arXiv subjects

Runbo Li

Publications and source records attributed to Runbo Li.

At least 19 recordsLinked to original sources

arXiv2026

Primes in arithmetic progressions to large moduli and refinements of Harman's sieve

We study the average distribution of primes of size $x$ in arithmetic progressions to moduli larger than $x^{\frac{1}{2}}$. Using arithmetic information from the works of many authors together with different variants of the original Harman's sieve, we construct suitable majorants and minorants for the prime indicator function $\mathbb{1}_{p}(n)$ that satisfy Bombieri--Vinogradov type mean value theorems with different types of moduli. Specifically, we obtain some mean value theorems for primes with bilinear forms of moduli up to $x^{\frac{9}{17}}$ or with trilinear forms of moduli up to $x^{\frac{17}{32}}$. As a by-product, we obtain new upper and lower bounds for $\pi(x; q, a)$ that hold for almost all moduli $q$.

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arXiv2026

On the largest prime factor of integers in short intervals III

Using Watt's mean value theorem and a delicate sieve decomposition, the author shows that the interval $[x, x+x^{\frac{1}{2}+\varepsilon}]$ contains an integer with a prime factor larger than $x^{\frac{35}{36}-\varepsilon}$ for sufficiently large $x$. This gives a solution with $\gamma = \frac{1}{36}$ to the Exercise 5.1 in Harman's monograph and improves the previous record of the author proved in 2024, where $\gamma = \frac{1}{26.5}$ is obtained.

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arXiv2025

An average Brun-Titchmarsh theorem and shifted primes with a large prime factor

The author studies an average version of Brun-Titchmarsh theorem with large moduli. Using Maynard's recent breakthrough on the Bombieri-Friedlander-Iwaniec type triple convolution estimates, we refine the previous result of Baker and Harman (1996). As an application, we improve a result of Baker and Harman (1998) on shifted primes with a large prime factor, showing that the largest prime factor of $p - 1$ is larger than $p^{0.679}$ for infinitely many primes $p$.

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arXiv2025

On the exceptional set in the $abc$ conjecture

The $abc$ conjecture states that there are only finitely many triples of coprime positive integers $(a,b,c)$ such that $a+b=c$ and $\operatorname{rad}(abc) < c^{1-\epsilon}$ for any $\epsilon > 0$. Using the optimized methods in a recent work of Browning, Lichtman and Ter\"av\"ainen, we showed that the number of those triples with $c \leqslant X$ is $O\left(X^{56/85+\varepsilon}\right)$ for any $\varepsilon > 0$, where $\frac{56}{85} \approx 0.658824$. This constitutes an improvement of the previous bound $O\left(X^{33/50}\right)$.

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arXiv2025

Largest square divisors of shifted primes

The author shows that there are infinitely many primes $p$ such that for any nonzero integer $a$, $p-a$ is divisible by a square $d^2 > p^{\frac{1}{2}+\frac{1}{700}}$. The exponent $\frac{1}{2}+\frac{1}{700}$ improves Merikoski's $\frac{1}{2}+\frac{1}{2000}$. Many powerful devices in Harman's sieve are used for this improvement.

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arXiv2025

On almost primes in Piatetski-Shapiro sequences

The author proves that for $0.9985 < \gamma < 1$, there exist infinitely many primes $p$ such that $[p^{1/\gamma}]$ has at most 5 prime factors counted with multiplicity. This gives an improvement upon the previous results of Banks-Guo-Shparlinski and Xue-Li-Zhang.

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arXiv2025

Primes in arithmetic progressions to smooth moduli: A minorant version

The author prove that there exists a function $\rho(n)$ which is a minorant for the prime indicator function $\mathbb{1}_{p}(n)$ and has distribution level $\frac{10}{19}$ in arithmetic progressions to smooth moduli. This refines the previous results of Baker--Irving and Stadlmann.

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arXiv2025

On prime-producing sieves and distribution of $\alpha p-\beta$ mod $1$

The author proves that there are infinitely many primes $p$ such that $\| \alpha p - \beta \| < p^{-\frac{28}{87}}$, where $\alpha$ is an irrational number and $\beta$ is a real number. This sharpens a result of Jia (2000) and provides a new triple $(\gamma, \theta, \nu)=(\frac{59}{87}, \frac{28}{87}, \frac{1}{29})$ that can produce special primes in Ford and Maynard's work on prime-producing sieves. Our minimum amount of Type-II information required ($\nu = \frac{1}{29}$) is less than any previous work on this topic using only traditional Type-I and Type-II information.

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arXiv2025

A note on variants of Buchstab's identity

The author proves variants of Buchstab's identity on sieve functions, refining the previous work on new iteration rules of Brady. The main tool used in the proof is a special form of combinatorial identities related to the binomial coefficients. As a by--product, the author obtains better inequalities of $F_{\kappa}(s)$ and $f_{\kappa}(s)$ for dimensions $\kappa > 1$.

