Bailey, David H. (ed.) et al., From analysis to visualization. A celebration of the life and legacy of Jonathan M. Borwein, Callaghan, Australia, September 25–29, 2017. Cham: Springer. Springer Proc. Math. Stat. 313, 423-439 (2020); correction ibid. 313, C1 (2020).
The paper is a good demonstration that the theory of random walks has beautiful connections to a large number of exotic topics of mathematics. Those who study random walks will eventually encounter topics like arithmetic differential equations, algebraic groups, Mahler measure, special function theory, modular functions, and so on.
The first part of the paper studies uniform random walks. An $$N$$-step uniform planar random walk starts at the origin and consists of $$N$$ steps of length 1 each taken into a uniformly random direction. Let $$X_N$$ be the distance to the origin after $$N$$ steps. The $$s$$-th moment of $$X_N$$ is denoted by $$W_N(s)$$.
We learn from the third section that $$W_N(s)$$ is a Mahler measure of the polynomial $$x_1+\cdots+x_N$$ at $$s$$, and that $$W_N'(0)$$ is the logarithmic Mahler measure of the same polynomial. Based on this theory, one can deduce interesting closed form expressions for $$W_k'(0)$$, for small integers $$k$$.
The second part studies generic two-step random walks, that is, random walks where $$X=e^{2\pi i\theta_1}X_1+e^{2\pi i\theta_2}X_2$$, where $$\theta_i$$ are uniformly distributed on $$[0,1]$$, and $$X_i$$ are independent random variables on $$[0,\infty[$$ $$(i=1,2)$$. The finding of the $$W_N'(0)$$ values is described in details in Sections 4–7.
The last part of the paper studies additional relations between Mahler measures and the $$W_N'(0)$$ values for small $$N$$.
For the entire collection see [Zbl 1442.00024].

### MSC:

 33E05 Elliptic functions and integrals 33C20 Generalized hypergeometric series, $${}_pF_q$$
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### References:

