These fragments can be gathered together to make amulets, and produce gold when doing so. The gold provided depends on your total skill level and how many fragments you are combining/making into amulets. It can also be increased with a transmutation tablet in your inventory. You will get gold for combining fragments and making amulets.
Right-click 'Combine fragments' takes all of the fragments in your inventory, regardless of size, and attempts to make the highest number fragment it can. For example, if you have the following:
Right-click 'Combine fragments' will give you 3 alchemist's amulet fragment (10) and one alchemist's amulet fragment (5).
Right-click 'Transform fragments' will give you either a shining alchemist's amulet (for fragments below 10) and a radiant alchemist's amulet for an alchemist's amulet fragment (10).
The shining alchemist's amulet is purely decorative and has no benefits, the radiant alchemist's amulet however, will allow you to cast the High-Level Alchemy spell, without the use of runes, up to 15 times a day and offers 1 free teleport to The Grand Exchange per day.
It should be noted that charges are global across all amulets - you cannot do more than 15 uses a day if you have more than 1 amulet.
Before April 17th 2013, a group of 9 or fewer fragments would transform into an alchemist's amulet (uncharged), but transforming a group of 10 fragments would produce an alchemist's amulet (charged). These two amulets can no longer be made but are still usable for players who own them.