The distribution of totients

TL;DR

This paper refines and simplifies the proofs of key results on the distribution of Euler's totient function, providing clearer arguments, fixing minor errors, and offering new precise methods to relate sums to volumes.

Contribution

It presents a streamlined version of the 1998 article with improved proofs, corrected errors, and new methods for analyzing the distribution of totients.

Findings

Simplified proofs of main theorems

Corrected a small error in the lower bound proof

Introduced a new precise method relating sums to volumes

Abstract

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.