Upper bound of $\left| \sum_{n=1}^N e^{2\pi i\psi(n+1)} \right|$, where $\psi(x)$ is the digamma function

Question

Let $\psi(x)$ the digamma function, see its definition and relation with harmonic functions from this Wikipedia.

Question. I am interested about if is known how to find an upper bound of $$ \left| \sum_{n=1}^N e^{2\pi i\psi(n+1)} \right|, \tag{1}$$ as $\leq N\Delta$, where $\Delta=\Delta(N)$ is little and satisfies $\Delta(N)\to 0$ as $N\to\infty$. Thanks you in advance.

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