found a first working solution

Author: SaturnFromTitan

doc/source/whatsnew/v1.1.0.rst

@@ -56,6 +56,7 @@ Other API changes
 - :meth:`Series.describe` will now show distribution percentiles for ``datetime`` dtypes, statistics ``first`` and ``last``
   will now be ``min`` and ``max`` to match with numeric dtypes in :meth:`DataFrame.describe` (:issue:`30164`)
 - :meth:`Groupby.groups` now returns an abbreviated representation when called on large dataframes (:issue:`1135`)
+- :class:`Series` constructor will default to construct an :class:`Index`, rather than an :class:`RangeIndex` when constructed with empty data, matching the behaviour of ``data=None``.
 
 Backwards incompatible API changes
 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

pandas/core/series.py

@@ -299,7 +299,12 @@ def __init__(
             if index is None:
                 if not is_list_like(data):
                     data = [data]
-                index = ibase.default_index(len(data))
+
+                if is_empty_data(data):
+                    index = Index([])
+                else:
+                    index = ibase.default_index(len(data))
+
             elif is_list_like(data):
 
                 # a scalar numpy array is list-like but doesn't

pandas/tests/groupby/test_grouping.py

@@ -641,7 +641,7 @@ def test_groupby_empty(self):
         gr = s.groupby([])
 
         result = gr.mean()
-        tm.assert_series_equal(result, s)
+        tm.assert_series_equal(result, s, check_index_type=False)
 
         # check group properties
         assert len(gr.grouper.groupings) == 1

pandas/tests/io/json/test_pandas.py

@@ -678,7 +678,7 @@ def test_series_roundtrip_empty(self, orient, numpy):
         else:
             expected.index = expected.index.astype(float)
 
-        tm.assert_series_equal(result, expected)
+        tm.assert_series_equal(result, expected, check_index_type=False)
 
     @pytest.mark.parametrize("numpy", [True, False])
     def test_series_roundtrip_timeseries(self, orient, numpy):

pandas/tests/reshape/test_concat.py

@@ -2202,7 +2202,7 @@ def test_concat_empty_series_timelike(self, tz, values):
             }
         )
         result = concat([first, second], axis=1)
-        tm.assert_frame_equal(result, expected)
+        tm.assert_frame_equal(result, expected, check_index_type=False)
 
     def test_default_index(self):
         # is_series and ignore_index

pandas/tests/series/test_constructors.py

@@ -127,37 +127,40 @@ def test_constructor_empty(self, input_class):
         with tm.assert_produces_warning(DeprecationWarning, check_stacklevel=False):
             empty = Series()
             empty2 = Series(input_class())
-
-        # these are Index() and RangeIndex() which don't compare type equal
-        # but are just .equals
-        tm.assert_series_equal(empty, empty2, check_index_type=False)
+        tm.assert_series_equal(empty, empty2)
+        assert type(empty.index) is Index
 
         # With explicit dtype:
         empty = Series(dtype="float64")
         empty2 = Series(input_class(), dtype="float64")
-        tm.assert_series_equal(empty, empty2, check_index_type=False)
+        tm.assert_series_equal(empty, empty2)
+        assert type(empty.index) is Index
 
         # GH 18515 : with dtype=category:
         empty = Series(dtype="category")
         empty2 = Series(input_class(), dtype="category")
-        tm.assert_series_equal(empty, empty2, check_index_type=False)
+        tm.assert_series_equal(empty, empty2)
+        assert type(empty.index) is Index
 
         if input_class is not list:
             # With index:
             with tm.assert_produces_warning(DeprecationWarning, check_stacklevel=False):
                 empty = Series(index=range(10))
                 empty2 = Series(input_class(), index=range(10))
             tm.assert_series_equal(empty, empty2)
+            assert type(empty.index) is pd.RangeIndex
 
             # With index and dtype float64:
             empty = Series(np.nan, index=range(10))
             empty2 = Series(input_class(), index=range(10), dtype="float64")
             tm.assert_series_equal(empty, empty2)
+            assert type(empty.index) is pd.RangeIndex
 
             # GH 19853 : with empty string, index and dtype str
             empty = Series("", dtype=str, index=range(3))
             empty2 = Series("", index=range(3))
             tm.assert_series_equal(empty, empty2)
+            assert type(empty.index) is pd.RangeIndex
 
     @pytest.mark.parametrize("input_arg", [np.nan, float("nan")])
     def test_constructor_nan(self, input_arg):

pandas/tests/series/test_missing.py

@@ -1151,7 +1151,8 @@ def test_interpolate_corners(self, kwargs):
         s = Series([np.nan, np.nan])
         tm.assert_series_equal(s.interpolate(**kwargs), s)
 
-        s = Series([], dtype=object).interpolate()
+        index = pd.RangeIndex.from_range(range(0, 0))
+        s = Series([], dtype=object, index=index).interpolate()
         tm.assert_series_equal(s.interpolate(**kwargs), s)
 
     def test_interpolate_index_values(self):