Sum of digamma asymptotic error terms of an arithmetic series

We define an S function as the sum of the asymptotic error terms of digamma function of an arithmetic series, $S(a) \equiv \sum_{n=1}^\infty \left[\ln\frac{n}{a} - \frac{a}{2n}-\psi\left(\frac{n}{a}\right)\right]$, and show a few properties of it. Using the S function, we construct a real and positive $\phi$ function. Riemann hypothesis holds if $\tilde{\phi}(k)$, the complex Fourier transform of $\phi$, has only real zeros.