The series limit of sum_k 1/[k log k (log log k)²]

2009/02/04 by Richard J. Mathar, Mathar, Richard J.
Mathematics · #40-04 #40A25 #65B10 #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Mathematics and Applications #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.0902.0789

openalex publication_date 2009/02/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The slowly converging series sum_k=3ⁱnfinity 1/[k * log k * (log log k)^a] is evaluated to 38.4067680928 at a=2. After some initial terms, the infinite tail of the sum is replaced by the integral of the associated interpolating function, which is available in simple analytic form. Biases that originate from the difference between the smooth area under the function and the corresponding Riemann sum are corrected by standard means. The cases a=3 and a=4 are computed in the same manner.

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