[Paper Review] On logarithmic integrals, harmonic sums and variations

Ming Hao Zhao|arXiv (Cornell University)|Nov 27, 2019

Advanced Mathematical Identities19 references8 citations

TL;DR

This paper presents a comprehensive evaluation of logarithmic integrals (LIs), harmonic sums (ESs), and their quadratic and polylogarithmic variants (QLIs, QPLIs, QESs) up to weight 5 using non-MZV methods, with additional results via MZV theory. It derives closed-form solutions for 85 LIs, 89 ESs, 263 PLIs, 193 QLIs, 172 QPLIs, 83 QESs, and related families, establishing connections to hypergeometric series and nonhomogeneous integrals through advanced integration techniques and special function identities.

ABSTRACT

Based on various non-MZV approaches we evaluate certain logarithmic integrals and harmonic sums. More specifically, 85 LIs, 89 ESs, 263 PLIs, 28 non-alt QESs, 39 GESs, 26 BESs,14 IBESs with weight no more than 5, 193 QLIs, 172 QPLIs, 83 QESs, 7 QBESs, 3 IQBESs with weight no more than 4. With help of MZV theory we evaluate other 10 QESs with weight no more than 4. High weight hypergeometric series, nonhomogeneous integrals and other results are also established.

Motivation & Objective

• To systematically evaluate logarithmic integrals (LIs), polylog integrals (PLIs), and their quadratic and harmonic sum variants (QESs, GESs, BESs) up to weight 5 without relying on multiple zeta value (MZV) theory.
• To extend these evaluations to higher-weight cases using MZV theory, particularly for 10 previously intractable QESs with weight ≤4.
• To establish connections between these integrals and nonhomogeneous integrals, nested sums, and generalized hypergeometric functions.
• To solve two longstanding conjectures: Borwein’s binomial sum conjecture and Au’s general LI conjecture, using advanced integral techniques.
• To demonstrate the boundary of elementary methods and the necessity of deeper algebraic structures like QMZV (quadratic MZVs) for higher-weight problems.

Proposed method

• Employs brute force integration, integration by parts, fractional and Weierstrass substitutions, and contour integration to evaluate LIs and PLIs.
• Applies hypergeometric identities, double integration, and parametric integration techniques to derive closed forms for QLIs and QPLIs.
• Utilizes Feynman’s trick on Mellin transforms and beta derivatives to handle parametric families of integrals.
• Employs Valean’s master formula and series expansion methods to evaluate QPLIs and QESs with specific parameter classes.
• Applies symmetric relations, partial fractions, and Fourier expansions to simplify and solve ES and QES families.
• Integrates MZV theory for higher-weight cases where elementary methods fail, particularly for 10 QESs with weight ≤4.

Experimental results

Research questions

• RQ1 What closed-form evaluations can be obtained for logarithmic integrals (LIs) and polylog integrals (PLIs) of weight ≤5 using non-MZV techniques?
• RQ2 How can quadratic log integrals (QLIs) and quadratic polylog integrals (QPLIs) be systematically evaluated up to weight 4?
• RQ3 What is the role of MZV theory in solving the remaining 10 QESs with weight ≤4 that resist elementary methods?
• RQ4 Can nonhomogeneous integrals and generalized Euler sums (GES, BES, IBES) be expressed in terms of known special functions and polylogarithmic constants?
• RQ5 What are the implications of solving Borwein’s and Au’s conjectures for the broader theory of hypergeometric and logarithmic integrals?

Key findings

• The paper derives closed-form expressions for 85 convergent logarithmic integrals (LIs) of weight ≤5 using non-MZV methods.
• It evaluates 89 Euler sums (ESs), 263 polylog integrals (PLIs), 193 quadratic log integrals (QLIs), and 172 quadratic polylog integrals (QPLIs) with weight ≤4.
• For 83 quadratic Euler sums (QESs), closed forms are obtained via non-MZV techniques, while 10 additional QESs with weight ≤4 are resolved using MZV theory.
• The paper solves Borwein’s conjecture on binomial sums and Au’s conjecture on general logarithmic integrals, both previously open problems.
• It establishes connections between high-weight hypergeometric series and integrals, including representations via generalized hypergeometric functions $_pF_q$.
• The results demonstrate that beyond weight 5, irreducible constants emerge, indicating the limits of polylogarithmic closed forms and the necessity of QMZV theory for deeper unification.