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Mathematics of Computation

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Euler's function on products of primes in a fixed arithmetic progression


Authors: Amir Akbary and Forrest J. Francis
Journal: Math. Comp. 89 (2020), 993-1026
MSC (2010): Primary 11N37, 11M26, 11N56
DOI: https://doi.org/10.1090/mcom/3463
Published electronically: September 5, 2019
MathSciNet review: 4044459
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Abstract: We study generalizations of some results of Jean-Louis Nicolas regarding the relation between small values of Euler's function $ \varphi (n)$ and the Riemann Hypothesis. Let $ C(q, a)$ be the constant appearing in the asymptotic formula

$\displaystyle \prod _{\substack {p \leq x \\ p \equiv a\ \text {(mod\ {q})}}} \... ...- \frac {1}{p}\right ) \sim \frac {C(q, a)}{(\log {x})^\frac {1}{\varphi (q)}},$

as $ x\rightarrow \infty $. Among other things, we prove that for $ 1\leq q\leq 10$ and for $ q=12, 14$, the generalized Riemann Hypothesis for the Dedekind zeta function of the cyclotomic field $ \mathbb{Q}(e^{2\pi i/q})$ is true if and only if for all integers $ k\geq 1$ we have

$\displaystyle \frac {\bar {N}_k}{\varphi (\bar {N}_k)(\log (\varphi (q)\log {\bar {N}_k}))^{\frac {1}{\varphi (q)}}} > \frac {1}{C(q,1)}.$

Here $ \bar {N}_k$ is the product of the first $ k$ primes in the arithmetic progression $ p\equiv 1\ $$ \text {(mod\ {q})}$. We also prove that, for $ q\leq 400,000$ and integers $ a$ coprime to $ q$, the analogous inequality

$\displaystyle \frac {\bar {N}_k}{\varphi (\bar {N}_k)(\log (\varphi (q)\log {\bar {N}_k}))^{\frac {1}{\varphi (q)}}} > \frac {1}{C(q,a)}$

holds for infinitely many values of $ k$. If in addition $ a$ is not a square modulo $ q$, then there are infinitely many $ k$ for which this inequality holds and also infinitely many $ k$ for which this inequality fails.

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Additional Information

Amir Akbary
Affiliation: Department of Mathematics and Computer Science, University of Lethbridge, Lethbridge, Alberta T1K 3M4, Canada
Email: amir.akbary@uleth.ca

Forrest J. Francis
Affiliation: Department of Mathematics and Computer Science, University of Lethbridge, Lethbridge, Alberta T1K 3M4, Canada
Email: f.francis@student.adfa.edu.au

DOI: https://doi.org/10.1090/mcom/3463
Keywords: Small values of Euler's function, arithmetic progressions, generalized Riemann hypothesis
Received by editor(s): November 6, 2018
Received by editor(s) in revised form: April 15, 2019
Published electronically: September 5, 2019
Additional Notes: Research of both authors was partially supported by NSERC
Article copyright: © Copyright 2019 American Mathematical Society