Connection to Theta Functions
\zeta(s)=\frac{\pi^{\frac{s}{2}}}{s(s-1)\Gamma\left(\frac{1}{2}s\right)}+\frac{\pi^{\frac{s}{2}}}{\Gamma\left(\frac{1}{2}s\right)}\int_{1}^{\infty}\frac{\omega(x)}{x}\left(x^{\frac{s}{2}}+x^{\frac{1-s}{2}}\right)dx
The Riemann Zeta Function is significant in many areas of mathematics, including the study of Theta Functions. The formula presented here provides an introductory yet intriguing insight into this topic; for example, part of the same reasoning used in the proof appears in Riemann’s “On the number of Primes less than a given quantity”.
Our demonstration heavily utilizes Poisson’s Summation Formula.
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