## A fast algorithm to compute the Ramanujan-Deninger gamma function and some number-theoretic applications

HTML articles powered by AMS MathViewer

- by
Alessandro Languasco and Luca Righi
**HTML**| PDF - Math. Comp.
**90**(2021), 2899-2921 Request permission

## Abstract:

We introduce a fast algorithm to compute the Ramanujan-Deninger gamma function and its logarithmic derivative at positive values. Such an algorithm allows us to greatly extend the numerical investigations about the Euler-Kronecker constants $\mathfrak {G}_q$, $\mathfrak {G}_q^+$ and $M_q=\max _{\chi \ne \chi _0} \vert L^\prime /L(1,\chi )\vert$, where $q$ is an odd prime, $\chi$ runs over the primitive Dirichlet characters $\bmod \ q$, $\chi _0$ is the principal Dirichlet character $\bmod \ q$ and $L(s,\chi )$ is the Dirichlet $L$-function associated to $\chi$. Using such algorithms we obtained that $\mathfrak {G}_{50 040 955 631} =-0.16595399\dotsc$ and $\mathfrak {G}_{50 040 955 631}^+ =13.89764738\dotsc$ thus getting a new negative value for $\mathfrak {G}_q$.

Moreover we also computed $\mathfrak {G}_q$, $\mathfrak {G}_q^+$ and $M_q$ for every prime $q$, $10^6< q\le {10^{7}}$, thus extending the results by Languasco [Res. Number Theory 7 (2021), no. 1, Paper No. 2]. As a consequence we obtain that both $\mathfrak {G}_q$ and $\mathfrak {G}_q^+$ are positive for every odd prime $q$ up to ${10^{7}}$ and that $\frac {17}{20} \log \log q< M_q < \frac {5}{4} \log \log q$ for every prime $1531 < q\le {10^{7}}$. In fact the lower bound holds true for $q>13$. The programs used and the results here described are collected at the following address http://www.math.unipd.it/ languasc/Scomp-appl.html.

