= Solution
We use the <Van der Corput sum-integral lemma>. Put $e(u)=e^{2\pi iu}$ and $I_h=\int_a^b e(f(x)-hx)\,dx$. The <Fourier series> of the periodization of $1_{[a,b]}(x)e(f(x))$ gives
$$
\sum_{a\le n\le b}'e(f(n))=\lim_{H\to\infty}\sum_{|h|\le H}I_h,
$$
where integer endpoints have half weight. This is the <Dirichlet-Jordan convergence theorem> for a piecewise smooth, or more generally bounded-variation, periodic function. Here $f$ is $C^1$, so the periodized function has <bounded variation>. Changing to the requested endpoint convention costs at most one.
Write $u=f'$. For $h\ne0$, $|u-h|\ge|h|-\delta>0$. Since $u$ is continuous and <monotone>, the reciprocal has <bounded variation>, and <integration by parts> in the Riemann-Stieltjes sense yields
$$
I_h=\left[\frac{e(f(x)-hx)}{2\pi i(u(x)-h)}\right]_a^b-\frac1{2\pi i}\int_a^b e(f(x)-hx)\,d\!\left(\frac1{u(x)-h}\right).
$$
The variation of the reciprocal is at most $2\delta/(h^2-\delta^2)$. Summing over $h\ne0$ gives $O((1-\delta)^{-1})$, separating $|h|=1$ and using convergence of $\sum_{h\ge2}h^{-2}$. For each endpoint, use
$$
\frac1{u-h}=-\frac1h+\frac{u}{h(u-h)}.
$$
The symmetric partial sums of the first term are a constant multiple of $\sum_{h=1}^H\sin(2\pi hx)/h$, uniformly bounded in $H$ and $x$; this standard <Fourier series> bound follows by splitting at $h\asymp1/\|x\|$ and applying <Abel summation> to the remaining sine sum. The second term is absolutely summable with bound $O((1-\delta)^{-1})$. The same bound therefore holds for the whole sum of the $h\ne0$ integrals. Since $I_0$ is the ordinary integral,
$$
\boxed{\sum_{a<n\le b}e(f(n))=\int_a^be(f(x))\,dx+O\bigl((1-\delta)^{-1}\bigr).}
$$
No second derivative is required; monotonicity supplies the needed variation estimate.
For the <Hardy-Littlewood approximation to the Riemann zeta function>, take $f(w)=-t\log w/(2\pi)$. On $w\ge x\ge|t|/\pi$, $|f'(w)|\le1/2$ and $f'$ is <monotone>. The proved lemma says that the difference between the partial sum of $w^{-it}$ and its integral over $(x,Y]$ is $O(1)$ uniformly in $Y$. Weighted <Abel summation> with the decreasing weight $w^{-\sigma}$ then makes the weighted difference $O(x^{-\sigma})$, since its total variation on $[x,\infty)$ is $x^{-\sigma}$. Initially for $\sigma>1$, the tail integral is $x^{1-s}/(s-1)$. The bounded primitive of the discrepancy gives a <locally uniformly convergent> weighted discrepancy integral for every $\sigma>0$, continuing the identity to that region. Thus, away from the <pole>,
$$
\boxed{\zeta(s)=\sum_{n\le x}n^{-s}+\frac{x^{1-s}}{s-1}+O(x^{-\sigma}),\qquad x\ge|t|/\pi.}
$$
If $t=0$ the ordinary sum-integral comparison supplies the same estimate. At $s=1$ the formula is understood meromorphically. It approximates the <Riemann zeta function> by a finite <Dirichlet polynomial>, transfers <exponential sum> estimates to bounds in the <critical strip>, yields elementary near-one bounds for $\zeta$ and its <derivative>, and supports estimates for the <mean value of Dirichlet polynomials> and numerical calculations.
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