Tests of conjectures on multiple Watson values

2015/04/29 by Broadhurst, David
#FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1504.08007

Abstract

I define multiple Watson values (MWVs) as iterated integrals, on the interval x∈[0,1], of the 6 differential forms A=dlog(x), B=-dlog(1-x), T=-dlog(1-z₁x), U=-dlog(1-z₂x), V=-dlog(1-z₃x) and W=-dlog(1-z₄x), where z₁=γ², z₂=γ/(1+γ), z₃=γ²/(1-γ) and z₄=γ=2sin(π/14) solves the cubic (1-γ²)(1-γ)=γ. Following a suggestion by Pierre Deligne, I conjecture that the dimension of the space of \mathbb Z-linearly independent MWVs of weight w is the number D_w generated by 1/(1-2x-x²-x³)=1+∑_w>0D_w x^w. This agrees with 6639 integer relation searches, of dimensions up to D₅+1=85, performed at 2000-digit precision, for w<6.

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