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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Euler’s function on products of primes in a fixed arithmetic progression
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by Amir Akbary and Forrest J. Francis HTML | PDF
Math. Comp. 89 (2020), 993-1026 Request permission

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 \[ \prod _{\substack {p \leq x \\ p \equiv a\ \text {(mod\ {q})}}} \left (1 - \frac {1}{p}\right ) \sim \frac {C(q, a)}{(\log {x})^\frac {1}{\varphi (q)}},\] 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 \[ \frac {\overline {N}_k}{\varphi (\overline {N}_k)(\log (\varphi (q)\log {\overline {N}_k}))^{\frac {1}{\varphi (q)}}} > \frac {1}{C(q,1)}.\] Here $\overline {N}_k$ 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 \[ \frac {\overline {N}_k}{\varphi (\overline {N}_k)(\log (\varphi (q)\log {\overline {N}_k}))^{\frac {1}{\varphi (q)}}} > \frac {1}{C(q,a)}\] 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.
References
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Additional Information
  • Amir Akbary
  • Affiliation: Department of Mathematics and Computer Science, University of Lethbridge, Lethbridge, Alberta T1K 3M4, Canada
  • MR Author ID: 650700
  • 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
  • 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
  • © Copyright 2019 American Mathematical Society
  • Journal: Math. Comp. 89 (2020), 993-1026
  • MSC (2010): Primary 11N37, 11M26, 11N56
  • DOI: https://doi.org/10.1090/mcom/3463
  • MathSciNet review: 4044459