A series involving a product of four consecutive harmonic numbers

2025/07/10 by Chen, Wilson J., Nguyen, Vincent
#11J72 #40A05 #40C99 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2507.19502

Abstract

In correspondence with Goldbach, Euler began investigating series of the form ∑_{k ≥ 1} k^-m(1 + 2^-n + ⋯ + k^-n), which are known today as Euler sums. For the case where n=1 and m ≥ 2, Euler was able to obtain a closed form in terms of zeta values. We use elementary techniques in the spirit of Euler to evaluate the series ∑_{k ≥ 1} \fracH_k H_k+1 H_k+2 H_k+3k(k+1)(k+2)(k+3), where H_k := 1 + (1)/(2) + ⋯ + (1)/(k) is the kth harmonic number, in terms of zeta values. The closed form is a potential counterexample to a conjecture of Furdui and Sîntămărian. We relate this problem to conjectures regarding irrationality and ℚ-linear independence of zeta values.

Citations

• Dirichlet type extensions of Euler sums
• Explicit Evaluations for Several Variants of Euler Sums
• On harmonic numbers and nonlinear Euler sums

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• New Results for Euler Sums
• Euler sums of generalized harmonic numbers and connected extensions
• Euler sums of generalized hyperharmonic numbers
• Four types of variant Euler harmonic sums
• On an infinite number of new families of odd-type Euler sums