Home | Current | Past volumes | About | Login | Notify | Contact | Search
 Probability Surveys > Vol. 11 (2014) open journal systems 

Distribution of the sum-of-digits function of random integers: A survey

Louis H. Y. Chen, Department of Mathematics, National University of Singapore
Hsien-Kuei Hwang, Institute of Statistical Science, Institute of Information Science, Academia Si
Vytas Zacharovas, Dept. Mathematics & Informatics, Vilnius University

We review some probabilistic properties of the sum-of-digits function of random integers. New asymptotic approximations to the total variation distance and its refinements are also derived. Four different approaches are used: a classical probability approach, Stein’s method, an analytic approach and a new approach based on Krawtchouk polynomials and the Parseval identity. We also extend the study to a simple, general numeration system for which similar approximation theorems are derived.

AMS 2000 subject classifications: Primary 60F05, 60C05; secondary 62E17, 11N37, 11K16.

Keywords: Sum-of-digits function, Stein’s method, Gray codes, total variation distance, numeration systems, Krawtchouk polynomials, digital sums, asymptotic normality

Creative Common LOGO

Full Text: PDF

Chen, Louis H. Y., Hwang, Hsien-Kuei, Zacharovas, Vytas, Distribution of the sum-of-digits function of random integers: A survey, Probability Surveys, 11, (2014), 177-236 (electronic). DOI: 10.1214/12-PS213.


[1]   Agnarsson, G., On the number of hypercubic bipartitions of an integer. Discrete Math., 313(24):2857–2864, 2013. MR3115296

[2]   Alkauskas, G., Dirichlet series associated with strongly q-multiplicative functions. Ramanujan J., 8(1):13–21, 2004. MR2068427

[3]   Allouche, J.-P. and Shallit, J., Automatic Sequences. Cambridge University Press, Cambridge, 2003. MR1997038

[4]   Barat, G., Berthé, V., Liardet, P., and Thuswaldner, J., Dynamical directions in numeration. Ann. Inst. Fourier (Grenoble), 56(7):1987–2092, 2006. MR2290774

[5]   Barbour, A. D., Stein’s method and poisson process convergence. J. Appl. Probab., 25:175–184, 1988. MR0974580

[6]   Barbour, A. D., Stein’s method for diffusion approximations. Probab. Th. Related Fields, 84(3):297–322, 1990. MR1035659

[7]   Barbour, A. D. and Chen, L. H. Y., On the binary expansion of a random integer. Statist. Probab. Lett., 14(3):235–241, 1992. MR1173624

[8]   Barbour, A. D., Holst, L., and Janson, S., Poisson Approximation. The Clarendon Press, Oxford University Press, New York, 1992. MR1163825

[9]   Bassily, N. L. and Kátai, I., Distribution of the values of q-additive functions on polynomial sequences. Acta Math. Hungar., 68(4):353–361, 1995. MR1333478

[10]   Bassily, N. L. and Kátai, I., Distribution of consecutive digits in the q-ary expansions of some subsequences of integers. In Proceedings of the XVI Seminar on Stability Problems for Stochastic Models, Part II (Eger, 1994), volume 78, pages 11–17, 1996. MR1381030

[11]   Bellman, R. and Shapiro, H. N., On a problem in additive number theory. Ann. of Math. (2), 49:333–340, 1948. MR0023864

[12]   Berthé, V. and Rigo, M., editors, Combinatorics, Automata and Number Theory. Cambridge University Press, 2010. MR2742574

[13]   Bowden, J., Special Topics in Theoretical Arithmetic. J. Bowden, Garden City, New York, 1936.

