Sums of certain fractional parts

TL;DR

This paper establishes an improved upper bound for sums of fractional parts of smooth functions, leveraging Weyl's bound and Popov's technique, with applications in analytic number theory.

Contribution

It introduces a novel upper bound for fractional part sums by combining Weyl's bound and Popov's method, advancing analytic number theory techniques.

Findings

Improved upper bound for fractional part sums

Enhanced main term estimation using Weyl's bound

Application potential in analytic number theory

Abstract

In this note, an upper bound for the sum of fractional parts of certain smooth functions is established. Such sums arise naturally in numerous problems of analytic number theory. The main feature is here an improvement of the main term due to the use of Weyl's bound for exponential sums and a device used by Popov.

Equations

$$\sum _{N<n⩽2N}\psi (f(n))$$

$$\sum _{N<n⩽2N}\psi (f(n))\ll \frac{N}{L}+\sum _{\ell ⩽L}\frac{1}{\ell }\left|\sum _{N<n⩽2N}e(\ell f(n))\right|$$

$$\sum _{N<n⩽2N}\psi (f(n)){\ll }_{k}N{\lambda }_k^{\frac{1}{2^k-1}}+N^{1-2^{1-k}}\log N+N^{1-2^{3-k}+2^{4-2k}}{\lambda }_k^{-2^{1-k}}$$

$$\sum _{N<n⩽2N}\psi (f(n))\ll N{\lambda }_2^{1/3}+{\lambda }_2^{-1/2},$$

$$\sum _{N<n⩽2N}\psi (f(n))\ll N{\lambda }_3^{1/7}+N^{3/4}\log N+N^{1/4}{\lambda }_3^{-1/4},$$

$$\sum _{N<n⩽2N}\psi (f(n))\ll N{\lambda }_4^{1/15}+N^{7/8}\log N+N^{9/16}{\lambda }_4^{-1/8}.$$

$$x_1^j+\cdots +x_s^j=y_1^j+\cdots +y_s^j\left(1⩽j⩽k\right)$$

$$N^{-\epsilon }\sum _{N<n⩽2N}\psi \left(f(n)\right){\ll }_{\epsilon ,k}N{\lambda }_k^{\frac{1}{k^2-k+1}}+N^{1-\frac{1}{k(k-1)}}+N^{1-\frac{2}{k(k-1)}}{\lambda }_k^{-\frac{2}{k^2(k-1)}}$$

$$N^{-\epsilon }\sum _{N<n⩽2N}\psi (f(n)){\ll }_{\epsilon }N{\lambda }_3^{1/7}+N^{5/6}+N^{2/3}{\lambda }_3^{-1/9},$$

$$N^{-\epsilon }\sum _{N<n⩽2N}\psi (f(n)){\ll }_{\epsilon }N{\lambda }_4^{1/13}+N^{11/12}+N^{5/6}{\lambda }_4^{-1/24}.$$

$$\sum _{N<n⩽2N}\psi (P(n))$$

$${\lambda }_j⩽\left|f^{(j)}(x)\right|⩽c_j{\lambda }_j\text{with}{\lambda }_k=s_{k}N{\lambda }_{k+1}.$$

$$N^{-\epsilon }\sum _{N<n⩽2N}\psi \left(f(n)\right){\ll }_{k,\epsilon }N{\lambda }_k^{2^{1-k}}+N{\lambda }_k^{\frac{2^{1-k}}{k}}{\left(N^k{\lambda }_k\right)}^{-\frac{2}{k{d}_k}}+N{\left(N^k{\lambda }_k\right)}^{-\frac{2}{d_k}}.$$

$$N^{-\epsilon }\sum _{N<n⩽2N}\psi \left(f(n)\right){\ll }_{\epsilon }N{\lambda }_2^{1/2}+N^{7/8}{\lambda }_2^{3/16}+N^{3/4}{\lambda }_2^{-1/8},$$

$$N^{-\epsilon }\sum _{N<n⩽2N}\psi \left(f(n)\right){\ll }_{\epsilon }N{\lambda }_3^{1/4}+N^{18/19}{\lambda }_3^{5/76}+N^{16/19}{\lambda }_3^{-1/19},$$

