2017/03/11 by Steven Charlton, Charlton, Steven
Mathematics · #11G55 #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Mathematics and Applications #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1703.03961
openalex publication_date 2017/03/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we will give an account of Dan's reduction method for reducing the weight n multiple logarithm I1,1,…,1(x1, x2, …, xn) to an explicit sum of lower depth multiple polylogarithms in ≤ n - 2 variables. We provide a detailed explanation of the method Dan outlines, and we fill in the missing proofs for Dan's claims. This establishes the validity of the method itself, and allows us to produce a corrected version of Dan's reduction of I1,1,1,1 to I3,1 's and I4 's. We then use the symbol of multiple polylogarithms to answer Dan's question about how this reduction compares with his earlier reduction of I1,1,1,1 , and his question about the nature of the resulting functional equation of I3,1 . Finally, we apply the method to I1,1,1,1,1 at weight 5 to first produce a reduction to depth ≤ 3 integrals. Using some functional equations from our thesis, we further reduce this to I3,1,1 , I3,2 and I5 , modulo products. We also see how to reduce I3,1,1 to I3,2 , modulo δ (modulo products and depth 1 terms), and indicate how this allows us to reduce I1,1,1,1,1 to I3,2 's only, modulo δ.