Explicit results on exponential sums¶

Collecting references: [Daboussi and Rivat, 2001].

1. Bounds with the first derivative¶

We start with the Kusmin-Landau Lemma.

Theorem

Let $f$ be a function over $[a, b]$ such that $f^\prime$ is monotonic and satisfies $\theta \le f^\prime(u)\le 1-\theta$ for some $\theta \in(0,1/2]$. Then $\displaystyle \biggl|\sum_{a\le n\le b} e(f(n))\biggr| \le \cot\frac{\pi\theta}{2}\le \frac{2}{\pi \theta}.$

2. Bounds with the second derivative¶

Here is a corrected version of Lemma 3 of [Cheng and Graham, 2004], see Lemma 2.3 of [Patel, 2022].

Theorem (2004)

Let $f$ be a real-valued function with two continuous derivatives on $[N+1, N+L]$. Suppose there are $W > 1$ and $\lambda > 1$ such that $1 \le W |f^{\prime\prime}(x)| \le \lambda$ for every $x\in [N+1, N+L]$. Then we have $\displaystyle \biggl|\sum_{n= N+1}^{N+L} \exp( 2i\pi f(n)) \biggr| \le 2\biggl(\frac{L \lambda}{W} +2\biggr) \biggl(2\sqrt{\frac{W}{\pi}} + 1\biggr).$

3. Bounds with the third derivative¶

Here is Lemma 1.2 of: [Hiary, 2016]. See Lemma 2.3 [Patel, 2022].

Theorem (2016)

Let $f$ be a real-valued function with three continuous derivatives on $[N+1, N+L]$. Suppose there are $W > 1$ and $\lambda > 1$ such that $1 \le W |f^{\prime\prime\prime}(x)| \le \lambda$ for every $xin [N+1, N+L]:math:. Then, for any `eta > 0$, we have :math:displaystyle biggl|sum_{n= N+1}^{N+L} exp( 2ipi f(n)) biggr|^2 le (LW^{-1/3} +eta) (alpha L + beta W^{2/3})` where $\displaystyle \alpha = \frac{1}{\eta} +\frac{64\lambda}{75} \sqrt{\eta + W^{-1/3}}+\frac{\lambda\eta}{W^{1/3}} +\frac{\lambda}{W^{2/3}},$ and $\displaystyle \beta = \frac{65}{15\sqrt{\eta}} + \frac{3}{W^{1/3}}.$

4. Bounds with higher derivatives¶

See Lemma 3.1 and 3.2 of [Patel, 2022].

5. Iterated Van der Corput Inequality¶

During the proof of Lemma 8.6 in: [Granville and Ramaré, 1996] one finds the next inequality.

Theorem (1996)

Let $f$ be a real-valued function with $k+1$ continuous derivatives on $(A, B]$ and let $N$ be a lower bound for the number of integers in $(A,B]$. The quantity $\displaystyle \biggl|\frac{1}{8N} \sum_{A < n\le B} \exp(2 i \pi f(n))\biggr|^{2^k}$ is bounded above by $\displaystyle \frac{1}{8}\biggl(\frac{1}{Q} + \frac{1}{Q^{2-2^{1-k}}} \sum_{r_1 =1}^{Q2^{-0}} \sum_{r_2 =1}^{Q2^{-1}} \cdots \sum_{r_k =1}^{Q2^{-k+1}} \biggl| \frac{1}{N} \sum_{A < n \le B-r_1-r_2-\cdots-r_k} \exp(\pm 2i\pi f_{r_1,\cdots,r_k}(n)) \biggr| \biggr)$ where the function $f_{r_1,\cdots,r_k}$ satisfies $\displaystyle \forall t,\ \exists y\in[t, t + r_1 + \cdots + r_k], \quad f^{\prime}_{r_1,\cdots, r_k}(t) = r_1r_2\cdots r_k f^{(k+1)}(y).$

6. Explicit Poisson Formula¶

Here is a consequence of the main theorem of [Karatsuba and Korolëv, 2007].

Theorem (2007)

Suppose $f^\prime$ is decreasing on $[N+1,N+L]$ and set $f^\prime(N+L)=\alpha$ and $f^\prime(N) = \beta$. For integer $\nu\in(\alpha, \beta]$, let $x_\nu$ be the solution to $f^\prime(x)=\nu$. Suppose further that $\lambda_2\le |f^{\prime\prime}(x)|\le h_2\lambda_2$ and $\lambda_3\le |f^{\prime\prime\prime}(x)|\le h_3\lambda_3$. Then $\displaystyle \sum_{n=N+1}^{N+L} \exp(2i\pi f(n)) = \sum_{\alpha < \nu\le \beta} \frac{\exp(2i\pi (f(x_\nu)-\nu x_\nu-1/8))}{\sqrt{f^{\prime\prime}(x_\nu)}} +\mathcal{E}$ where $\displaystyle |\mathcal{E}|\le \frac{40}{\sqrt{\pi}}\lambda^{-1/2} + \frac{3+2h_2}{\pi} \log(\beta-\alpha+2) + 2.9 h_2h_3^{1/5}L(\lambda_2\lambda_3)^{1/5} +1.9.$