I watched a BlackPen RedPen video on the Riemann Zeta function in which he derived the Integral Representation of it from the Gamma Function, the link is here https://www.youtube.com/watch?v=ctG4YgMs74w, It is Called the Bose Integral and it appears in Statistical Mechanics.I also viewed the Wikipedia page https://en.wikipedia.org/wiki/Riemann_zeta_function, It shows the Euler Product formula proven by the legendary mathematician Leonhard Euler. But my question is this, How the Integral Representation of the Riemann Zeta Function is linked with the Product Formula of Euler??
Link between Integral and Product representation of Zeta Function
calculusriemann-zeta
Best Answer
If you question strictly speaks about any integral representation of Riemann zeta function, then there are many which includes the one you mentioned (Bose integral). Another integral representation is the one mentioned by Riemann in his famous paper "Ueber die anzahl der primzahlen unter einer gegenbenen grosse".
Actually, if you see the proof of Bose integral, it uses the following representation of zeta function. $$\zeta(s)=\sum_{k\ge 0}\frac{1}{k^s}$$ For all $\Re(s)\gt 0$. Then he does the substitution on integral definition of Gamma function and finally relates it with zeta function by above mentioned definition. Which is very useful for analytic continuation. But the Euler product formula is to be found from above mentioned definition of zeta function. So it's not possible that integral has some other possible relation with product formula. The only relation that exists will be established by that summation definition of zeta function only.
But, there are some other definitions of zeta function like, $$\ln\zeta(s)=\int_{0}^{\infty} x^{-s-1}\Pi(x) dx$$ Where, $\Pi(x)$ denotes the Riemann prime counting function. You can go through the paper i mentioned which makes it obvious that this is directly linked with Euler product formula.