How Euler Could Have Done It: Euler and a direct proof for the functional equation for the Riemann zeta-function

a On page 22 in his book [13] Hardy credited Landau and Cahen as the first who mentioned that Euler had stated the functional equation of the Riemann -function in his paper [7].b The same conjecture of ( 2) is also found in his paper "Exercitationes analyticae" [8] (E432: "Analytical Exercises").But he did not give a proof there either.