[14]   Brown, T. C., Powers of digital sums. Fibonacci Quart., 32(3):207–210, 1994. MR1285747

[15]   Bush, L. E., An asymptotic formula for the average sum of the digits of integers. Amer. Math. Monthly, 47:154–156, 1940. MR0001225

[16]   Chen, F.-J., A problem in the r-adic representation of positive integers (Chinese). J. Nanjing University (Natural Sciences), 40(1):89–93, 2004. MR2370534

[17]   Chen, L. H. Y., Poisson approximation for dependent trials. Ann. Probability, 3(3):534–545, 1975. MR0428387

[18]   Chen, L. H. Y., Fang, X., and Shao, Q.-M., From Stein identities to moderate deviations. Ann. Probab., 41(1):262–293, 2013. MR3059199

[19]   Chen, L. H. Y. and Shao, Q.-M., Stein’s method for normal approximation. In An Introduction to Stein’s Method, volume 4 of Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., pages 1–59. Singapore Univ. Press, Singapore, 2005. MR2235448

[20]   Chen, L. H. Y. and Soon, S. Y. T., On the number of ones in the binary expansion of a random integer. Unpublished manuscript, 1994.

[21]   Chen, W.-M., Hwang, H.-K., and Chen, G.-H., The cost distribution of queue-mergesort, optimal mergesorts, and power-of-2 rules. J. Algorithms, 30(2):423–448, 1999. MR1671856

[22]   Cheo, P.-H. and Yien, S.-C., A problem on the k-adic representation of positive integers. Acta Math. Sinica, 5:433–438, 1955. MR0075979

[23]   Clements, G. F. and Lindström, B., A sequence of (±1)determinants with large values. Proc. Amer. Math. Soc., 16:548–550, 1965. MR0178001

[24]   Cooper, C. and Kennedy, R. E., Digital sum sums. J. Inst. Math. Comput. Sci. Math. Ser., 5(1):45–49, 1992. MR1182467

[25]   Cooper, C. N. and Kennedy, R. E., A generalization of a theorem by Cheo and Yien concerning digital sums. Internat. J. Math. Math. Sci., 9(4):817–820, 1986. MR0870542

[26]   Coquet, J., Power sums of digital sums. J. Number Theory, 22(2):161–176, 1986. MR0826949

[27]   Dartyge, C., Luca, F., and Stănică, P., On digit sums of multiples of an integer. J. Number Theory, 129(11):2820–2830, 2009. MR2549536

[28]   Deheuvels, P. and Pfeifer, D., A semigroup approach to Poisson approximation. Ann. Probab., 14(2):663–676, 1986. MR0832029

[29]   Delange, H., Sur les fonctions q-additives ou q-multiplicatives. Acta Arith., 21:285–298 (errata insert), 1972. MR0309891

[30]   Delange, H., Sur la fonction sommatoire de la fonction “somme des chiffres”. Enseignement Math. (2), 21(1):31–47, 1975. MR0379414

[31]   Diaconis, P., The distribution of leading digits and uniform distribution mod 1. Ann. Probability, 5(1):72–81, 1977. MR0422186

[32]   Diaconis, P., Group Representations in Probability and Statistics. Institute of Mathematical Statistics Lecture Notes—Monograph Series, 11. Institute of Mathematical Statistics, Hayward, CA, 1988. MR0964069

[33]    Diaconis, P., Graham, R. L., and Morrison, J. A., Asymptotic analysis of a random walk on a hypercube with many dimensions. Random Structures Algorithms, 1(1):51–72, 1990. MR1068491

[34]   Dickson, L. E., History of the Theory of Numbers. Vol. I: Divisibility and Primality. Chelsea Publishing Co. (unaltered reprintings of the 1919 original), New York, 1966.

[35]   d’Ocagne, M., Sur certaines sommations arithmétiques. Jornal de Sciencias Mathematicas e Astronomicas (de M. Gomes Teixeira. Coimbre), 7:117–128, 1886.

