Abstract
In this chapter we give a formula that describes Bernoulli numbers in terms of Stirling numbers. This formula will be used to prove a theorem of Clausen and von Staudt in the next chapter. As an application of this formula, we also introduce an interesting algorithm to compute Bernoulli numbers.
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Notes
- 1.
Colin Maclaurin (born in February 1698 in Kilmodan, Cowal, Argyllshire, Scotland—died on June 14, 1746 in Edinburgh, Scotland).
- 2.
Differentiable M times and the Mth derivative is continuous.
- 3.
It is unknown whether γ = 0.5772156649015328606065120900824⋯ is an irrational number or not.
- 4.
Georg Friedrich Bernhard Riemann (born on September 17, 1826 in Breselenz, Germany—died on July 20, 1866 in Selasca, Italy).
- 5.
André Weil (born on May 6, 1906 in Paris, France—died on August 6, 1998 in Princeton, USA).
- 6.
Since π is a transcendental number [71] , ζ(2k) are all transcendental numbers.
- 7.
Roger Apéry (born on November 14, 1916 in Ruen, France—died on December 18, 1994 in Caen, France).
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Ibukiyama, T., Kaneko, M. (2014). The Euler–Maclaurin Summation Formula and the Riemann Zeta Function. In: Bernoulli Numbers and Zeta Functions. Springer Monographs in Mathematics. Springer, Tokyo. https://doi.org/10.1007/978-4-431-54919-2_5
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