$$N^{-\epsilon }\sum _{N<n⩽2N}\psi \left(f(n)\right){\ll }_{\epsilon }N{\lambda }_4^{1/8}+N^{43/44}{\lambda }_4^{9/352}+N^{10/11}{\lambda }_4^{-1/44}.$$

$$\sum _{M<n⩽M+N}\min \left(H,\frac{1}{\Vert n\alpha \Vert }\right)\ll HN\alpha +\left(N+{\alpha }^{-1}\right)\log H+H.$$

$$\left|\sum _{N<n⩽N_1}{a}_n\right|⩽\frac{1}{H}\left|\sum _{N<n⩽N_1-H}\sum _{h⩽H}{a}_{n+h}\right|+H.$$

$${\alpha }_n:=\{\begin{array}{ll}{a}_n,& \text{if }N<n⩽N_1\\ 0,& \text{otherwise}.\end{array}$$

$$\sum _{N<n⩽N_1}{a}_n$$

$$\left|\sum _{N<n⩽N_1}{a}_n\right|⩽\frac{1}{H}\left|\sum _{N<n⩽N_1-H}\sum _{h⩽H}{a}_{n+h}\right|+\frac{H^2}{H}$$

$$N{\lambda }_k^{2^{1-k}}+N{\left(N^k{\lambda }_k\right)}^{-2/d_k}>N.$$

$$\sum _{N<n⩽2N}\psi \left(f(n)\right)$$

$$r_{n,k}(h):=\frac{1}{k!}{\int }_n^{n+h}{\left(n+h-t\right)}^k{f}^{(k+1)}(t)\text{d}t.$$

$$\sum _{h⩽H-1}\left|r_{n,k}(h+1)-r_{n,k}(h)\right|<\frac{c_{k+1}}{(k+1)!}{H}^{k+1}{\lambda }_{k+1}=\frac{c_{k+1}}{s_k(k+1)!}{H}^{k+1}{N}^{-1}{\lambda }_k$$

$$\sum _{N<n⩽2N}\psi \left(f(n)\right)\ll \frac{N}{L}+\frac{1}{H}\sum _{N<n⩽2N}\left\{\sum _{\ell ⩽L}\left(\frac{1}{\ell }+H^{k+1}{N}^{-1}{\lambda }_k\right)\underset{H_1⩽H}{\max }\left|\sum _{h⩽H_1}e\left(\ell g_n(h)\right)\right|\right\}+H\log L.$$

$$L⩽H^{-k-1}N{\lambda }_k^{-1}$$

$$\sum _{N<n⩽2N}\psi \left(f(n)\right)\ll \frac{N}{L}+\frac{1}{H}\sum _{N<n⩽2N}{\mathcal{S}}_{H,L}(n)+H\log L$$

$${\mathcal{S}}_{H,L}(n):=\sum _{\ell ⩽L}\frac{1}{\ell }\underset{H_1⩽H}{\max }\left|\sum _{h⩽H_1}e\left(\ell g_n(h)\right)\right|$$

$${\mathcal{S}}_{H,L}(n)$$

$$\frac{1}{\ell }⩽\frac{k!H^{k-1}}{m}<\frac{k!}{j}\left(j⩾1\right)$$

$$(HL{)}^{-\epsilon }{\mathcal{S}}_{H,L}(n)$$

Full text

1. Introduction and main result

Le $\{x\}$ be the fractional part of $x\in\mathbb{R}$ and $\psi(x):=\{x\}-\frac{1}{2}$ be the first Bernoulli function. Sums of the shape

[TABLE]

where $N$ is a large number and $f$ is a smooth function, are of great importance in analytic number theory (see [1, 6, 9] for instance). A large amount of problems are reduced to obtaining a non-trivial bound for the sum (1), such as, among others, the Dirichlet divisor problem, the Gauss circle problem, the problem of the gaps between $k$-free numbers or the distribution of squarefull numbers.