[36]   Doran, R., The Gray code. J. Universal Comput. Sci., 13(11):1573–1597, 2007. MR2390238

[37]   Drazin, M. P. and Griffith, J. S., On the decimal representation of integers. Proc. Cambridge Philos. Soc., 48:555–565, 1952. MR0049959

[38]   Drmota, M., The joint distribution of q-additive functions. Acta Arith., 100(1):17–39, 2001. MR1864623

[39]   Drmota, M., Fuchs, M., and Manstavičius, E., Functional limit theorems for digital expansions. Acta Math. Hungar., 98(3):175–201, 2003. MR1956755

[40]   Drmota, M. and Gajdosik, J., The distribution of the sum-of-digits function. J. Théor. Nombres Bordeaux, 10(1):17–32, 1998. MR1827283

[41]   Dumont, J.-M. and Thomas, A., Systèmes de numération et fonctions fractales relatifs aux substitutions. Theoret. Comput. Sci., 65(2):153–169, 1989. MR1020484

[42]   Dumont, J.-M. and Thomas, A., Digital sum moments and substitutions. Acta Arith., 64(3):205–225, 1993. MR1225425

[43]   Dumont, J. M. and Thomas, A., Gaussian asymptotic properties of the sum-of-digits function. J. Number Theory, 62(1):19–38, 1997. MR1430000

[44]   Ettestad, D. J. and Carbonara, J. O., Formulas for the number of states of an interesting finite cellular automaton and a connection to Pascal’s triangle. J. Cell. Autom., 5(1–2):157–166, 2010. MR2583067

[45]   Fang, Y., A theorem on the k-adic representation of positive integers. Proc. Amer. Math. Soc., 130(6):1619–1622 (electronic), 2002. MR1887007

[46]   Flajolet, P. and Golin, M., Mellin transforms and asymptotics. The mergesort recurrence. Acta Inform., 31(7):673–696, 1994. MR1300060

[47]    Flajolet, P., Grabner, P., Kirschenhofer, P., Prodinger, H., and Tichy, R. F., Mellin transforms and asymptotics: Digital sums. Theoret. Comput. Sci., 123(2):291–314, 1994. MR1256203

[48]   Flajolet, P. and Ramshaw, L., A note on Gray code and odd-even merge. SIAM J. Comput., 9(1):142–158, 1980. MR0557835

[49]   Foster, D. M. E., Estimates for a remainder term associated with the sum of digits function. Glasgow Math. J., 29(1):109–129, 1987. MR0876156

[50]   Foster, D. M. E., A lower bound for a remainder term associated with the sum of digits function. Proc. Edinburgh Math. Soc. (2), 34(1):121–142, 1991. MR1093181

[51]   Foster, D. M. E., Averaging the sum of digits function to an even base. Proc. Edinburgh Math. Soc. (2), 35(3):449–455, 1992. MR1187007

[52]   Gelfond, A. O., Sur les nombres qui ont des propriétés additives et multiplicatives données. Acta Arith., 13:259–265, 1967/1968. MR0220693

[53]   Gilbert, E. N., Games of identification or convergence. SIAM Review, 4(1):16–24, 1962.

[54]   Gittenberger, B. and Thuswaldner, J. M., Asymptotic normality of b-additive functions on polynomial sequences in the gaussian number field. Journal of Number Theory, 84(2):317–341, 2000. MR1796518

[55]   Glaisher, J. W. L., On the residue of a binomial-theorem coefficient with respect to a prime modulus. Quart. J. Pure and Appl. Math., 30:150–156, 1899.

