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Article Dans Une Revue Hardy-Ramanujan Journal Année : 2022

Proof of the functional equation for the Riemann zeta-function

Résumé

In this article, we shall prove a result which enables us to transfer from finite to infinite Euler products. As an example, we give two new proofs of the infinite product for the sine function depending on certain decompositions. We shall then prove some equivalent expressions for the functional equation, i.e. the partial fraction expansion and the integral expression involving the generating function for Bernoulli numbers. The equivalence of the infinite product for the sine functions and the partial fraction expansion for the hyperbolic cotangent function leads to a new proof of the functional equation for the Riemann zeta function.
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Dates et versions

hal-03251104 , version 1 (06-06-2021)
hal-03251104 , version 2 (05-01-2022)

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Citer

Jay Mehta, P. -y Zhu. Proof of the functional equation for the Riemann zeta-function. Hardy-Ramanujan Journal, 2022, Special commemorative volume in honour of Srinivasa Ramanujan - 2021, Volume 44 - Special Commemorative volume in honour of Srinivasa Ramanujan - 2021, pp.143 -- 151. ⟨10.46298/hrj.2022.7663⟩. ⟨hal-03251104v2⟩

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