Multiple \(q\)-zeta brackets. (English) Zbl 1312.11069

Zudilin, Wadim

Mathematics 3, No. 1, 119-130 (2015).

Summary: The multiple zeta values (MZVs) possess a rich algebraic structure of algebraic relations, which is conjecturally determined by two different (shuffle and stuffle) products of a certain algebra of noncommutative words. In a recent work, H. Bachmann constructed a \(q\)-analogue of the MZVs – the so-called bi-brackets – for which the two products are dual to each other, in a very natural way [“Generating series of multiple divisor sums and other interesting \(q\)-series”. Talk slides. University of Bristol (2014)]. We overview Bachmann’s construction and discuss the radial asymptotics of the bi-brackets, its links to the MZVs, and related linear (in)dependence questions of the \(q\)-analogue.

MSC:

11M32

Multiple Dirichlet series and zeta functions and multizeta values

Keywords:

multiple zeta value; \(q\)-analogue; multiple divisor sum; double shuffle relations; linear independence; radial asymptotics

References:

[1]	Bradley, Multiple q-zeta values, J. Algebra. 283 pp 752– (2005) · Zbl 1114.11075
[2]	Okuda, On relations for the multiple q-zeta values, Ramanujan J. 14 pp 379– (2007) · Zbl 1211.11099
[3]	Castillo Medina, Unfolding the double shuffle structure of q-multiple zeta values, Bull. Austral. Math. Soc. (to appear); Preprint (2014) · Zbl 1379.11076
[4]	Bachmann, The algebra of generating functions for multiple divisor sums and applications to multiple zeta values, Preprint (2014)
[5]	Bachmann, A short note on a conjecture of Okounkov about a q-analogue of multiple zeta values, Preprint (2014)
[6]	Bachmann, Generating Series of Multiple Divisor Sums and Other Interesting q-Series (2014)
[7]	Pupyrev, Linear and algebraic independence of q-zeta values, Math. Notes 78 pp 563– (2005) · Zbl 1160.11338
[8]	Flajolet, Analytic Combinatorics (2009)
[9]	Zagier, The Mellin transform and other useful analytic techniques. Appendix to Zeidler, E, Quantum Field Theory I: Basics in Mathematics and Physics. A Bridge Between Mathematicians and Physicists pp 305– (2006)
[10]	Zudilin, Algebraic relations for multiple zeta values, Russ. Math. Surv. 58 pp 1– (2003) · Zbl 1171.11323
[11]	Brown, Tate motives over \(\mathbb{Z}\), Ann. Math. 175 pp 949– (2012) · Zbl 1278.19008