Sums of digits and the Hurwitz zeta function

Abstract

Let s ₂(n) denote the sum of the binary digits of n. Then it is easily seen that

$$\sum\limits_{n = 1}^{ + \infty } {\frac{{s_2 (n)}}{{n(n + 1)}} = 2Log2.}$$

We prove that, if s _B (n) is the sum of the digits of n in base B, and if a _w,B (n) is the number of (possibly overlapping) occurrences of the word w in the B-ary expansion of n, then the series

$$\sum\limits_{n = 1}^{ + \infty } {s_B (n)\left( {\frac{1}{{n^s }} - \frac{1}{{(n + 1)^s }}} \right) } and \sum\limits_{n = 1}^{ + \infty } {a_{w,B} (n)\left( {\frac{1}{{n^s }} - \frac{1}{{(n + 1)^s }}} \right) }$$

can be expressed in terms of the Riemann zeta function or of the Hurwitz zeta function.

This allows us to show that

$$\sum\limits_1^{ + \infty } {\frac{{s_B (n)}}{{n(n + 1)}} = \frac{B}{{B - 1}}LogB}$$

but also to give formulas like

$$\sum\limits_1^{ + \infty } {\frac{{s_2 (n)(2n + 1)}}{{n^2 (n + 1)^2 }} = \frac{{\pi ^2 }}{9}.}$$

Moreover we give a general expression for the series

$$\sum\limits_{n = 1}^{ + \infty } {\frac{{a_{w,B} (n)}}{{n(n + 1)}}.}$$

For example one has:

$$\sum\limits_{n = 1}^{ + \infty } {\frac{{a_{11,2} (n)}}{{n(n + 1)}} = \frac{3}{2}Log2 - \frac{\pi }{2}} and \sum\limits_{n = 1}^{ + \infty } {\frac{{a_{01,2} (n)}}{{n(n + 1)}} = \frac{1}{2}Log2 + \frac{\pi }{4}.}$$