Summation of series defined by counting blocks of digits

AbstractWe discuss the summation of certain series defined by counting blocks of digits in the B-ary expansion of an integer. For example, if s2(n) denotes the sum of the base-2 digits of n, we show that ∑n⩾1s2(n)/(2n(2n+1))=(γ+log4π)/2. We recover this previous result of Sondow and provide several generalizations