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, $S(a)\equiv {\sum }_{n=1}^{\mathrm{\infty }}[\ln \frac{n}{a}-\frac{a}{2n}-\psi \left(\frac{n}{a}\right)]$, 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.