 [1] Adamchik, V.S.: Integral and series representations for Catalan’s constant. Unpublished note. http://www.cs.cmu.edu/ adamchik/articles/catalan.htm · Zbl 1009.33006 [2] Ahlgren, S., Berndt, B.C., Yee, A.Y., Zaharescu, A.: Integrals of Eisenstein series and derivatives of $$L$$-functions. Int. Math. Res. Not. 2002(32), 1723-1738 (2002) · Zbl 1086.11023 [3] Akatsuka, H.: Zeta Mahler measures. J. Number Theory 129(11), 2713-2734 (2009) · Zbl 1235.11096 [4] Almkvist, G., van Straten, D., Zudilin, W.: Generalizations of Clausen’s formula and algebraic transformations of Calabi-Yau differential equations. Proc. Edinb. Math. Soc. 54(2), 273-295 (2011) · Zbl 1223.33007 [5] Bailey, D.H., Borwein, J.M.: Hand-to-hand combat with multi-thousand-digit integrals. J. Comput. Sci. 3, 77-86 (2012) [6] Bailey, D.H., Borwein, J.M., Broadhurst, D.J., Glasser, M.L.: Elliptic integral evaluations of Bessel moments and applications. J. Phys. A 41(20), 5203-5231 (2008) · Zbl 1152.33003 [7] Berndt, B.C., Zaharescu, A.: An integral of Dedekind eta-functions in Ramanujan’s lost notebook. J. Reine Angew. Math. 551, 33-39 (2002) · Zbl 1001.11017 [8] Borwein, J.M.: A short walk can be beautiful. J. Humanist. Math. 6(1), 86-109 (2016) [9] Borwein, J.M., Nuyens, D., Straub, A., Wan, J.: Some arithmetic properties of short random walk integrals. Ramanujan J. 26(1), 109-132 (2011) · Zbl 1233.60024 [10] Borwein, J.M., Straub, A., Wan, J., Zudilin, W.: Densities of short uniform random walks, with an appendix by D. Zagier. Can. J. Math. 64(5), 961-990 (2012) · Zbl 1296.33011 [11] Borwein, J.M., Straub, A., Wan, J.: Three-step and four-step random walk integrals. Exp. Math. 22(1), 1-14 (2013) · Zbl 1268.33005 [12] Boyd, D.: Mahler’s measure and special values of $$L$$-functions. Exp. Math. 7(1), 37-82 (1998) · Zbl 0932.11069 [13] Boyd, D., Lind, D., Rodriguez-Villegas, F., Deninger, C.: The many aspects of Mahler’s measure. Final report of the Banff workshop 03w5035 (26 April-1 May 2003). http://www.birs.ca/workshops/2003/03w5035/report03w5035.pdf [14] Broadhurst, D.: Feynman integrals, $$L$$-series and Kloosterman moments. Commun. Number Theory Phys. 10(3), 527-569 (2016) · Zbl 1362.81072 [15] Chan, H.H., Wan, J., Zudilin, W.: Legendre polynomials and Ramanujan-type series for $$1/\pi$$. Isr. J. Math. 194(1), 183-207 (2013) · Zbl 1357.11123 [16] Cohen, H.: Personal communication (23 March 2018) [17] Duke, W., Imamo $$\bar{\rm{g}}$$ lu, Ö.: On a formula of Bloch. Funct. Approx. 37(1), 109-117 (2007) · Zbl 1213.11141 [18] Guillera, J.: A family of Ramanujan-Orr formulas for $$1/\pi$$. Integral Transforms Spec. Funct. 26(7), 531-538 (2015) · Zbl 1432.33014 [19] Paşol, V., Zudilin, W.: A study of elliptic gamma function and allies. Res. Math. Sci. 5(4), Art. 39, p 11 (2018) · Zbl 1440.11063 [20] Rogers, M.D.: A study of inverse trigonometric integrals associated with three-variable Mahler measures, and some related identities. J. Number Theory 121, 265-304 (2006) · Zbl 1132.11032 [21] Rogers, M.D., Straub, A.: A solution of Sun’s \$520 challenge concerning $$520/\pi$$. Int. J. Number Theory 9, 1273-1288 (2013) · Zbl 1277.33002 [22] Rogers, M.D., Zudilin, W.: On the Mahler measure of $$1+X+1/X+Y+1/Y$$. Int. Math. Res. Not. 2014(9), 2305-2326 (2014) · Zbl 1378.11091 [23] Shinder, E., Vlasenko, M.: Linear Mahler measures and double $$L$$-values of modular forms. J. Number Theory 142, 149-182 (2014) · Zbl 1298.11100 [24] Smyth, C.J.: On measures of polynomials in several variables. Bull. Aust. Math. Soc. 23, 49-63 (1981) · Zbl 0442.10034 [25] Sun, Z.-W.: List of conjectural series for powers of $$\pi$$ and other constants (2014). arXiv:1102.5649v47 [math.CA] [26] Takloo-Bighash, R.: A remark on a paper of S. Ahlgren, B.C. Berndt, A.J. Yee, and A. Zaharescu: “Integrals of Eisenstein series and derivatives of $$L$$-functions” [2]. Int. J. Number Theory 2(1), 111-114 (2006) [27] Wan, J.G.: Personal communication (26 July 2011) [28] Wan, J.G.: Series for $$1/\pi$$ using Legendre’s relation. Integral Transforms Spec. Funct. 25(1), 1-14 (2014) · Zbl 1283.33012 [29] Wan, J.G., Zudilin, W.: Generating functions of Legendre polynomials: a tribute to Fred Brafman. J. Approx. Theory 164, 488-503 (2012) · Zbl 1242.33018 [30] Zhou, Y.: On Borwein’s conjectures for planar uniform random walks. J Austral. Math. Soc. 107(3), 392-411 (2019) · Zbl 1472.60081 [31] Zhou, Y.: Wick rotations, Eichler integrals, and multi-loop Feynman diagrams. Commun. Number Theory Phys. 12(1), 127-192 (2018) · Zbl 1393.81029 [32] Zudilin, W.: Period(d)ness of $$L$$-values. In: Borwein, J.M., et al. (eds.) Number Theory and Related Fields, In Memory of Alf van der Poorten. Springer Proceedings in Mathematics and Statistics, vol. 43, pp. 381-395. Springer, New York (2013) · Zbl 1316.11038 [33] Zudilin, W.: A generating function of the squares of Legendre polynomials. Bull. Aust. Math. Soc. 89(1), 125-131 (2014) · Zbl 1334.33022
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