[Paper Review] A variant of the prime number theorem

Kui Liu, Jie Wu+1 more|arXiv (Cornell University)|May 23, 2021

Analytic Number Theory Research11 references4 citations

TL;DR

This paper establishes a new error term in a variant of the prime number theorem involving the von Mangoldt function averaged over the integer parts of $ x/n $. By refining estimates on 3-dimensional exponential sums using Vaughan’s identity and advanced exponential sum techniques, the authors improve the error exponent from $ 97/203 \approx 0.4778 $ to $ 9/19 \approx 0.4736 $, achieving $ O_{\varepsilon}(x^{9/19 + \varepsilon}) $, which represents a quantitative sharpening of prior results in this arithmetic averaging problem.

ABSTRACT

Let $Λ(n)$ be the von Mangoldt function, and let $[t]$ be the integral part of real number $t$. In this note, we prove that for any $\varepsilon>0$ the asymptotic formula $$ \sum_{n\le x} Λ\Big(\Big[\frac{x}{n}\Big]\Big) = x\sum_{d\ge 1} \frac{Λ(d)}{d(d+1)} + O_{\varepsilon}\big(x^{9/19+\varepsilon}\big) \qquad (x o\infty)$$ holds. This improves a recent result of Bordellès, which requires $\frac{97}{203}$ in place of $\frac{9}{19}$.

Motivation & Objective

• To improve the error term in the asymptotic formula for $ \sum_{n \leq x} \Lambda\left(\left\lfloor \frac{x}{n}\right\rfloor\right) $, a variant of the prime number theorem.
• To refine the exponent in the error term beyond previous results, particularly surpassing the $ 97/203 $ bound established by Bordellès.
• To apply advanced exponential sum techniques, especially in three dimensions, to achieve a tighter error estimate.
• To demonstrate that the new exponent $ 9/19 $ is superior to those derived from classical exponent pair hypotheses, even under Riemann Hypothesis assumptions.
• To provide a sharper asymptotic formula for the sum of $ \Lambda $ over the floor of $ x/n $, contributing to the understanding of prime distribution in structured sequences.

Proposed method

• Utilizes Vaughan’s identity to decompose the sum $ \sum_{D < d \leq 2D} \Lambda(d) g(d) $ into type I and type II sums, enabling finer control over exponential sums.
• Applies a novel estimate for 3-dimensional exponential sums $ \mathfrak{S}_{\delta,3} $, derived from a generalized version of Lemma 2.1 on Diophantine inequalities.
• Employs Vaaler’s trigonometric polynomial approximation for the fractional part function $ \psi(t) = \{t\} - 1/2 $, allowing effective handling of error terms via exponential sum bounds.
• Optimizes the choice of parameters such as $ H $, $ M $, $ N $, and $ D $ across dyadic intervals to balance competing error terms in the sum decomposition.
• Combines estimates from both 3D and 2D exponential sum bounds (from [1] and [5]) to derive a composite error term that dominates the total error in $ S_2(x) $.
• Applies dyadic decomposition to the sum over $ d $, breaking the range $ N < d \leq x/N $ into dyadic intervals $ D_j = x/(2^j N) $, and applies the main estimate to each.

Experimental results

Research questions

• RQ1 Can the error term in the asymptotic formula for $ \sum_{n \leq x} \Lambda\left(\left\lfloor \frac{x}{n}\right\rfloor\right) $ be improved beyond the current known exponent $ 97/203 $?
• RQ2 What is the best possible exponent $ \theta $ such that $ \sum_{n \leq x} \Lambda\left(\left\lfloor \frac{x}{n}\right\rfloor\right) = x \sum_{d \geq 1} \frac{\Lambda(d)}{d(d+1)} + O_{\varepsilon}(x^{\theta + \varepsilon}) $?
• RQ3 Can 3-dimensional exponential sum estimates yield a better error bound than 2-dimensional ones in this context?
• RQ4 Is the exponent $ 9/19 $ optimal under current analytic number theory techniques, particularly when compared to results under the Riemann Hypothesis?
• RQ5 Can the method of exponential sums, combined with Vaughan’s identity, be systematically extended to other arithmetic functions beyond $ \Lambda(n) $?

Key findings

• The authors establish the asymptotic formula $ \sum_{n \leq x} \Lambda\left(\left\lfloor \frac{x}{n}\right\rfloor\right) = x \sum_{d \geq 1} \frac{\Lambda(d)}{d(d+1)} + O_{\varepsilon}(x^{9/19 + \varepsilon}) $, which improves upon the previous best-known exponent $ 97/203 \approx 0.4778 $.
• The new error exponent $ 9/19 \approx 0.4736 $ is strictly smaller than $ 97/203 $, representing a nontrivial improvement in the error term's growth rate.
• The improvement is achieved through a refined analysis of 3-dimensional exponential sums, particularly via a new bound on $ \mathfrak{S}_{\delta,3} $, which dominates the error in the sum over $ d $.
• The method avoids reliance on the Riemann Hypothesis and still yields a better exponent than the $ 35/71 \approx 0.493 $ bound from Ma and Wu, and even outperforms the $ 28/59 \approx 0.4745 $ bound under the exponent pair hypothesis.
• The result is robust under the choice of $ N = x^{9/19} $, which balances the error terms in $ S_1(x) $ and $ S_2(x) $, leading to the optimal total error of $ O_{\varepsilon}(x^{9/19 + \varepsilon}) $.
• The analysis confirms that the 3D exponential sum estimate is the dominant term in the error, and its optimization leads to the final bound, with the remaining terms being absorbed into the main error.