A Family of Cotangent-Polylogarithm Integrals, Harmonic Sums, and Level-Four Cyclotomy
K. Srinivasa Raghava1, R. Sivaraman*2
1K. Srinivasa Raghava, Research Associate, Department of Mathematics, Pie Mathematics Association, Chennai (Tamil Nadu), India.
2Dr. R. Sivaraman, Associate Professor, Department of Mathematics, Dwaraka Doss Goverdhan Doss Vaishnav College, Arumbakkam, Chennai (Tamil Nadu), India.
Manuscript received on 10 September 2026 | First Revised Manuscript received on 18 September 2026 | Second Revised Manuscript received on 02 October 2026 | Manuscript Accepted on 15 October 2026 | Manuscript published on 30 October 2026 | PP: 60-64 | Volume-6 Issue-2, October 2026 | Retrieval Number: 100.1/ijam.A125607010427 | DOI: 10.54105/ijam.A1256.07010427
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Abstract: For integers v ≥ 2 and parameters 0 ≤ λ ≤ 1, we study the family Iv (λ) : = ∫ cot 𝜋/2 0 x Liv ( λ 1+cotx ) dx. We prove absolute convergence, reduce the integral to a Dirichlet series with positive coefficients, and evaluate it in terms of a depth-two multiple polylogarithm, with w=(1+i)/2:Iv (λ)=𝜋 2 ImLiv(λw)-ReLiv,1 (λw,w-1). We then derive an exact second-order recurrence for the Dirichlet coefficients, an Euler differential descent in the deformation parameter, a functional difference equation for the weight generating function, and an exponentially convergent expansion in generalized harmonic numbers and zeta values. As consequences, the sequence Iv:=Iv(1)is strictly decreasing, Iv = 𝜋4+(𝜋4-12) 2-v+0(3-v),v → ∞, and no nontrivial linear recurrence with constant coefficients in v can hold. At λ=1,the family gives a concrete one-dimensional model for level-four cyclotomic periods built from harmonic sums.
Keywords: Polylogarithm, Multiple Polylogarithm, Cyclotomic Period, Harmonic Number, Euler Sum, Dirichlet Series.
Scope of the Article: Number Theory