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Sums of digits and the Hurwitz zeta function

Part of the Lecture Notes in Mathematics book series (LNM,volume 1434)

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:

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

Keywords

  • Functional Equation
  • Zeta Function
  • Riemann Zeta Function
  • Dartmouth College
  • High Transcendental Function

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

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© 1990 Springer-Verlag

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Allouche, JP., Shallit, J. (1990). Sums of digits and the Hurwitz zeta function. In: Nagasaka, K., Fouvry, E. (eds) Analytic Number Theory. Lecture Notes in Mathematics, vol 1434. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0097122

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  • DOI: https://doi.org/10.1007/BFb0097122

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-52787-9

  • Online ISBN: 978-3-540-47147-9

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