3.3 $T[x]$: Log-Step Staircase Function

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$T[x]$ – A staircase function which takes a step of $\log[n]$ at each positive integer $n$.

The log-step staircase function $T[x]$ is defined as follows.

$\quad T[x]=\sum _{n=1}^{\lfloor x\rfloor}\log[n]$


The following plot illustrates $T[x]$ (blue) takes a step of $\log[n]$ at each positive integer $x=n$. The function $x\,(\log[x]-1)-\frac{1}{2}\left(\log\left[\frac{1}{2}\right]-1\right)$ (orange) is shown as a reference.

Log-Step Staircase Function $T[x]$

The log-step staircase function $T[x]$ is related to the unit-step staircase function $S[x]$ via their first-order derivatives as follows.

$\quad T'[x]=\log[x]\ S'[x]$


The log-step staircase function $T[x]$ and it’s first-order derivative $T'[x]$ are related to the first-order derivative $\zeta'[s]$ of the Riemann zeta function as follows.

$\quad\zeta'[s]=-s\int_0^\infty T[x]\,x^{-s-1} dx\,,\quad\Re[s]>1$

$\quad \zeta'[s]=-\int_0^\infty T'[x]\,x^{-s} dx\,,\quad\Re[s]>1$