Wenig bekannte Fakten über Reinugung Basel.

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, we can conclude much more about this class of zeta constants. In particular, since π displaystyle pi

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which in our situation equates to the limiting recurrence relation (or generating function convolution, or product) expanded as

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The agreement he observed gave him sufficient confidence to announce his result to the mathematical community.

using the method of elementary symmetric polynomials. Namely, we have a recurrence relation between the elementary symmetric polynomials and the power sum polynomials given as on this page by

By Vieta's formulas we can calculate the sum of the roots directly by examining the first two coefficients of the polynomial, and this comparison shows that

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Using the Weierstrass factorization theorem, it can also Beryllium shown that the right-hand side is Reinugung Basel the product of linear factors given by its roots, just as we do for finite polynomials (which Euler assumed as a heuristic for expanding an infinite degree polynomial in terms of its roots, but rein fact is not always true for general P ( x ) displaystyle P(x)

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Euler generalised the Schwierigkeit considerably, and his ideas were taken up years later by Bernhard Riemann in his seminal 1859 paper "On the Number of Primes Less Than a Given Magnitude", in which he defined his zeta function and proved its basic properties. The problem is named after Basel, hometown of Euler as well as of the Bernoulli family who unsuccessfully attacked the Harte nuss.

Other notably special identities and representations of this constant appear rein the sections below. Series representations[edit]

The Riemann zeta function ζ(s) is one of the most significant functions rein mathematics because of its relationship to the distribution of the prime numbers. The zeta function is defined for any complex number s with real part greater than 1 by the following formula:

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to Apéry's proof. Hinein particular, he documents recurrence relations for almost integer sequences converging to the constant and continued fractions for the constant. Other continued fractions for this constant include[16]

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