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Given the canonical infinite product representation (Weierstrass form) of the gamma function, $$\Gamma(z)= \left [ze^{\gamma z}\prod_{m=1}^{\infty} \left ( 1+ \frac{z}{m} \right)e^{-z/m} \right ]^{-1} $$where $\gamma$ is the Euler-Mascheroni constant, and the meromorphic continuation of the Riemann-zeta function $\xi(z)=\frac {1}{2}\int_{t=1}^{\infty}(\vartheta(it)-1)(t^{z/2}+t^{(1-z)/2} )\frac{dt}{t}= \pi^{-z/2}\Gamma(\frac{z}{2})\zeta(z)$; $\vartheta(z)$ being the Jacobi-theta function, then is there an analytic expression that can be imposed on the complex variable $z$ that satisfies the following condition: $|\zeta(z)|<|\xi(z)|$, and conversely (i.e. $|\zeta(z)|>|\xi(z)|$)?

Best Regards

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  • $\begingroup$ Hint: Use Weierstrass factorization theorem that asserts an entire function can be represented by a product involving its zeroes $\endgroup$ – mwomath May 17 '16 at 21:14

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