)
+    elif dtypes.issubdtype(result_dtype, np.unsignedinteger):
+      result_dtype = dtypes.canonicalize_dtype(np.uint64)
+    elif dtypes.issubdtype(result_dtype, np.integer):
+      result_dtype = dtypes.canonicalize_dtype(np.int64)
+
   if upcast_f16_for_computation and dtypes.issubdtype(result_dtype, np.inexact):
     computation_dtype = _upcast_f16(result_dtype)
   else:
@@ -146,6 +157,9 @@ def _cast_to_bool(operand):
     warnings.filterwarnings("ignore", category=np.ComplexWarning)
     return lax.convert_element_type(operand, np.bool_)
 
+def _cast_to_numeric(operand):
+  return _promote_dtypes_numeric(operand)[0]
+
 
 def _ensure_optional_axes(x):
   def force(x):
@@ -159,34 +173,46 @@ def _ensure_optional_axes(x):
     force, x, "The axis argument must be known statically.")
 
 
-@partial(api.jit, static_argnames=('axis', 'dtype', 'keepdims'), inline=True)
-def _reduce_sum(a, axis: Optional[Union[int, Tuple[int, ...]]] = None,
-                dtype=None, out=None, keepdims=None, initial=None, where=None):
-  return _reduction(a, "sum", np.sum, lax.add, 0,
+# TODO(jakevdp) change promote_integers default to False
+_PROMOTE_INTEGERS_DOC = """
+promote_integers : bool, default=True
+    If True, then integer inputs will be promoted to the widest available integer
+    dtype, following numpy's behavior. If False, the result will have the same dtype
+    as the input. ``promote_integers`` is ignored if ``dtype`` is specified.
+"""
+
+
+@partial(api.jit, static_argnames=('axis', 'dtype', 'keepdims', 'promote_integers'), inline=True)
+def _reduce_sum(a, axis: Optional[Union[int, Tuple[int, ...]]] = None, dtype=None,
+                out=None, keepdims=None, initial=None, where=None, promote_integers=True):
+  return _reduction(a, "sum", np.sum, lax.add, 0, preproc=_cast_to_numeric,
                     bool_op=lax.bitwise_or, upcast_f16_for_computation=True,
                     axis=axis, dtype=dtype, out=out, keepdims=keepdims,
-                    initial=initial, where_=where, parallel_reduce=lax.psum)
+                    initial=initial, where_=where, parallel_reduce=lax.psum,
+                    promote_integers=promote_integers)
 
-@_wraps(np.sum, skip_params=['out'])
+@_wraps(np.sum, skip_params=['out'], extra_params=_PROMOTE_INTEGERS_DOC)
 def sum(a, axis: Optional[Union[int, Tuple[int, ...]]] = None, dtype=None,
-        out=None, keepdims=None, initial=None, where=None):
+        out=None, keepdims=None, initial=None, where=None, promote_integers=True):
   return _reduce_sum(a, axis=_ensure_optional_axes(axis), dtype=dtype, out=out,
-                     keepdims=keepdims, initial=initial, where=where)
+                     keepdims=keepdims, initial=initial, where=where,
+                     promote_integers=promote_integers)
 
 
-@partial(api.jit, static_argnames=('axis', 'dtype', 'keepdims'), inline=True)
-def _reduce_prod(a, axis: Optional[Union[int, Tuple[int, ...]]] = None,
-                 dtype=None, out=None, keepdims=None, initial=None, where=None):
-  return _reduction(a, "prod", np.prod, lax.mul, 1,
+@partial(api.jit, static_argnames=('axis', 'dtype', 'keepdims', 'promote_integers'), inline=True)
+def _reduce_prod(a, axis: Optional[Union[int, Tuple[int, ...]]] = None, dtype=None,
+                 out=None, keepdims=None, initial=None, where=None, promote_integers=True):
+  return _reduction(a, "prod", np.prod, lax.mul, 1, preproc=_cast_to_numeric,
                     bool_op=lax.bitwise_and, upcast_f16_for_computation=True,
                     axis=axis, dtype=dtype, out=out, keepdims=keepdims,
-                    initial=initial, where_=where)
+                    initial=initial, where_=where, promote_integers=promote_integers)
 
-@_wraps(np.prod, skip_params=['out'])
+@_wraps(np.prod, skip_params=['out'], extra_params=_PROMOTE_INTEGERS_DOC)
 def prod(a, axis: Optional[Union[int, Tuple[int, ...]]] = None, dtype=None,
-         out=None, keepdims=None, initial=None, where=None):
+         out=None, keepdims=None, initial=None, where=None, promote_integers=True):
   return _reduce_prod(a, axis=_ensure_optional_axes(axis), dtype=dtype,
-                      out=out, keepdims=keepdims, initial=initial, where=where)
+                      out=out, keepdims=keepdims, initial=initial, where=where,
+                      promote_integers=promote_integers)
 