[56]   Glaser, A., History of Binary and Other Nondecimal Numeration. Tomash Publishers, Los Angeles, Calif., second edition, 1981. MR0666393

[57]   Grabner, P. J., Completely q-multiplicative functions: the Mellin transform approach. Acta Arith., 65(1):85–96, 1993. MR1239244

[58]   Grabner, P. J. and Hwang, H.-K., Digital sums and divide-and-conquer recurrences: Fourier expansions and absolute convergence. Constr. Approx., 21(2):149–179, 2005. MR2107936

[59]   Grabner, P. J., Kirschenhofer, P., Prodinger, H., and Tichy, R. F., On the moments of the sum-of-digits function. In Applications of Fibonacci Numbers, Vol. 5 (St. Andrews, 1992), pages 263–271. Kluwer Acad. Publ., Dordrecht, 1993. MR1271366

[60]   Graham, R. L., On primitive graphs and optimal vertex assignments. Ann. New York Acad. Sci., 175:170–186, 1970. MR0269533

[61]   Greene, D. H. and Knuth, D. E., Mathematics for the Analysis of Algorithms. Modern Birkhäuser Classics. Birkhäuser Boston Inc., Boston, MA, 2008. MR2381155

[62]   Hadjicostas, P. and Lakshmanan, K. B., Recursive merge sort with erroneous comparisons. Discrete Appl. Math., 159(14):1398–1417, 2011. MR2823899

[63]   Hart, S., A note on the edges of the n-cube. Discrete Math., 14(2):157–163, 1976. MR0396293

[64]   Hata, M. and Yamaguti, M., The Takagi function and its generalization. Japan J. Appl. Math., 1(1):183–199, 1984. MR0839313

[65]   Heppner, E., Über die Summe der Ziffern natürlicher Zahlen. Ann. Univ. Sci. Budapest. Eötvös Sect. Math., 19:41–43 (1977), 1976. MR0506024

[66]   Hofer, R., Larcher, G., and Pillichshammer, F., Average growth-behavior and distribution properties of generalized weighted digit-block-counting functions. Monatsh. Math., 154(3):199–230, 2008. MR2413302

[67]   Holmes, S., Stein’s method for birth and death chains. In Stein’s Method: Expository Lectures and Applications, volume 46 of IMS Lecture Notes Monogr. Ser., pages 45–67. Inst. Math. Statist., Beachwood, OH, 2004. MR2118602

[68]   Hong, Z. and Sedgewick, R., Notes on merging networks (preliminary version). In Proc. ACM Symposium on Theory of Computing, pages 296–302, 1982.

[69]   Ifrah, G., The Universal History of Numbers. John Wiley & Sons Inc., New York, 2000. From prehistory to the invention of the computer, Translated from the 1994 French original by David Bellos, E. F. Harding, Sophie Wood and Ian Monk. MR1725387

[70]   Ismail, M. E. H., Classical and Quantum Orthogonal Polynomials in One Variable, volume 98 of Encyclopedia of Mathematics and Its Applications. Cambridge University Press, Cambridge, 2009. MR2542683

[71]   Kano, H., On the sums of digits in integers. Proc. Japan Acad. Ser. A Math. Sci., 67(5):148–150, 1991. MR1114959

[72]   Kátai, I., On the sum of digits of primes. Acta Math. Acad. Sci. Hungar., 30(1–2):169–173, 1977. MR0472747

[73]   Kátai, I. and Mogyoródi, J., On the distribution of digits. Publ. Math. Debrecen, 15:57–68, 1968. MR0236139

[74]   Kennedy, R. E. and Cooper, C. N., An extension of a theorem by Cheo and Yien concerning digital sums. Fibonacci Quart., 29(2):145–149, 1991. MR1119401

[75]   Kirschenhofer, P., On the variance of the sum of digits function. In Number-Theoretic Analysis (Vienna, 1988–89), volume 1452 of Lecture Notes in Math., pages 112–116. Springer, Berlin, 1990. MR1084640

[76]   Kirschenhofer, P. and Prodinger, H., Subblock occurrences in positional number systems and Gray code representation. J. Inform. Optim. Sci., 5(1):29–42, 1984. MR0737164

[77]   Klavžar, S., Milutinović, U., and Petr, C., Stern polynomials. Adv. in Appl. Math., 39(1):86–95, 2007. MR2319565

[78]   Knuth, D. E., Art of Computer Programming, Volume 2: Seminumerical Algorithms. Addison-Wesley, third edition, November 1997.

