Sunday, 29 March 2015

calculus - Elementary derivation of certian identites related to the Riemannian Zeta function and the Euler-Mascheroni Constant

Is the proof of these identities possible, only using elementary differential and integral calculus? If it is, can anyone direct me to the proofs? ( or give a hint for the solution )



1)0ex2lnxdx=14(γ+2ln2)π



2)0exln2xdx=γ2+π26




3) γ=1011+xn=1x2n1dx



and lastly,



4) ζ(s)=e(log(2π)1γ/2)s2(s1)Γ(1+s/2)ρ(1sρ)es/ρ



I personally think the last is obtained from a simple use of the Weierstrass factorization theorem. I'm unsure as to what substitution is used.



γ is the Euler-Mascheroni constant and ζ(s) is the Riemannian Zeta function.




Thanks in advance.

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