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arXiv2024

Primes in almost all short intervals

The author sharpens a result of Jia (1996), showing that the interval $[n, n+n^{\frac{1}{21.5}+\varepsilon}]$ contains prime numbers for almost all $n$. Watt's mean value bound, a delicate sieve decomposition and more accurate estimates for integrals are used to good effect.

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arXiv2024

On the largest prime factor of quadratic polynomials

Let $x$ denote a sufficiently large integer. We show that the recent result of Grimmelt and Merikoski actually yields the largest prime factor of $n^2 +1$ is greater than $x^{1.317}$ infinitely often. As an application, we give a new upper bound for the number of integers $n \leqslant x$ which $n^2 +1$ has a primitive divisor.

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arXiv2024

On Chen's theorem, Goldbach's conjecture and almost prime twins II

Let $N$ denote a sufficiently large even integer and $x$ denote a sufficiently large integer, we define $D_{1,2}(N)$ as the number of primes $p$ that such that $N - p$ has at most 2 prime factors. In this paper, we show that $D_{1,2}(N) \geqslant 1.9728 \frac{C(N) N}{(\log N)^2}$, which is rather near to the asymptotic constant $2$ in Hardy--Littlewood conjecture for Goldbach's conjecture. We also get similar results on twin prime problem and additive representations of integers. The proof combines various techniques in sieve methods, such as weighted sieve, Chen's switching principle, new distribution levels proved by Lichtman and Pascadi, Chen's double sieve and Harman's sieve.

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arXiv2024

A remark on large even integers of the form $p+P_3$

Let $N$ denotes a sufficiently large even integer, $p$ denotes a prime and $P_{r}$ denotes an integer with at most $r$ prime factors. In this paper, we study the solutions of the equation $N-p=P_3$ and consider two special cases where $p$ is small, and $p,P_3$ are within short intervals.

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arXiv2024

Hybrid estimation of single exponential sums, exceptional characters and primes in short intervals

We provide a new hybrid estimation of single exponential sums, combining Van der Corput, Huxley and Bourgain's result. We also focus on primes in short intervals $(x-x^{\alpha},x]$ under the assumption of the existence of exceptional Dirichlet characters and get a small improvement of a 2004 result of Friedlander and Iwaniec. By using our new estimation of exponential sums, we extend the previous admissible range $0.4937 \leqslant \alpha \leqslant 1$ to $0.4923 \leqslant \alpha \leqslant 1$.

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arXiv2023

A remark on the distribution of $\sqrt{p}$ modulo one involving primes of special type II

Let $P_{r}$ denote an integer with at most $r$ prime factors counted with multiplicity. In this paper we prove that for some $\lambda < \frac{1}{12}$, the inequality $\{\sqrt{p}\}<p^{-\lambda}$ has infinitely many solutions in primes $p$ such that $p+2=P_r$, where $r= 4, 5, 6, 7$. Specially, when $r = 4$ we obtain $\lambda = \frac{1}{15.1}$, which improves Cai's $\frac{1}{15.5}$.

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arXiv2023

On the upper and lower bound orders of almost prime triples

A Hardy-Littlewood triple is a 3-tuple of integers with the form $(n, n+2, n+6)$. In this paper, we study Hardy-Littlewood triples of the form $(p, P_{a}, P_{b})$ and improve the upper and lower bound orders of it, where $p$ is a prime and $P_{r}$ has at most $r$ prime factors. Our new results generalize and improve the previous results.

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arXiv2023

Remarks on additive representations of natural numbers

For two relatively prime square-free positive integers $a$ and $b$, we study integers of the form $a p+b P_{2}$ and give a new lower bound for the number of such representations, where $a p$ and $b P_{2}$ are both square-free, $p$ denote a prime, and $P_{2}$ has at most two prime factors. We also consider some special cases where $p$ is small, $p$ and $P_2$ are within short intervals, $p$ and $P_2$ are within arithmetical progressions and a Goldbach-type upper bound result. Our new results generalize and improve previous results.

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arXiv2023

On some problems of primes with the floor function

Let $\left[x\right]$ be the largest integer not exceeding $x$. For $0<\theta \leq 1$, let $\pi_{\theta}(x)$ denote the number of integers $n$ with $1 \leq n \leq x^{\theta}$ such that $\left[\frac{x}{n}\right]$ is prime and $S_{\mathbb{P}}(x)$ denote the number of primes in the sequence $\left\{\left[\frac{x}{n}\right]\right\}_{n \geqslant 1}$. In this paper, we obtain the asymptotic formula $$ \pi_{\theta}(x)=\frac{x^{\theta}}{(1-\theta) \log x}+O\left(x^{\theta}(\log x)^{-2}\right) $$ provide that $\frac{435}{923}<\theta<1$, and prove that $$ S_{\mathbb{P}}(x)=x\sum_{p} \frac{1}{p(p+1)}+O_{\varepsilon}\left(x^{435/923+\varepsilon}\right) $$ for $x \rightarrow \infty$. Thus improve the previous result due to Ma, Wu and the author.

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