[79]   Kobayashi, Z., Digital sum problems for the Gray code representation of natural numbers. Interdiscip. Inform. Sci., 8(2):167–175, 2002. MR1972038

[80]   Kobayashi, Z. and Sekiguchi, T., On a characterization of the standard Gray code by using it edge type on a hypercube. Inform. Process. Lett., 81(5):231–237, 2002. MR1879645

[81]   Krüppel, M., De Rham’s singular function, its partial derivatives with respect to the parameter and binary digital sums. Rostock. Math. Kolloq., 64:57–74, 2009. MR2605000

[82]   Kummer, E. E., Über die Ergänzungssätze zu den allgemeinen Reciprocitätsgesetzen. J. Reine Angew. Math., 44:93–146, 1852.

[83]   Laczay, B. and Ruszinkó, M., Collision channel with multiplicity feedback. In E. Biglieri and L. Györfi, editors, Proceedings of the NATO Advanced Study Institute on Coding and Analysis of Multiple Access Channels. Theory and Practice, volume D. 10, pages 250–270. IOS Press, 2007.

[84]   Lagarias, J. C., The Takagi function and its properties. In Functions in Number Theory and Their Probabilistic Aspects, RIMS Kôkyûroku Bessatsu, B34, pages 153–189. Res. Inst. Math. Sci. (RIMS), Kyoto, 2012. MR3014845

[85]   Legendre, A., Théorie des Nombres. Firmin Didot Frères, fourth edition, 1900.

[86]   Li, S.-Y. R., Binary trees and uniform distribution of traffic cutback. J. Comput. System Sci., 32(1):1–14, 1986. MR0844201

[87]   Lindström, B., On a combinatorial problem in number theory. Canad. Math. Bull., 8:477–490, 1965. MR0181604

[88]   Loh, W.-L., Stein’s method and multinomial approximation. Ann. Appl. Probab., 2(3):536–554, 1992. MR1177898

[89]   Lucas, É., Sur les congruences des nombres eulériens et des coefficients différentiels des fonctions trigonométriques, suivant un module premier. Bull. Soc. Math. France, 6:49–54, 1878. MR1503769

[90]   MacWilliams, F. J. and Sloane, N. J. A., The Theory of Error-Correcting Codes. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.

[91]   Madritsch, M. and Pethʺo , A., Asymptotic normality of additive functions on polynomial sequences in canonical number systems. J. Number Theory, 131(9):1553–1574, 2011. MR2802135

[92]   Madritsch, M. G., Asymptotic normality of b-additive functions on polynomial sequences in number systems. Ramanujan J., 21(2):181–210, 2010. MR2593247

[93]   Manstavičius, E., Probabilistic theory of additive functions related to systems of numeration. In New Trends in Probability and Statistics, Vol. 4 (Palanga, 1996), pages 413–429. VSP, Utrecht, 1997. MR1653626

[94]   Mauclaire, J.-L., Sur la répartition des fonctions q-additives. J. Théor. Nombres Bordeaux, 5(1):79–91, 1993. MR1251228

[95]   Mauclaire, J.-L. and Murata, L., On q-additive functions. I. Proc. Japan Acad. Ser. A Math. Sci., 59(6):274–276, 1983. MR0718620

[96]   Mauclaire, J.-L. and Murata, L., On q-additive functions. II. Proc. Japan Acad. Ser. A Math. Sci., 59(9):441–444, 1983. MR0732606

[97]   Mauduit, C. and Rivat, J., Propriétés q-multiplicatives de la suite nc, c > 1. Acta Arith., 118(2):187–203, 2005. MR2141049

[98]   Mauduit, C. and Rivat, J., La somme des chiffres des carrés. Acta Math., 203(1):107–148, 2009. MR2545827

