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Summation of Series Defined by Counting Blocks of Digits

Abstract

We discuss the summation of certain series defined by counting blocks of digits in the BB-ary expansion of an integer. For example, if s2(n)s_2(n) denotes the sum of the base-2 digits of nn, we show that n1s2(n)/(2n(2n+1))=(γ+log4π)/2\sum_{n \geq 1} s_2(n)/(2n(2n+1)) = (\gamma + \log \frac{4}{\pi})/2. We recover this previous result of Sondow in math.NT/0508042 and provide several generalizations.Comment: 12 pages, Introduction expanded, references added, accepted by J. Number Theor

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