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Additional info for Seminaire Bourbaki 1969-1970, Exposes 364-381
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Unterschiedliche Versionen dieser originellen Idee wurden immer wieder entdeckt und publiziert, unter anderem von F. Holme (1970), I. Papadimitriou (1973), und von Ransford (1982), der ihn damals John Scholes zuschrieb. Beweis. Der erste Schritt besteht in einer bemerkenswerten Relation zwischen Werten der quadrierten Kotangens-Funktion. Es gilt n¨amlich f¨ur alle m ≥ 1 cot2 π 2m+1 + cot2 2π 2m+1 + . . + cot2 mπ 2m+1 = 2m(2m−1) . 6 (1) Um dies zu beweisen, beginnen wir mit der Relation eix = cos x + i sin x.
Mit q > 1 gilt offenbar qQ = 1 + 1 q + 1 q2 + ... = 1 + Q und damit 1 Q = . q−1 der ungef¨ahr nb ist, also f¨ur große n eben nicht ganzzahlig sein kann. Er ist b n¨amlich gr¨oßer als n+1 und kleiner als nb , wie man aus dem Vergleich mit einer geometrischen Reihe sieht: 1 < n+1 < 1 1 1 + + + ... n + 1 (n + 1)(n + 2) (n + 1)(n + 2)(n + 3) 1 1 1 1 + + + ... = . -Trick nicht einmal ausreicht, um zu zeigen, dass e2 irrational ist. Das ist eine st¨arkere √ Aussage: 2 ist ein Beispiel einer Zahl, die irrational ist, ihr Quadrat aber nicht.
B ORWEIN & K. D ILCHER : Pi, Euler numbers, and asymptotic expansions, Amer. Math. Monthly 96 (1989), 681-687. [4] S. F ISCHLER : Irrationalit´e de valeurs de zˆeta (d’apr`es Ap´ery, Rivoal, . . ), Bourbaki Seminar, No. 910, November 2002; Ast´erisque 294 (2004), 27-62. [5] J. C. L AGARIAS : An elementary problem equivalent to the Riemann hypothesis, Amer. Math. Monthly 109 (2002), 534-543. [6] W. J. L E V EQUE : Topics in Number Theory, Vol. I, Addison-Wesley, Reading MA 1956. [7] A. M. YAGLOM & I.