![{\displaystyle f(x)=-{\frac {1}{2\pi i}}{\frac {1}{\log x}}\int \limits _{a-\infty i}^{a+\infty i}{\frac {d{\frac {\log \zeta (s)}{s}}}{ds}}x^{s}ds}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf90322c867bb381fafb757994c4020484035dca)
umzuformen.
Da
![{\displaystyle -\log \prod {\frac {s}{2}}=\lim \left(\sum _{n=1}^{n=m}\log \left(1+{\frac {s}{2n}}\right)-{\frac {s}{2}}\log m\right),\quad \mathrm {f{\ddot {u}}r} \;m=\infty ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f978cb5b42651512af61b7d870638149bec8b862)
also
![{\displaystyle -{\frac {d{\frac {1}{s}}\log \prod {\frac {s}{2}}}{ds}}=\sum _{1}^{\infty }{\frac {d{\frac {1}{s}}\log \left(1+{\frac {s}{2n}}\right)}{ds}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d29c37c9af461901c64da9455c409a0b914ba0e)
so erhalten dann sämmtliche Glieder des Ausdrucks für
mit Ausnahme von
![{\displaystyle {\frac {1}{2\pi i}}{\frac {1}{\log x}}\int \limits _{a-\infty i}^{a+\infty i}{\frac {1}{ss}}\log \xi (0)x^{s}ds=\log \xi (0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/837889f77b4c6b64eed2c8dd9dd65c61ca576515)
die Form
![{\displaystyle \pm {\frac {1}{2\pi i}}{\frac {1}{\log x}}\int \limits _{a-\infty i}^{a+\infty i}{\frac {d\left({\frac {1}{s}}\log \left(1-{\frac {s}{\beta }}\right)\right)}{ds}}x^{s}ds.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da7019857a3b3577cfe8bcd4c759fa1e119c6a47)
Nun ist aber
![{\displaystyle {\frac {d\left({\frac {1}{s}}\log \left(1-{\frac {s}{\beta }}\right)\right)}{d\beta }}={\frac {1}{(\beta -s)\beta }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8763a22332cb9c5db5f7863fd1072565e7850700)
und, wenn der reelle Theil von
größer als der reelle Theil von
ist,
![{\displaystyle {\begin{aligned}{\frac {1}{2\pi i}}\int \limits _{a-\infty i}^{a+\infty i}{\frac {x^{s}ds}{(\beta -s)\beta }}&={\frac {x^{\beta }}{\beta }}&=\int \limits _{\infty }^{x}x^{\beta -1}dx,\\&\;{\text{oder}}&=\int \limits _{0}^{x}x^{\beta -1}dx,\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a6f295bcfabb07d87e2eed67eb846232f028274)
je nachdem der reelle Theil von
negativ oder positiv ist. Man hat daher
![{\displaystyle {\frac {1}{2\pi i}}{\frac {1}{\log x}}\int \limits _{a-\infty i}^{a+\infty i}{\frac {d\left({\frac {1}{s}}\log \left(1-{\frac {s}{\beta }}\right)\right)}{ds}}x^{s}ds}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b62e5b2265d4569f16b27faf6d59d128580b7c89)