 
 @partial(api.jit, static_argnames=('axis', 'keepdims'), inline=True)
diff --git a/tests/lax_numpy_test.py b/tests/lax_numpy_test.py
index c89dcfaa5..eaa4cf931 100644
--- a/tests/lax_numpy_test.py
+++ b/tests/lax_numpy_test.py
@@ -407,6 +407,11 @@ JAX_REDUCER_NO_DTYPE_RECORDS = [
     op_record("ptp", 1, number_dtypes, nonempty_shapes, jtu.rand_default, []),
 ]
 
+JAX_REDUCER_PROMOTE_INT_RECORDS = [
+    op_record("prod", 1, all_dtypes, all_shapes, jtu.rand_small_positive, []),
+    op_record("sum", 1, all_dtypes, all_shapes, jtu.rand_default, []),
+]
+
 JAX_ARGMINMAX_RECORDS = [
     op_record("argmin", 1, default_dtypes, nonempty_shapes, jtu.rand_some_equal, []),
     op_record("argmax", 1, default_dtypes, nonempty_shapes, jtu.rand_some_equal, []),
@@ -941,6 +946,49 @@ class LaxBackedNumpyTests(jtu.JaxTestCase):
     self._CheckAgainstNumpy(np_fun, jnp_fun, args_maker, rtol=tol)
     self._CompileAndCheck(jnp_fun, args_maker)
 
+  @parameterized.named_parameters(itertools.chain.from_iterable(
+      jtu.cases_from_list(
+        {"testcase_name": "{}_inshape={}_axis={}_keepdims={}_initial={}_promote_integers={}".format(
+            rec.test_name.capitalize(),
+            jtu.format_shape_dtype_string(shape, dtype), axis, keepdims, initial, promote_integers),
+        "rng_factory": rec.rng_factory, "shape": shape, "dtype": dtype,
+        "np_op": getattr(np, rec.name), "jnp_op": getattr(jnp, rec.name),
+        "initial": initial, "axis": axis, "keepdims": keepdims, "inexact": rec.inexact,
+        "promote_integers": promote_integers}
+        for shape in rec.shapes for dtype in rec.dtypes
+        for axis in list(range(-len(shape), len(shape))) + [None]
+        for initial in [0, 1] for keepdims in [False, True]
+        for promote_integers in [True, False]
+        if jtu.is_valid_shape(shape, dtype))
+      for rec in JAX_REDUCER_PROMOTE_INT_RECORDS))
+  def testReducerPromoteInt(self, np_op, jnp_op, rng_factory, shape, dtype, axis,
+                            keepdims, initial, inexact, promote_integers):
+    rng = rng_factory(self.rng())
+    is_bf16_nan_test = dtype == jnp.bfloat16 and rng_factory.__name__ == 'rand_some_nan'
+    @jtu.ignore_warning(category=RuntimeWarning,
+                        message="Degrees of freedom <= 0 for slice.*")
+    @jtu.ignore_warning(category=np.ComplexWarning)
+    def np_fun(x):
+      x = np.asarray(x)
+      if inexact:
+        x = x.astype(dtypes.to_inexact_dtype(x.dtype))
+      x_cast = x if not is_bf16_nan_test else x.astype(np.float32)
+      res = np_op(x_cast, axis, keepdims=keepdims, initial=initial)
+      res = res if not is_bf16_nan_test else res.astype(jnp.bfloat16)
+      print(f"res.dtype = {res.dtype}")
+      if not promote_integers and dtypes.issubdtype(res.dtype, np.integer):
+        res = res.astype(dtypes.to_numeric_dtype(x.dtype))
+      return res
+
+    jnp_fun = lambda x: jnp_op(x, axis, keepdims=keepdims, initial=initial, promote_integers=promote_integers)
+    jnp_fun = jtu.ignore_warning(category=jnp.ComplexWarning)(jnp_fun)
+    args_maker = lambda: [rng(shape, dtype)]
+    tol = {jnp.bfloat16: 3E-2}
+    print(jnp_fun(*args_maker()))
+    print(np_fun(*args_maker()))
+    self._CheckAgainstNumpy(np_fun, jnp_fun, args_maker, rtol=tol)
+    self._CompileAndCheck(jnp_fun, args_maker)
+
   @parameterized.named_parameters(itertools.chain.from_iterable(
       jtu.cases_from_list(
         {"testcase_name": "{}_inshape={}_axis={}_keepdims={}".format(
['tests/lax_numpy_test.py', 'jax/_src/numpy/reductions.py']