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Some identities involving the Riemann zeta-function. II. (English) Zbl 0614.10013

[Part I, see the first author and A. Siva Rama Sarma, ibid. 10, 602-607 (1979; Zbl 0399.10003).]

Die Autoren untersuchen Ausdrücke der Form f*g(2n) mit f,g{ζ,ζ ¯,σ,σ ¯,t,t ¯} und f*g*h(2n) mit f,g,h{ζ,σ,t}· Hierbei bedeuten ζ(s)= n=1 n -s die Riemannsche Zeta-Funktion, σ(s)=(1-2 1-s )ζ(s),t(s)=(ζ(s)+σ(s)),f ¯(2n)=(2n-1)f(2n) und f*g(2n)= k=1 n-1 f(2k)g(2n-2k)· Dies führt zu Gleichungen der Art ζ*ζ ¯(2n)=(n-1)(2n+1)ζ(2n)

4 a+b+c=n,a,b,c1 ζ(2a)ζ(2b)ζ(2c)=4ζ*ζ*ζ(2n)=(n+1)(2n+1)ζ(2n)-6ζ(2)ζ(2n-2)·

Die Beweise stützen sich auf Identitäten in Ableitungen und Potenzen von cot πz=1/z-2 n=1 ζ(2n)z 2n-1 sowie den Gleichungen f*g ¯=f*g+f*g ¯+f ¯*g, f*f ¯(2n)=(n-1)f*f(2n).

Reviewer: D.Leitmann

11B39Fibonacci and Lucas numbers, etc.
11M06ζ(s) and L(s,χ)