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

HTML articles powered by AMS MathViewer

- 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

- Michael A. Bennett, Greg Martin, Kevin O’Bryant, and Andrew Rechnitzer,
*Explicit bounds for primes in arithmetic progressions*, Illinois J. Math.**62**(2018), no. 1-4, 427–532. MR**3922423**, DOI 10.1215/ijm/1552442669 - Kevin Ford, Florian Luca, and Pieter Moree,
*Values of the Euler $\phi$-function not divisible by a given odd prime, and the distribution of Euler-Kronecker constants for cyclotomic fields*, Math. Comp.**83**(2014), no. 287, 1447–1476. MR**3167466**, DOI 10.1090/S0025-5718-2013-02749-4 - A. Fröhlich and M. J. Taylor,
*Algebraic number theory*, Cambridge Studies in Advanced Mathematics, vol. 27, Cambridge University Press, Cambridge, 1993. MR**1215934** - G. H. Hardy and E. M. Wright,
*An introduction to the theory of numbers*, 6th ed., Oxford University Press, Oxford, 2008. Revised by D. R. Heath-Brown and J. H. Silverman; With a foreword by Andrew Wiles. MR**2445243** - A. E. Ingham,
*The distribution of prime numbers*, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1990. Reprint of the 1932 original; With a foreword by R. C. Vaughan. MR**1074573** - A. Languasco and A. Zaccagnini,
*A note on Mertens’ formula for arithmetic progressions*, J. Number Theory**127**(2007), no. 1, 37–46. MR**2351662**, DOI 10.1016/j.jnt.2006.12.015 - A. Languasco and A. Zaccagnini,
*On the constant in the Mertens product for arithmetic progressions. II. Numerical values*, Math. Comp.**78**(2009), no. 265, 315–326. MR**2448709**, DOI 10.1090/S0025-5718-08-02148-0 - Alessandro Languasco and Alessandro Zaccagnini,
*Computing the Mertens and Meissel-Mertens constants for sums over arithmetic progressions*, Experiment. Math.**19**(2010), no. 3, 279–284. With an appendix by Karl K. Norton. MR**2743571**, DOI 10.1080/10586458.2010.10390624 - William J. LeVeque,
*Fundamentals of number theory*, Dover Publications, Inc., Mineola, NY, 1996. Reprint of the 1977 original. MR**1382656** - The LMFDB Collaboration,
*The L-functions and Modular Forms Database*, 2017. [Online; accessed 20 October 2017]. *Maple 2017.0*, Maplesoft, a division of Waterloo Maple Inc., Waterloo, Ontario.- Franz Mertens,
*Ein Beitrag zur analytischen Zahlentheorie*, J. Reine Angew. Math.**78**(1874), 46–62 (German). MR**1579612**, DOI 10.1515/crll.1874.78.46 - Kevin S. McCurley,
*Explicit estimates for the error term in the prime number theorem for arithmetic progressions*, Math. Comp.**42**(1984), no. 165, 265–285. MR**726004**, DOI 10.1090/S0025-5718-1984-0726004-6 - F. Johansson and others,
*mpmath: a Python library for arbitrary-precision floating-point arithmetic (version 0.18)*, 2013. http://mpmath.org/. - Hugh L. Montgomery and Robert C. Vaughan,
*Multiplicative number theory. I. Classical theory*, Cambridge Studies in Advanced Mathematics, vol. 97, Cambridge University Press, Cambridge, 2007. MR**2378655** - M. Ram Murty,
*Problems in analytic number theory*, Graduate Texts in Mathematics, vol. 206, Springer-Verlag, New York, 2001. Readings in Mathematics. MR**1803093**, DOI 10.1007/978-1-4757-3441-6 - Władysław Narkiewicz,
*The development of prime number theory*, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2000. From Euclid to Hardy and Littlewood. MR**1756780**, DOI 10.1007/978-3-662-13157-2 - Jean-Louis Nicolas,
*Petites valeurs de la fonction d’Euler*, J. Number Theory**17**(1983), no. 3, 375–388 (French, with English summary). MR**724536**, DOI 10.1016/0022-314X(83)90055-0 - Jean-Louis Nicolas,
*Small values of the Euler function and the Riemann hypothesis*, Acta Arith.**155**(2012), no. 3, 311–321. MR**2983456**, DOI 10.4064/aa155-3-7 - David J. Platt,
*Numerical computations concerning the GRH*, Math. Comp.**85**(2016), no. 302, 3009–3027. MR**3522979**, DOI 10.1090/mcom/3077 - D. J. Platt and O. Ramaré,
*Explicit estimates: from $\Lambda (n)$ in arithmetic progressions to $\Lambda (n)/n$*, Exp. Math.**26**(2017), no. 1, 77–92. MR**3599008**, DOI 10.1080/10586458.2015.1123124 - Olivier Ramaré and Robert Rumely,
*Primes in arithmetic progressions*, Math. Comp.**65**(1996), no. 213, 397–425. MR**1320898**, DOI 10.1090/S0025-5718-96-00669-2 - J. Barkley Rosser and Lowell Schoenfeld,
*Approximate formulas for some functions of prime numbers*, Illinois J. Math.**6**(1962), 64–94. MR**137689** - The Sage Developers,
*Sagemath, the Sage Mathematics Software System (Version 8.3)*, 2018. http://www.sagemath.org. - Kenneth S. Williams,
*Mertens’ theorem for arithmetic progressions*, J. Number Theory**6**(1974), 353–359. MR**364137**, DOI 10.1016/0022-314X(74)90032-8

## 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