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Mathematics > Number Theory
Title: Un développement asymptotique des sommes harmoniques de Kempner-Irwin
(Submitted on 21 Apr 2024 (v1), last revised 27 Apr 2024 (this version, v2))
Abstract: Let $I(b,d,k)$ be the subseries of the harmonic series keeping only those denominators having exactly $k$ occurrences of the digit $d$ in base $b$. We prove the existence of an asymptotic development to all orders in descending powers of $b$, for $d$ and $k$ fixed, of $I(b,d,k)-b \log(b)$ or rather, if $k=0$ and $d>0$, of $I(b,d,0)- (b\log(b) - b\log(1+1/d))$. We give explicitly, according to cases, the first four or first five terms. The coefficients of these developments involve the values of the Riemann zeta function at positive integers.
Submission history
From: Jean-François Burnol [view email][v1] Sun, 21 Apr 2024 20:19:22 GMT (33kb)
[v2] Sat, 27 Apr 2024 19:34:52 GMT (33kb)
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