3.2 $S[x]$: Unit-Step Staircase Function

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$S[x]$ – A staircase function which takes a unit-step at each positive integer.

The unit-step staircase function $S[x]$ is defined as follows.

$\quad S[x]=\sum_{n=1}^{\lfloor x\rfloor}\theta[x-n]=\lfloor x\rfloor$


The following plot illustrates $S[x]$ (blue) takes a unit step at each positive integer. The linear function $x$ (orange) is shown as a reference.

Unit-Step Staircase Function $S[x]$

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

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


The unit-step staircase function $S[x]$ and it’s first-order derivative $S'[x]$ are related to the Riemann zeta function $\zeta[s]$ as follows.

$\quad\zeta[s]=s\int_0^\infty S[x]\,x^{-s-1} dx\,,\quad\Re[s]>1$

$\quad\zeta[s]=\int_0^\infty S'[x]\,x^{-s} dx\,,\quad\Re[s]>1$