[99]   Mauduit, C. and Rivat, J., Sur un problème de Gelfond: la somme des chiffres des nombres premiers. Ann. of Math., 171(3):1591–1646, 2010. MR2680394

[100]   McIlroy, M. D., The number of 1’s in binary integers: Bounds and extremal properties. SIAM J. Comput., 3:255–261, 1974. MR0436687

[101]   Mehrabian, A., Mitsche, D., and Prałat, P., On the maximum density of graphs with unique-path labelings. SIAM J. Discrete Math., 27(3):1228–1233, 2013. MR3072758

[102]   Mirsky, L., A theorem on representations of integers in the scale of r. Scripta Math., 15:11–12, 1949. MR0030991

[103]   Morrison, J. A., Weighted averages of Radon transforms on Z2k. SIAM J. Algebraic Discrete Methods, 7(3):404–413, 1986. MR0844043

[104]   Muramoto, K., Okada, T., Sekiguchi, T., and Shiota, Y., Digital sum problems for the p-adic expansion of natural numbers. Interdiscip. Inform. Sci., 6(2):105–109, 2000. MR1839805

[105]   Muramoto, K., Okada, T., Sekiguchi, T., and Shiota, Y., Power and exponential sums of digital sums with information per digits. Math. J. Toyama Univ., 26:35–44, 2003. MR2048391

[106]   Murata, L. and Mauclaire, J.-L., An explicit formula for the average of some q-additive functions. In Prospects of Mathematical Science (Tokyo, 1986), pages 141–156. World Sci. Publishing, Singapore, 1988. MR0948466

[107]   Okada, T., Sekiguchi, T., and Shiota, Y., Applications of binomial measures to power sums of digital sums. J. Number Theory, 52(2):256–266, 1995. MR1336748

[108]   Okada, T., Sekiguchi, T., and Shiota, Y., An explicit formula of the exponential sums of digital sums. Japan J. Indust. Appl. Math., 12(3):425–438, 1995. MR1356664

[109]   Okada, T., Sekiguchi, T., and Shiota, Y., A generalization of Hata-Yamaguti’s results on the Takagi function. II. Multinomial case. Japan J. Indust. Appl. Math., 13(3):435–463, 1996. MR1415064

[110]   Osbaldestin, A. H., Digital sum problems. In Fractals in the Fundamental and Applied Sciences, pages 307–328. Elsevier Science, B. V., North-Holland, Amsterdam, 1991.

[111]   Panny, W. and Prodinger, H., Bottom-up mergesort—A detailed analysis. Algorithmica, 14(4):340–354, 1995. MR1343320

[112]   Prodinger, H., Generalizing the sum of digits function. SIAM J. Algebraic Discrete Methods, 3(1):35–42, 1982. MR0644955

[113]   Prodinger, H., Nonrepetitive sequences and Gray code. Discrete Math., 43(1):113–116, 1983. MR0680311

[114]   Prodinger, H., A subword version of d’Ocagne’s formula. Utilitas Math., 24:125–129, 1983. MR0724766

[115]   Prodinger, H., Digits and beyond. In Mathematics and Computer Science, II (Versailles, 2002), Trends Math., pages 355–377. Birkhäuser, Basel, 2002. MR1940147

[116]   Roberts, J. B., On binomial coefficient residues. Canad. J. Math., 9:363–370, 1957. MR0086828

[117]   Roos, B., Binomial approximation to the Poisson binomial distribution: The Krawtchouk expansion. Theory Probab. Appl., 45(2):258–272, 2001. MR1967760

[118]   Sándor, J. and Crstici, B., Handbook of Number Theory. II. Kluwer Academic Publishers, Dordrecht, 2004. MR2119686

[119]   Savage, C., A survey of combinatorial Gray codes. SIAM Rev., 39(4):605–629, 1997. MR1491049

[120]   Schmid, J., The joint distribution of the binary digits of integer multiples. Acta Arith., 43(4):391–415, 1984. MR0756290

