On Benford’s law for multiplicative functions

Chandee, Vorrapan; Li, Xiannan; Pollack, Paul; Singha Roy, Akash

doi:10.1090/proc/16480

On Benford’s law for multiplicative functions
HTML articles powered by AMS MathViewer

by Vorrapan Chandee, Xiannan Li, Paul Pollack and Akash Singha Roy;

Proc. Amer. Math. Soc. 151 (2023), 4607-4619

DOI: https://doi.org/10.1090/proc/16480

Published electronically: July 14, 2023

HTML | PDF | Request permission

Abstract:

We provide a criterion to determine whether a real multiplicative function is a strong Benford sequence. The criterion implies that the $k$-divisor functions, where $k \neq 10^j$, and Hecke eigenvalues of newforms, such as Ramanujan tau function, are strong Benford. In contrast to some earlier work, our approach is based on Halász’s Theorem.

References

Theresa C. Anderson, Larry Rolen, and Ruth Stoehr, Benford’s law for coefficients of modular forms and partition functions, Proc. Amer. Math. Soc. 139 (2011), no. 5, 1533–1541. MR 2763743, DOI 10.1090/S0002-9939-2010-10577-4
S. Aursukaree and V. Chandee, Equidistribution of $\log (d(n))$, Proceedings of Annual Pure and Applied Mathematics Conference, Chulalongkorn University, Thailand, May 2016, 399–410.
Tom Barnet-Lamb, David Geraghty, Michael Harris, and Richard Taylor, A family of Calabi-Yau varieties and potential automorphy II, Publ. Res. Inst. Math. Sci. 47 (2011), no. 1, 29–98. MR 2827723, DOI 10.2977/PRIMS/31
F. Benford, The law of anomalous numbers, Proc. Amer. Philos. Soc. 78 (1938), no. 4, 551–572.
Hubert Delange, Quelques résultats nouveaux sur les fonctions additives, Colloque de Théorie des Nombres (Univ. Bordeaux, Bordeaux, 1969) Supplément au Bull. Soc. Math. France, Tome 99, Soc. Math. France, Paris, 1971, pp. 45–53 (French). MR 404186, DOI 10.24033/msmf.33
Pierre Deligne, La conjecture de Weil. I, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 273–307 (French). MR 340258, DOI 10.1007/BF02684373
Persi Diaconis, The distribution of leading digits and uniform distribution $\textrm {mod}$ $1$, Ann. Probability 5 (1977), no. 1, 72–81. MR 422186, DOI 10.1214/aop/1176995891
P. D. T. A. Elliott, Probabilistic number theory. I, Grundlehren der Mathematischen Wissenschaften, vol. 239, Springer-Verlag, New York-Berlin, 1979. Mean-value theorems. MR 551361, DOI 10.1007/978-1-4612-9989-9
Andrew Granville and K. Soundararajan, Large character sums: pretentious characters and the Pólya-Vinogradov theorem, J. Amer. Math. Soc. 20 (2007), no. 2, 357–384. MR 2276774, DOI 10.1090/S0894-0347-06-00536-4
Marie Jameson, Jesse Thorner, and Lynnelle Ye, Benford’s law for coefficients of newforms, Int. J. Number Theory 12 (2016), no. 2, 483–494. MR 3461444, DOI 10.1142/S1793042116500299
Alex V. Kontorovich and Steven J. Miller, Benford’s law, values of $L$-functions and the $3x+1$ problem, Acta Arith. 120 (2005), no. 3, 269–297. MR 2188844, DOI 10.4064/aa120-3-4
Jeffrey C. Lagarias and K. Soundararajan, Benford’s law for the $3x+1$ function, J. London Math. Soc. (2) 74 (2006), no. 2, 289–303. MR 2269630, DOI 10.1112/S0024610706023131
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
V. Kumar Murty, Modular forms and the Chebotarev density theorem. II, Analytic number theory (Kyoto, 1996) London Math. Soc. Lecture Note Ser., vol. 247, Cambridge Univ. Press, Cambridge, 1997, pp. 287–308. MR 1694997, DOI 10.1017/CBO9780511666179.019
Simon Newcomb, Note on the Frequency of Use of the Different Digits in Natural Numbers, Amer. J. Math. 4 (1881), no. 1-4, 39–40. MR 1505286, DOI 10.2307/2369148
Jean-Pierre Serre, Quelques applications du théorème de densité de Chebotarev, Inst. Hautes Études Sci. Publ. Math. 54 (1981), 323–401 (French). MR 644559
Jesse Thorner and Asif Zaman, A Chebotarev variant of the Brun-Titchmarsh theorem and bounds for the Lang-Trotter conjectures, Int. Math. Res. Not. IMRN 16 (2018), 4991–5027. MR 3848226, DOI 10.1093/imrn/rnx031
E. C. Titchmarsh, The theory of the Riemann zeta-function, 2nd ed., The Clarendon Press, Oxford University Press, New York, 1986. Edited and with a preface by D. R. Heath-Brown. MR 882550
Giancarlo Travaglini, Number theory, Fourier analysis and geometric discrepancy, London Mathematical Society Student Texts, vol. 81, Cambridge University Press, Cambridge, 2014. MR 3307692, DOI 10.1017/CBO9781107358379
Da Qing Wan, On the Lang-Trotter conjecture, J. Number Theory 35 (1990), no. 3, 247–268. MR 1062334, DOI 10.1016/0022-314X(90)90117-A
Siqi Zheng, Necessary and sufficient conditions for Benford sequences, Pi Mu Epsilon J. 13 (2013), no. 9, 553–561. MR 3155505