Euler discovered a recursion formula for the Riemann zeta function evaluated at the even integers. He also evaluated special Dirichlet series whose coefficients are the partial sums of the harmonic series. This paper introduces a new method for deducing Euler's formulas as well as a host of new relations, not only for the zeta function but for several allied functions.
Unable to display preview. Download preview PDF.
- 1.T.M. Apostol, Introduction to Analytic Number Theory, UndergraduateTexts in Mathematics, Springer-Verlag, New York, Fifth printing, 1998.Google Scholar
- 2.T.M. Apostol and T.H. Vu, "Dirichlet series related to the Riemann zeta function," Journal of Number Theory 19 (1984) 85–102.Google Scholar
- 3.D.H. Bailey, J.M. Borwein, and R. Girgensohn, "Experimental evaluation of Euler sums," Experimental Mathematics 3 (1994) 17–30.Google Scholar
- 4.T.J.I. Bromwich, An Introduction to the Theory of Infinite Series, 2nd edn., Macmillan and Co., London, 1942.Google Scholar
- 5.R.E. Crandall and J.P. Buhler, "On the evaluation of Euler sums," Experimental Mathematics 3 (1994) 275–285.Google Scholar
- 6.M.E. Hoffman, "The algebra of multiple harmonic series," Journal of Algebra 194 (1997) 477–495.Google Scholar
- 7.K. Knopp, Theory and Application of Infinite Series (R.C. Young, Transl.), 2nd edn. Hafner, New York, 1948.Google Scholar
- 8.L.J. Mordell, "On the evaluation of some multiple series," J. London Math. Soc. 33(2) (1958) 368–371.Google Scholar
- 9.S. Ramanujan, Note Books, Vol. 2, Tata Institute of Fundamental Research, Bombay, 1957.Google Scholar
- 10.G.T. Williams, "A new method of evaluating ς(2n," American Math. Monthly 60 (1953) 19–25.Google Scholar