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Euler’s formula for the zeta function at the positive even integers. (English) Zbl 1428.11147
Summary: We give a new proof of Euler’s formula for the values of the Riemann zeta function at the positive even integers. The proof involves estimating a certain integral of elementary functions two different ways and using a recurrence relation for the Bernoulli polynomials evaluated at $$\frac{1}{2}$$.
MSC:
 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$ 11B68 Bernoulli and Euler numbers and polynomials
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References:
 [1] 10.1090/S0002-9939-2010-10565-8 · Zbl 1223.40001 [2] 10.2307/2319041 · Zbl 0293.10001 [3] 10.2307/2690371 · Zbl 0526.01015 [4] ; Montgomery, Multiplicative number theory, I : Classical theory. Multiplicative number theory, I : Classical theory. Cambridge Studies in Advanced Mathematics, 97, (2007) · Zbl 1142.11001 [5] 10.1090/S0273-0979-07-01175-5 · Zbl 1135.01010 [6] 10.1007/978-0-8176-4571-7
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