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Jonathan M. Borwein Commemorative Conference

JBCC 2017: From Analysis to Visualization pp 423–439Cite as

Short Walk Adventures

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Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 313))

Abstract

We review recent development of short uniform random walks, with a focus on its connection to (zeta) Mahler measures and modular parametrisation of the density functions. Furthermore, we extend available ‘probabilistic’ techniques to cover a variation of random walks and reduce some three-variable Mahler measures, which are conjectured to evaluate in terms of L-values of modular forms, to hypergeometric form.

To the memory of Jon Borwein, who convinced us that a short walk can be adventurous

The original version of this chapter was revised: The cross-citations in square brackets has now been updated. The correction to this chapter is available at https://doi.org/10.1007/978-3-030-36568-4_28

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Change history

  • 12 May 2020

    We review recent development of short uniform random walks, with a focus on its connection to (zeta) Mahler measures and modular parametrisation of the density functions. Furthermore, we extend available ‘probabilistic’ techniques to cover a variation of random walks and reduce some three-variable Mahler measures, which are conjectured to evaluate in terms of L-values of modular forms, to hypergeometric form.

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Acknowledgements

We thank H. Cohen for supplying us with the numerical observation (10) whose origin remains completely mysterious to us. We also thank the referee for their valuable feedback.

Both authors acknowledge support of the Max Planck Society during their stays at the Max Planck Institute for Mathematics (Bonn, Germany) in the years 2015–2017. The second author is partially supported by Laboratory of Mirror Symmetry NRU HSE, RF government grant, ag. no. 14.641.31.0001.

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of South Alabama, 411 University Blvd N, MSPB 325, Mobile, AL, 36688, USA

    Armin Straub

  2. Department of Mathematics, IMAPP, Radboud University, PO Box 9010, 6500 GL, Nijmegen, Netherlands

    Wadim Zudilin

  3. School of Mathematical and Physical Sciences, The University of Newcastle, Callaghan, NSW, 2308, Australia

    Wadim Zudilin

  4. Laboratory of Mirror Symmetry and Automorphic Forms, National Research University Higher School of Economics, 6 Usacheva str., 119048, Moscow, Russia

    Wadim Zudilin

Authors
  1. Armin Straub
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  2. Wadim Zudilin
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Correspondence to Wadim Zudilin .

Editor information

Editors and Affiliations

  1. Lawrence Berkeley National Laboratory, Berkeley, CA, USA

    David H. Bailey

  2. Western University, London, ON, Canada

    Naomi Simone Borwein

  3. Mathematical Sciences Institute, Australian National University, Canberra, ACT, Australia

    Richard P. Brent

  4. School of IT and Mathematical Sciences, University of South Australia, Mawson Lakes, SA, Australia

    Regina S. Burachik

  5. School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, NSW, Australia

    Judy-anne Heather Osborn

  6. School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, NSW, Australia

    Brailey Sims

  7. Department of Mathematics, Western Michigan University, Kalamazoo, MI, USA

    Qiji J. Zhu

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Straub, A., Zudilin, W. (2020). Short Walk Adventures. In: Bailey, D., et al. From Analysis to Visualization. JBCC 2017. Springer Proceedings in Mathematics & Statistics, vol 313. Springer, Cham. https://doi.org/10.1007/978-3-030-36568-4_27

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