The general strategy is to use a truncated version of the expansion of $\psi$ in Fourier series, providing the inequality

[TABLE]

where $e(x):=e^{2i\pi x}$ and $L\geqslant 1$ is any integer parameter to be chosen optimally. The problem is henceforth reduced to estimating exponential sums, for which several methods have been developed by Weyl, van der Corput and Vinogradov. For instance, when $f\in C^{k}\left[N,2N\right]$ satisfies $\left|f^{(k)}(x)\right|\asymp\lambda_{k}<1$, van der Corput’s estimate (see [6, Theorem 2.8]) and the inequality (2) yield the bound

[TABLE]

when $k\geqslant 2$, the term $N^{1-2^{1-k}}\log N$ being removed in the case $k=2$. When $k\in\{2,3,4\}$, this respectively gives

[TABLE]

The term $N\lambda_{k}^{\frac{1}{2^{k}-1}}$ is usually called the main term, the other two terms being the secondary terms. For monomial functions, i.e. functions $f\in C^{\infty}\left[N,2N\right]$ such that $\left|f^{(k)}(x)\right|\asymp TN^{-k}$ for some $T\geqslant 1$ and for any positive integer $k$, van der Corput’s method of exponent pairs provides better results (see [6, Lemma 4.3] or [1, Corollary 6.35]).

Recently, using new bounds given in [12, Theorem 1.2] for the number $J_{s,k}(X)$ of integral solutions of the system

[TABLE]

with $1\leqslant x_{i},y_{i}\leqslant X$ ($1\leqslant i\leqslant s$), some improvements in exponential sums have appeared in the literature (see [11, 7]). If we use the main result in [7] combined with (2), we obtain

[TABLE]

where $k\in\mathbb{Z}_{\geqslant 3}$ and $f\in C^{k}\left[N,2N\right]$ is such that there exists $\lambda_{k}\in\left(0,1\right)$ such that $\left|f^{(k)}\right|\asymp\lambda_{k}$. This improves on the main term of (3) as soon as $k\geqslant 4$ and gives the same exponent when $k=3$:

[TABLE]

Any improvement of (3) or (4) when $k\in\{2,3,4\}$ may lead to new results in the aforementioned problems. The main purpose of this note is to improve the main term in the cases $k\geqslant 2$ for (3) and $k\in\{2,3,4,5\}$ for (4). To do this, we use Weyl’s differencing method and add a device due to Popov [10], also used in [5] to estimates the sums

[TABLE]

where $P(x)$ is a polynomial of degree $2$ or $3$ with small positive leading coefficient. The method was then generalized in [3] to any polynomial of degree $\geqslant 2$, and we use here the Weyl’s schift to extend the results to smooth functions. Unfortunately, as often in exponential sums estimates, the secondary terms remain too weak to be really efficients in practice. Nevertheless, the result below seems to be new and we think that it may be of interest.

Theorem 1**.**

Let $k,N\in\mathbb{Z}_{\geqslant 2}$, $f\in C^{k+1}\left[N,2N\right]$ such that there exist $\lambda_{k},\lambda_{k+1},s_{k}>0$ and $c_{k},c_{k+1}\geqslant 1$ such that, for any $x\in\left[N,2N\right]$ and any $j\in\{k,k+1\}$

[TABLE]

Define $d_{k}:=2^{k}(k+1)+2k$. Then, for any $\varepsilon>0$

[TABLE]

Note that, if $N^{k-1}\lambda_{k}^{2+2^{1-k}}\leqslant 1$, then the $2$nd term is absorbed by the $3$rd one. To compare with (3) and (4) in the cases $k\in\{2,3,4\}$, Theorem 1 respectively gives

[TABLE]

2. Technical lemmas

Lemma 2**.**

Let $M\in\mathbb{Z}_{\geqslant 0}$, $N\in\mathbb{Z}_{\geqslant 1}$, $H\in\mathbb{Z}_{\geqslant 4}$ and $\alpha>0$. Then

[TABLE]

Proof.

This is [3, Lemma 3.2]. ∎

Lemma 3** (Weyl’s shift).**

Let $N,N_{1}\in\mathbb{Z}_{\geqslant 1}$ such that $N<N_{1}\leqslant 2N$ and $a_{N+1},\dotsc,a_{N_{1}}\in\mathbb{C}$ satisfying $\left|a_{n}\right|\leqslant 1$. Then, for any $H\in\left\{1,\dotsc,N-1\right\}$

[TABLE]

Proof.

