John D. Cook • 6/27/2026

Writing down harmonic numbers

This article discusses the nth harmonic number, defined as the sum of reciprocals of the first n positive integers. It explains how to express Hn as a fraction using n! and the least common multiple, then provides asymptotic estimates for the number of digits in the numerator and denominator based on the prime number theorem. The author tests these estimates with examples for n=50 and n=1000 using Python code, showing close agreement between predicted and actual digit counts. The content is technical and mathematical, focusing on number theory and computational verification.

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#Python #Asymptotic Analysis #Fractions