I've been looking at the zeros of the incomplete zeta function $\zeta_{lower}(s, z)$ recently. $$ \zeta_{\mathrm{lower}}(s,z)=-\frac{{\Gamma(1-s)}}{2\pi i}\int_{z}^{\infty}\frac{{(-t)^{s-1}}}{e^{t}-1}dt $$ The subscript "lower" indicates that the contour of integration passes below the branch cut on the positive real axis. By zeros, I mean I mean the zeros in $z$, for fixed $s$. I find that this function can have an infinite number of complex zeros $z_k$, each of which is a function of the parameter $s$. The zeros (for fixed $s$) live on a smooth curve and have an accumulation point at the origin.

Let $s_0$ be any zero of the (original, complete) Riemann zeta function. A remarkable thing happens when $s$ is near $s_0$. The kth zero, $z_k(s)$ depends logarithmically on $(s_0-s)$. $$ \log(z_k) \approx \frac{1}{s-1} \log(s_0-s) + C $$ Assuming $|\operatorname{Im}(s)| >> |\operatorname{Re}(s)|$, the entire set $z_k$ rotate about the origin as $s \rightarrow s_0$. Imagine a (slightly bent) barn door rotating about its hinge.

For added fun, if you move $s$ around $s_0$ and return to the original point, $z_k$ will become $z_{k+1}$ or $z_{k-1}$ depending on the sense of the rotation in $s$.

I've verified these prediction numerically.

So my question is whether anyone has seen this before, for the zeta function or for other incomplete functions.

I've been looking at the zeros of the incomplete zeta function $\zeta_{lower}(s, z)$ recently. $$ \zeta_{\mathrm{lower}}(s,z)=-\frac{{\Gamma(1-s)}}{2\pi i}\int_{z}^{\infty}\frac{{(-t)^{s-1}}}{e^{t}-1}dt $$ The subscript "lower" indicates that the contour of integration passes below the branch cut on the positive real axis. By zeros, I mean I mean the zeros in $z$, for fixed $s$. I find that this function can have an infinite number of complex zeros $z_k$, each of which is a function of the parameter $s$. The zeros (for fixed $s$) live on a smooth curve and have an accumulation point at the origin.

Let $s_0$ be any zero of the (original, complete) Riemann zeta function. A remarkable thing happens when $s$ is near $s_0$. The kth zero, $z_k(s)$ depends logarithmically on $(s_0-s)$. $$ \log(z_k(s)) \approx \frac{1}{s-1} \log(s_0-s) + C_k(s_0) $$ Assuming $|\operatorname{Im}(s)| >> |\operatorname{Re}(s)|$, the entire set $z_k$ rotate about the origin as $s \rightarrow s_0$. Imagine a (slightly bent) barn door rotating about its hinge.

For added fun, if you move $s$ around $s_0$ and return to the original point, $z_k$ will become $z_{k+1}$ or $z_{k-1}$ depending on the sense of the rotation in $s$.

I've verified these prediction numerically down to a distance of $10^{-20}$ between $s$ and $s_0$.

So my question is whether anyone has seen this before, for the zeta function or for other incomplete functions.

I've been looking at the zeros of the incomplete zeta function $\zeta_{lower}(s, z)$ recently. $$ \zeta_{lower}(s,z)=-\frac{{\Gamma(1-s)}}{2\pi i}\int_{z}^{\infty}\frac{{(-t)^{s-1}}}{e^{t}-1}dt $$ The subscript "lower" indicates that the contour of integration passes below the branch cut on the positive real axis. By zeros, I mean I mean the zeros in $z$, for fixed $s$. I find that this function can have an infinite number of complex zeros $z_k$, each of which is a function of the parameter $s$. The zeros (for fixed $s$) live on a smooth curve and have an accumulation point at the origin.

Let $s_0$ be any zero of the (original, complete) Riemann zeta function. A remarkable thing happens when $s$ is near $s_0$. The kth zero, $z_k(s)$ depends logarithmically on $(s_0-s)$. $$ \log(z_k) \approx \frac{1}{s-1} \log(s_0-s) + C $$ Assuming $|Im(s)| >> |Re(s)|$, the entire set $z_k$ rotate about the origin as $s \rightarrow s_0$. Imagine a (slightly bent) barn door rotating about its hinge.

For added fun, if you move $s$ around $s_0$ and return to the original point, $z_k$ will become $z_{k+1}$ or $z_{k-1}$ depending on the sense of the rotation in $s$.

I've verified these prediction numerically.

So my question is whether anyone has seen this before, for the zeta function or for other incomplete functions.

I've been looking at the zeros of the incomplete zeta function $\zeta_{lower}(s, z)$ recently. $$ \zeta_{\mathrm{lower}}(s,z)=-\frac{{\Gamma(1-s)}}{2\pi i}\int_{z}^{\infty}\frac{{(-t)^{s-1}}}{e^{t}-1}dt $$ The subscript "lower" indicates that the contour of integration passes below the branch cut on the positive real axis. By zeros, I mean I mean the zeros in $z$, for fixed $s$. I find that this function can have an infinite number of complex zeros $z_k$, each of which is a function of the parameter $s$. The zeros (for fixed $s$) live on a smooth curve and have an accumulation point at the origin.

Let $s_0$ be any zero of the (original, complete) Riemann zeta function. A remarkable thing happens when $s$ is near $s_0$. The kth zero, $z_k(s)$ depends logarithmically on $(s_0-s)$. $$ \log(z_k) \approx \frac{1}{s-1} \log(s_0-s) + C $$ Assuming $|\operatorname{Im}(s)| >> |\operatorname{Re}(s)|$, the entire set $z_k$ rotate about the origin as $s \rightarrow s_0$. Imagine a (slightly bent) barn door rotating about its hinge.

For added fun, if you move $s$ around $s_0$ and return to the original point, $z_k$ will become $z_{k+1}$ or $z_{k-1}$ depending on the sense of the rotation in $s$.

I've verified these prediction numerically.

So my question is whether anyone has seen this before, for the zeta function or for other incomplete functions.