Let $\eta (s)={\sum }_{n=1}^{\infty }{(-1)}^{n+1}{n}^{-s}$ be the alternating zeta function. For a real number τ we define certain complex numbers $b_{M,m}(\tau )$ and consider finite Dirichlet series ${\upsilon }_M(\tau ,s)={\sum }_{m=1}^M{b}_{M,m}(\tau )m^{-s}$ and ${\eta }_N(\tau ,s)={\sum }_{M=1}^N{\upsilon }_M(\tau ,s)$. Computations demonstrate some remarkable properties of these finite Dirichlet series, but nothing was supported by a proof so far.

First, numerical data show that ${\eta }_N(\tau ,s)$ approximates $\eta (s)$ with high accuracy for s in the vicinity of $1/2+\mathrm{i}\tau$; this allows one to surmise that(*)$\eta (s)=\sum _{M=1}^{\infty }{\upsilon }_M(\tau ,s).$ Moreover, it looks that$\underset{M\to \infty }{\lim }\frac{\phantom{m}\overline{{\upsilon }_M(\tau ,1-\sigma +\mathrm{i}t)}\phantom{m}}{{\upsilon }_M(\tau ,\sigma +\mathrm{i}t)}=\frac{\eta (\sigma +\mathrm{i}t)}{\phantom{m}\overline{\eta (1-\sigma +\mathrm{i}t)}\phantom{m}};$ in other words, the individual summands in expected expansion $(⁎)$ satisfy with an increasing accuracy a counterpart of the classical functional equation.

Let ${\mathrm{\Upsilon }}_M(\tau ,\sigma +\mathrm{i}t)={\upsilon }_M(\tau ,\sigma +\mathrm{i}t)/\eta (\sigma +\mathrm{i}t)$. When M, τ, and either σ or t are fixed, and the fourth parameter varies, the plot of ${\mathrm{\Upsilon }}_M(\tau ,\sigma +\mathrm{i}t)$ on the complex plane contains numerous almost ideally circular arcs with geometrical parameters closely related to the non-trivial zeros of the zeta function.

让 $\eta (秒)={\sum }_{n=1}^{\infty }{(-1)}^{n+1}{n}^{-秒}$是交替 zeta 函数。对于实数τ，我们定义某些复数${乙}_{米,米}(\tau )$ 并考虑有限狄利克雷级数 ${ü}_{米}(\tau ,秒)={\sum }_{米=1}^{米}{乙}_{米,米}(\tau ){米}^{-秒}$ 和 ${\eta }_N(\tau ,秒)={\sum }_{米=1}^N{ü}_{米}(\tau ,秒)$. 计算证明了这些有限狄利克雷级数的一些显着性质，但迄今为止没有任何证据支持。

首先，数值数据表明 ${\eta }_N(\tau ,秒)$ 近似值 $\eta (秒)$对于s附近的高精度$1/2+\mathrm{一世}\tau$; 这使人们可以推测(*)$\eta (秒)=\sum _{米=1}^{\infty }{ü}_{米}(\tau ,秒).$ 而且，看起来$\underset{米\to \infty }{\mathrm{林}}\frac{\phantom{米}\overline{{ü}_{米}(\tau ,1-\sigma +\mathrm{一世}吨)}\phantom{米}}{{ü}_{米}(\tau ,\sigma +\mathrm{一世}吨)}=\frac{\eta (\sigma +\mathrm{一世}吨)}{\phantom{米}\overline{\eta (1-\sigma +\mathrm{一世}吨)}\phantom{米}};$换句话说，预期扩展中的单个被加数$(⁎)$ 以越来越高的精度满足经典函数方程的对应物。

让 ${\mathrm{⒈}}_{米}(\tau ,\sigma +\mathrm{一世}吨)={ü}_{米}(\tau ,\sigma +\mathrm{一世}吨)/\eta (\sigma +\mathrm{一世}吨)$. 当M、τ和σ或t是固定的，并且第四个参数变化时，图${\mathrm{⒈}}_{米}(\tau ,\sigma +\mathrm{一世}吨)$ 在复平面上包含许多几乎理想的圆弧，其几何参数与 zeta 函数的非平凡零点密切相关。