[121]   Schmidt, W. M., The joint distribution of the digits of certain integer s-tuples. In Studies in Pure Mathematics, pages 605–622. Birkhäuser, Basel, 1983. MR0820255

[122]   Schoutens, W., Stochastic Processes and Orthogonal Polynomials, volume 146 of Lecture Notes in Statistics. Springer-Verlag, New York, 2000. MR1761401

[123]   Shiokawa, I., On a problem in additive number theory. Math. J. Okayama Univ., 16:167–176, 1973/1974. MR0357352

[124]   Shiokawa, I., g-adical analogues of some arithmetical functions. Math. J. Okayama Univ., 17:75–94, 1974. MR0364069

[125]   Soon, Y.-T., Some Problems in Binomial and Compound Poisson Approximations. Ph.D. Thesis, National University of Singapore, 1993.

[126]   Stein, A. H., Exponential sums related to binomial coefficient parity. Proc. Amer. Math. Soc., 80(3):526–530, 1980. MR0581019

[127]   Stein, A. H., Exponential sums of sum-of-digit functions. Illinois J. Math., 30(4):660–675, 1986. MR0857218

[128]   Stein, C., A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. II: Probability Theory, pages 583–602. Univ. California Press, Berkeley, Calif., 1972. MR0402873

[129]   Stein, C., Approximate Computation of Expectations. Institute of Mathematical Statistics Lecture Notes—Monograph Series, 7. Institute of Mathematical Statistics, Hayward, CA, 1986. MR0882007

[130]   Steiner, W., The Distribution of Digital Expansions on Polynomial Sequences. Dissertation, TU-Wien, 2002.

[131]   Stolarsky, K. B., Power and exponential sums of digital sums related to binomial coefficient parity. SIAM J. Appl. Math., 32(4):717–730, 1977. MR0439735

[132]   Stolarsky, K. B., Integers whose multiples have anomalous digital frequencies. Acta Arith., 38(2):117–128, 1980/81. MR0604228

[133]   Szegʺo  , G., Orthogonal Polynomials. AMS, Providence, R.I., fourth edition, 1975. MR0372517

[134]   Tang, S. C., An improvement and generalization of Bellman-Shapiro’s theorem on a problem in additive number theory. Proc. Amer. Math. Soc., 14:199–204, 1963. MR0150082

[135]   Tenenbaum, G., Sur la non-dérivabilité de fonctions périodiques associées à certaines formules sommatoires. In The Mathematics of Paul Erdʺo  s, I, volume 13 of Algorithms Combin., pages 117–128. Springer, Berlin, 1997. MR1425180

[136]   Terras, A., Fourier Analysis on Finite Groups and Applications, volume 43 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge, 1999. MR1695775

[137]   Thim, J., Continuous Nowhere Differentiable Functions. Master Thesis, Luleå Tekniska Universitet, 2003.

[138]   Trollope, J. R., Generalized bases and digital sums. Amer. Math. Monthly, 74:690–694, 1967. MR0211950

[139]   Trollope, J. R., An explicit expression for binary digital sums. Math. Mag., 41:21–25, 1968. MR0233763

[140]   Wolfram, S., Statistical mechanics of cellular automata. Rev. Modern Phys., 55(3):601–644, 1983. MR0709077

[141]   Wolfram, S., Geometry of binomial coefficients. Amer. Math. Monthly, 91(9):566–571, 1984. MR0764797

[142]   Yu, X. Y., On the mean-value of the powers of digital sums. Kexue Tongbao (Chinese), 41(7):581–585, 1996. MR1418096

[143]   Zacharovas, V. and Hwang, H.-K., A Charlier-Parseval approach to Poisson approximation and its applications. Lithuanian Math. J., 50(1):88–119, 2010. MR2607681

Home | Current | Past volumes | About | Login | Notify | Contact | Search

Probability Surveys. ISSN: 1549-5787