Random number sequences and the first digit phenomenon
Dominique Schneider (Université du Littoral Côte d'Opale)
Abstract
The sequences of mantissa of positive integers and of prime numbers are known not to be distributed as Benford's law in the sense of the natural density. We show that we can correct this defect by selecting the integers or the primes by means of an adequate random process and we investigate the rate of convergence. Our main tools are uniform bounds for deterministic and random trigonometric polynomials. We then adapt the random process to prove the same result for logarithms and iterated logarithms of integers. Finally we show that, in many cases, the mantissa law of the $n$th randomly selected term converges weakly to the Benford's law.
Full Text: Download PDF | View PDF online (requires PDF plugin)
Pages: 1-17
Publication Date: October 4, 2012
DOI: 10.1214/EJP.v17-1900
References
- Benford, F.: The law of anomalous numbers. Proceedings of the American Philosophical Society 78, (1938), 551--572.
- Billingsley, P.: Probability and measure. Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons, New York-Chichester-Brisbane, 1979. xiv+515 pp. ISBN: 0-471-03173-9 MR0534323
- Cohen, G.; Cuny, C.: On random almost periodic trigonometric polynomials and applications to ergodic theory. Ann. Probab. 34 (2006), no. 1, 39--79. MR2206342
- Darroch, J. N.: On the distribution of the number of successes in independent trials. Ann. Math. Statist. 35 1964 1317--1321. MR0164359
- Drmota, M.; Tichy, R. F.: Sequences, discrepancies and applications. Lecture Notes in Mathematics, 1651. Springer-Verlag, Berlin, 1997. xiv+503 pp. ISBN: 3-540-62606-9 MR1470456
- Duncan, R. L.: Note on the initial digit. Fibonacci Quart. 7-5 1969 474--475.
- Eliahou, S.; Massé, B.; Schneider, D.: On the mantissa distribution of powers of natural and prime numbers. To appear in Acta Mathematica Hungarica, (2012), http://www.springerlink.com/content/37m42117885580mk/
- Fuchs, A.; Letta, G.: Le problème du premier chiffre décimal pour les nombres premiers. (French) [The first digit problem for primes] The Foata Festschrift. Electron. J. Combin. 3 (1996), no. 2, Research Paper 25, approx. 7 pp. (electronic). MR1392510
- Hamming, R. W.: On the distribution of numbers. Bell System Tech. J. 49 1970 1609--1625. MR0267809
- Hardy, G. H.: Divergent Series. Oxford, at the Clarendon Press, 1949. xvi+396 pp. MR0030620
- Janvresse E. and de la Rue T.: Averaging along uniform random integers. Uniform Distribution Theory 7-2, (2012), 61--76.
- Knuth, D. E.: The art of computer programming. Vol. 2. Seminumerical algorithms. Second edition. Addison-Wesley Series in Computer Science and Information Processing. Addison-Wesley Publishing Co., Reading, Mass., 1981. xiii+688 pp. ISBN: 0-201-03822-6 MR0633878
- Kuipers, L.; Niederreiter, H.: Uniform distribution of sequences. Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. xiv+390 pp. MR0419394
- Massé, B.; Schneider, D.: A survey on weighted densities and their connection with the first digit phenomenon. Rocky Mountain J. Math. 41 (2011), no. 5, 1395--1415. MR2838069
- Newcomb, S.: Note on the Frequency of Use of the Different Digits in Natural Numbers. Amer. J. Math. 4 (1881), no. 1-4, 39--40. MR1505286
- Nigrini M. J. and Mittermaier L. J.: The use of Benford's law as an aid in analytical procedures. Auditing: A Journal of Practice and Theory 16-2, (1997), 52--57.
- Petrov, V. V. Sums of independent random variables.: Translated from the Russian by A. A. Brown. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 82. Springer-Verlag, New York-Heidelberg, 1975. x+346 pp. MR0388499
- Rivat, J.; Tenenbaum, G. Constantes d'Erdos-Turán.: (French) [Erdos-Turan constants] Ramanujan J. 9 (2005), no. 1-2, 111--121. MR2166382
- Titchmarsh, E. C.: The theory of the Riemann zeta-function. Second edition. Edited and with a preface by D. R. Heath-Brown. The Clarendon Press, Oxford University Press, New York, 1986. x+412 pp. ISBN: 0-19-853369-1 MR0882550
- Wintner A.: On the cyclical distribution of the logarithms of the prime numbers. The Quarterly Journal of Mathematics 6-1, (1935), 65--68.
This work is licensed under a Creative Commons Attribution 3.0 License.