Define

[TABLE]

Then

[TABLE]

Since, in the first sum, $\alpha_{n+h}=a_{n+h}$, we get

[TABLE]

as asserted. ∎

3. Proof of Theorem 1

One may assume $N^{-k}<\lambda_{k}<1$, otherwise

[TABLE]

Using (2) and Lemma 3, we get for any $H,L\in\mathbb{Z}_{\geqslant 1}$ such that $H<N$

[TABLE]

with $g_{n}(h):=\frac{1}{k!}h^{k}f^{(k)}(n)+\dotsb+hf^{\,\prime}(n)$ and

[TABLE]

Note that

[TABLE]

so that, by partial summation

[TABLE]

Now assume

[TABLE]

so that

[TABLE]

where

[TABLE]

and set $\alpha_{n,k}:=\frac{1}{k!}\left|f^{(k)}(n)\right|$. From Weyl’s bound [9, p. 42], we get for any $\varepsilon>0$

[TABLE]

and Hölder’s inequality applied with exponent $\frac{2^{k}}{2^{k}-2}$ yields

[TABLE]

Following [10, (13),(14)] (see also [5]), notice that, in the innersum, we have

[TABLE]

so that

[TABLE]

and Lemma 2 and the crude bounds $\sigma(m)\ll m\log\log m$ and $\tau(m)\ll m^{\varepsilon}$ imply that

[TABLE]

Inserting in (6) and using $\alpha_{n,k}\asymp\lambda_{k}$, we get

[TABLE]

Considering (5), the choice of $L=\left\lfloor H^{-k-1}N\lambda_{k}^{-1}\right\rfloor$ gives

[TABLE]

The asserted result follows by choosing $H=4\left\lfloor\left(N^{2^{k}}\lambda_{k}^{-(2^{k}+2)}\right)^{1/d_{k}}\right\rfloor$, the extra term $H\log N$ being absorbed by the term $N^{1+\varepsilon}\left(N^{k}\lambda_{k}\right)^{-2/d_{k}}$ since $N^{k}\lambda_{k}>1$. Note that this latter hypothesis also ensures that $1<L<N^{k+1}$ and $H<N$, completing the proof. ∎

4. Extension to integer points close to smooth curves

In this section, let $\delta\in\left(0,\frac{1}{4}\right]$, $N\in\mathbb{Z}_{\geqslant 2}$ large, $f:\left[N,2N\right]\longrightarrow\mathbb{R}$ be any function, and define

[TABLE]

Since it is known (see [1, Exercise 4 p. 350] for instance) that, for any integer $1\leqslant L\leqslant 1+\left\lfloor(8\delta)^{-1}\right\rfloor$, we have

[TABLE]

the proof of Theorem 1 may easily be adapted in a similar way to get a bound for $\mathcal{R}(f,N,\delta)$ of the same kind.

Theorem 4**.**

Let $\delta\in\left(0,\frac{1}{4}\right]$, $k,N\in\mathbb{Z}_{\geqslant 2}$, $f\in C^{k+1}\left[N,2N\right]$ such that there exist $\lambda_{k},\lambda_{k+1},s_{k}>0$ and $c_{k},c_{k+1}\geqslant 1$ such that, for any $x\in\left[N,2N\right]$ and any $j\in\{k,k+1\}$

[TABLE]

Define $d_{k}:=2^{k}(k+1)+2k$. Then, for any $\varepsilon>0$

[TABLE]

This must be compared to the existing results of the theory. For instance, under the hypothesis $\left|f^{\,\prime\prime}\right|\asymp\lambda_{2}$, it is proved in [4] that

[TABLE]

In the cases $k=3$ or $k=4$, it is known from [8] that

[TABLE]

if $\left|f^{(j)}\right|\asymp\lambda_{j}$ for $j\in\{2,3\}$ such that $\lambda_{2}=N\lambda_{3}$, and

[TABLE]

if $\left|f^{(j)}\right|\asymp\lambda_{j}$ for $j\in\{3,4\}$ such that $\lambda_{3}=N\lambda_{4}$.