Abstract
Let φ denote Euler's φ function. For a fixed odd prime q we investigate the first and second order terms of the asymptotic series expansion for the number of n ≤ x such that q φ(n). Part of the analysis involves a careful study of the Euler-Kronecker constants for cyclotomic fields. In particular, we show that the Hardy-Littlewood conjecture about counts of prime k-tuples and a conjecture of Ihara about the distribution of these Euler- Kronecker constants cannot be both true.
Original language | English (US) |
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Pages (from-to) | 1447-1476 |
Number of pages | 30 |
Journal | Mathematics of Computation |
Volume | 83 |
Issue number | 287 |
DOIs | |
State | Published - May 2014 |
ASJC Scopus subject areas
- Algebra and Number Theory
- Computational Mathematics
- Applied Mathematics