## References

- Bruce C. Berndt,
*Ramanujan’s notebooks. Part I*, Springer-Verlag, New York, 1985. With a foreword by S. Chandrasekhar. MR**781125**, DOI 10.1007/978-1-4612-1088-7 - Henri Cohen,
*Number theory. Vol. II. Analytic and modern tools*, Graduate Texts in Mathematics, vol. 240, Springer, New York, 2007. MR**2312338** - Christopher Deninger,
*On the analogue of the formula of Chowla and Selberg for real quadratic fields*, J. Reine Angew. Math.**351**(1984), 171–191. MR**749681**, DOI 10.1515/crll.1984.351.171 - Karl Dilcher,
*Generalized Euler constants for arithmetical progressions*, Math. Comp.**59**(1992), no. 199, 259–282, S21–S24. MR**1134726**, DOI 10.1090/S0025-5718-1992-1134726-5 - Karl Dilcher,
*On generalized gamma functions related to the Laurent coefficients of the Riemann zeta function*, Aequationes Math.**48**(1994), no. 1, 55–85. MR**1277891**, DOI 10.1007/BF01837979 - 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 - M. Frigo and S. G. Johnson,
*The design and implementation of FFTW3*, Proceedings of the IEEE**93**(2005), 216–231. The C library is available at http://www.fftw.org. - Louis Gordon,
*A stochastic approach to the gamma function*, Amer. Math. Monthly**101**(1994), no. 9, 858–865. MR**1300491**, DOI 10.2307/2975134 - Nicholas J. Higham,
*The accuracy of floating point summation*, SIAM J. Sci. Comput.**14**(1993), no. 4, 783–799. MR**1223274**, DOI 10.1137/0914050 - Y. Ihara,
*The Euler-Kronecker invariants in various families of global fields*, in V. Ginzburg, ed., Algebraic Geometry and Number Theory: In Honor of Vladimir Drinfeld’s 50th Birthday, Progress in Mathematics**850**, Birkhäuser Boston, Cambridge, MA, 2006, 407–451. - Yasutaka Ihara,
*On “$M$-functions” closely related to the distribution of $L’/L$-values*, Publ. Res. Inst. Math. Sci.**44**(2008), no. 3, 893–954. MR**2451613** - Yasutaka Ihara, V. Kumar Murty, and Mahoro Shimura,
*On the logarithmic derivatives of Dirichlet $L$-functions at $s=1$*, Acta Arith.**137**(2009), no. 3, 253–276. MR**2496464**, DOI 10.4064/aa137-3-6 - Fredrik Johansson and Iaroslav V. Blagouchine,
*Computing Stieltjes constants using complex integration*, Math. Comp.**88**(2019), no. 318, 1829–1850. MR**3925487**, DOI 10.1090/mcom/3401 - W. Kahan,
*Further remarks on reducing truncation errors*, Communications of the ACM**8**(1965), 40. - Jeffrey C. Lagarias,
*Euler’s constant: Euler’s work and modern developments*, Bull. Amer. Math. Soc. (N.S.)**50**(2013), no. 4, 527–628. MR**3090422**, DOI 10.1090/S0273-0979-2013-01423-X - Youness Lamzouri,
*The distribution of Euler-Kronecker constants of quadratic fields*, J. Math. Anal. Appl.**432**(2015), no. 2, 632–653. MR**3378382**, DOI 10.1016/j.jmaa.2015.06.065 - Y. Lamzouri and A. Languasco,
*Small values of $L^\prime /L(1,\chi )$*, (2020), arXiv:2005.10714, to appear in Experimental Mathematics, https://doi.org/10.1080/10586458.2021.1927255 - Alessandro Languasco,
*Efficient computation of the Euler-Kronecker constants of prime cyclotomic fields*, Res. Number Theory**7**(2021), no. 1, Paper No. 2, 22. MR**4194178**, DOI 10.1007/s40993-020-00213-1 - Alessandro Languasco,
*Numerical verification of Littlewood’s bounds for $|L(1,\chi )|$*, J. Number Theory**223**(2021), 12–34. MR**4213696**, DOI 10.1016/j.jnt.2020.12.017 - Pieter Moree,
*Irregular behaviour of class numbers and Euler-Kronecker constants of cyclotomic fields: the log log log devil at play*, Irregularities in the distribution of prime numbers, Springer, Cham, 2018, pp. 143–163. MR**3837472** - 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** - The PARI Group,
*PARI/GP version 2.13.1*, Bordeaux, 2021. Available from http://pari.math.u-bordeaux.fr/. - James C. Schatzman,
*Accuracy of the discrete Fourier transform and the fast Fourier transform*, SIAM J. Sci. Comput.**17**(1996), no. 5, 1150–1166. MR**1404866**, DOI 10.1137/S1064827593247023

## Additional Information

**Alessandro Languasco**- Affiliation: Università di Padova, Dipartimento di Matematica “Tullio Levi-Civita”, Via Trieste 63, 35121 Padova, Italy
- MR Author ID: 354780
- ORCID: 0000-0003-2723-554X
- Email: alessandro.languasco@unipd.it
**Luca Righi**- Affiliation: Università di Padova, Dipartimento di Matematica “Tullio Levi-Civita”, Via Trieste 63, 35121 Padova, Italy
- ORCID: 0000-0002-4522-7511
- Email: righi@math.unipd.it
- Received by editor(s): May 24, 2020
- Received by editor(s) in revised form: February 20, 2021
- Published electronically: August 12, 2021
- Additional Notes: The calculations here described in Section \ref{sect-implementation} were performed using the University of Padova Strategic Research Infrastructure Grant 2017: “CAPRI: Calcolo ad Alte Prestazioni per la Ricerca e l’Innovazione”, http://capri.dei.unipd.it.

The first author is the corresponding author - © Copyright 2021 American Mathematical Society
- Journal: Math. Comp.
**90**(2021), 2899-2921 - MSC (2020): Primary 33-04, 11-04; Secondary 33E20, 11Y16, 11Y60
- DOI: https://doi.org/10.1090/mcom/3668
- MathSciNet review: 4305373