What does the number three mean in the gen_server docs?

This section is to be read with the gen_server(3) manual page in stdlib

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I think this is some man-pages related convention - these have 9 different sections to be be more readable, so gen_server(3) denotes section 3 (library calls section) of gen_server manual. Run man man for more.

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man man, Where to run this command? Erlang shell?

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No, operating system shell. These days you may have to install man first.

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It’s the UNIX convention for naming entries in the UNIX manual.
% man man
at the UNIX command prompt will tell you about it.
% man 1 tags
will tell you about the UNIX ‘tags’ command.
% man 3erl tags
will tell you about the Erlang ‘tags’ module.
You only really need the number in cases like this where the
same name occurs in multiple sections of the manual.
Erlang has manual pages in sections 1 (programs), 3erl (modules),
and 7 (MIBs).