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

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.

I've created this question and has no more context, to know what are the method and approach that you can use.

• Can you find a non-trivial upper bound for $$\left|\sum_{n=1}^{N}e^{2\pi i\log(n)}\right|$$ ? $\log(n)$ (just like $\psi(n+1)$) is slowly increasing. – Jack D'Aurizio Jan 3 '18 at 22:32
• Now I don't know how to find such non-trivial upper bound. Obviously I know that my sum is related to harmonic numbers $H_n$, with $n\geq 1$. Maybe I can to search what about the example that you are saying @JackD'Aurizio (how works the example with the logarithm). – user243301 Jan 3 '18 at 22:36
• As a rule of thumb, it is best to be able to tackle simple problems before wondering about more complex ones. Exponential sums are very important in additive number theory, I suggest you to have a look at Vinogradov - The method of trigonometrical sums in the theory of numbers. – Jack D'Aurizio Jan 3 '18 at 22:39
• I was interested in this example because it is the same example that the exponential sum for harmonic numbers $H_n$. Many thanks @JackD'Aurizio – user243301 Jan 3 '18 at 22:40