Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The distribution of totients   New version of my 1998 article. The method of proof of the main results
follows the original, but there are many simplifications/streamlining of
arguments, especially Lemma 3.6 (new Lemma 3.7). Fixed small error in proof of
lower bound for V_k(x) (see the paragraph after (5.20)), fixed the statement
and proof of Theorem 3. New, precise way to relate sums to volumes (Lemmas 3.1,
3.9) and provided full details in Section 6 of the proofs of Theorems 10-13.
Slightly different versions of Theorems 10,11,12,14 (qualitatively the same)
with simpler proofs. Sections 7 and 8 combined (new section 7). A few
definitions, such as S-normal, changed slightly. Some reorganization of
material.

25
The categories of this paper are: patt-sol
20
The categories of this paper are: acc-phys
22
The categories of this paper are: chao-dyn
1
The categories of this paper are: cs
14
The categories of this paper are: math-ph
32
The categories of this paper are: atom-ph
24
The categories of this paper are: dg-ga
3
The categories of this paper are: physics
16
The categories of this paper are: stat
21
The categories of this paper are: adap-org
5
The categories of this paper are: gr-qc
37
The categories of this paper are: supr-con
26
The categories of this paper are: bayes-an
13
The categories of this paper are: quant-ph
6
The categories of this paper are: astro-ph
15
The categories of this paper are: nucl-ex