via the following functions: $$f(x) = f(x;\vec{\gamma},\vec{\delta})  := \sum_{k=1}^K{{\left\lfloor {\frac{x}{\gamma_k}} \right\rfloor}} -\sum_{l=1}^L{{\left\lfloor {\frac{x}{\delta_l}} \right\rfloor}}.
\label{eqn.function}$$

The bridge between $f(x;\vec{\gamma},\vec{\delta}) $ and $v(n;\vec{\gamma},\vec{\delta})$ was uncovered by Landau.

The ratio is integral for all $n$ if and only if the function is nonnegative for all real positive $x$.

Under the balancing condition, the function $f$ is easily seen to be periodic, of period dividing $M(\vec{\gamma},\vec{\delta}) := \operatorname{lcm}(\gamma_1,\cdots,\gamma_K,\delta_1,\cdots,\delta_L )$. We will present additional properties[^3] of $f$ in the course of the proof of Theorem \[thm.height1hook\], in Section \[sec.height1hook\].

The importance of functions of type comes from the Beurling-Nyman criterion for the Riemann Hypothesis. This asserts that the Riemann zeta function $\zeta(s)$ has no zeros in the half-plane $\sigma > \frac{p-1}{p}$ if and only if for any $\epsilon >0$ there exists a function $\tilde{f}(x) = \sum_{n=1}^N c_n {{\left\lfloor {\alpha_n x} \right\rfloor}} 
$ (with $\sum c_n \alpha_n =0$ and $0 \le \alpha_n \le 1$ for all $n$) such that $$\left(\int_{1}^\infty  \left|\frac{1-\tilde{f}(x)}{x}\right|^p {\mathrm{d}}x\right)^{1/p} < \epsilon.$$ In other words, the Riemann Hypothesis is true if and only if linear combinations of similar to can successfully approximate the constant function 1 in the $L^2$-norm ($p=2$ gives $\frac{p-1}{p} = \frac12$).

An example of $\vec{\delta}$ and $\vec{\gamma}$ would be $(30,1)$ 