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Steve Fan

Publications and source records attributed to Steve Fan.

16 recordsLinked to original sources
arXiv2026

Strongly complete sets and a conjecture of Erd\H{o}s

A set $A\subseteq\mathbb{N}$ is called $\textit{complete}$ if every sufficiently large integer can be written as a sum of distinct elements of $A$. It is $\textit{strongly complete}$ if it remains complete after one deletes finitely many elements from it. Building on recent work of Bergelson and Simmons and that of Griesmer, we establish a new strong-completeness criterion exploiting a three-component partition of a given set. As an application, we show that $A$ is strongly complete whenever \[ \big|A\cap(2^k,2^{k+1}]\big|\ge5 \] for every sufficiently large $k\in\mathbb{N}$, and \[ \sum_{a\in A}\|a\theta\|=\infty, \quad\forall\theta\in\mathbb{R}\setminus\mathbb{Z}. \] In particular, this resolves a 1961 conjecture of Erd\H{o}s. The new strong-completeness criterion also enables us to make progress on a 1996 problem of Burr, Erd\H{o}s, Graham, and Li concerning strong completeness of mixed power sets by refining a previous result of Bergelson and Simmons. Besides, we study the polynomially perturbed ray set \[ \{\lfloor t\alpha^n\rfloor,\lfloor t\alpha^n\rfloor+P(n):n\in\mathbb{N}\}, \] which combines the polynomial set $\{P(n):n\in\mathbb{N}\}$ and the single-ray set $\{\lfloor t\alpha^n\rfloor:n\in\mathbb{N}\}$ both previously considered by Graham, and show that it is strongly complete for any $t>0$ and $\alpha\in(0,2)$ and any primitive integer-valued polynomial $P$. The machinery developed for the proof of this result also yields other interesting applications.

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arXiv2026

The van der Corput property for sums of two squares

Let $S_N=\{1\le d\le N:d=x^2+y^2\text{ for some }x,y\in\mathbb Z\}.$ We prove a power-saving form of the van der Corput property for $S_N$. As a consequence, we obtain a strong S\'{a}rk\"{o}zy-type result: if $A\subseteq [N]$ has no nonzero difference equal to a sum of two squares, then $|A|\ll_\varepsilon N^{7/8+\varepsilon}$ for every $\varepsilon>0$, improving upon an earlier quasipolynomial bound due to Rice. The shape of this bound is optimal, as a construction of Younis yields a set $A\subseteq [N]$ with $|A|\gg N^{1/2}$ such that $(A-A)\cap S_N=\emptyset$.

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arXiv2026

On the asymptotic density of the ordered pairs $(a,b)$ of positive integers such that $\gcd(ab,a+b)=\gcd(a,b)$

Consider the arithmetic function of two variables $f(a,b)= \gcd(ab,a+b)/\gcd(a,b)$, investigated in a recent preprint. We deduce asymptotic formulas for sums of the form $\sum_{a,b\le x} h(f(a,b))$, where $h$ belongs to a certain class of arithmetic functions. In particular, we obtain an asymptotic formula for the number of ordered pairs $(a,b)\in {\Bbb N}^2$ such that $a,b\le x$ and $f(a,b)=m$, where $m\in {\Bbb N}$ is fixed. This shows that in the case $m=1$ the corresponding density is the quadratic class number constant $C= \prod_p (1-1/(p^2(p+1))) \doteq 0.881513$.

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arXiv2026

Extensions of the Furstenberg-S\'ark\"ozy theorem via the arithmetic level-$d$ inequality

Green and Sawhney recently obtained a quasipolynomial bound in the Furstenberg--S\'ark\"ozy theorem for square differences by proving an ``arithmetic level-d'' inequality, thereby yielding a greatly improved density increment scheme. We apply their method to treat general intersective polynomials $h\in\mathbb{Z}[x]$. In particular, let \[ D(h(\mathbb{N}),X):= \max{|A|:\ A\subseteq [1,X]\cap\mathbb{N} \text{and}\ (A-A)\cap h(\mathbb{N})\subseteq\{0\}}. \] We prove that for every $0<\mu<1/2$ there are constants $c_0, X_{\text{min}}>0$ depending on $h$ and $\mu$ such that for every $X>X_{\text{min}}$, \[D(h(\mathbb{N}), X)\leq Xe^{-c_0(\log X)^\mu}.\] This is the best quantitative upper bound presently known for sets lacking intersective polynomial differences, improving upon the work of Arala. In order to achieve the admissible exponent range $0<\mu<1/2$, we use sieve methods to develop novel exponential sum estimates in the style of Rice, and we use the ``random sparsification'' procedure of Green and Sawhney.

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arXiv2025

A family of analogues to the Robin criterion

The Robin criterion states that the Riemann hypothesis is equivalent to the inequality $\sigma(n) < e^\gamma n \log \log n$ for all $n>5040$, where $\sigma(n)$ is the sum of divisors of $n$, and $\gamma$ is the Euler--Mascheroni constant. Define the family of functions \[ \sigma^{[k]} (n):=\sum_{[d_1,\dots,d_k]=n}d_1\dots d_k \] where $[d_1, \dots, d_k]$ is the least common multiple of $d_1, \dots, d_k$. These functions behave asymptotically like $\sigma(n)^k$ as $k\to\infty$. We prove the following analogue of the Robin criterion: for any $k \geq 2$, the Riemann hypothesis holds if and only if $\sigma^{[k]} (n) < \frac{(e^\gamma n \log \log n)^k}{\zeta(k)}$ for all $n > 2162160$, where $\zeta$ is the Riemann zeta function.

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arXiv2025

S\'ark\"ozy's theorem in $\mathbb{F}_q[t]$ via the van der Corput property

Fix a positive prime power $q$, and let $\mathbb{F}_q[t]$ be the ring of polynomials over the finite field $\mathbb{F}_q$ with $\text{char}(\mathbb{F}_q)>2$. Suppose $A \subseteq \{f \in \mathbb{F}_q[t]: \text{deg } f \leq N\}$ contains no pair of elements whose difference is of the form $P-1$ with $P$ irreducible. Adapting Green's approach to S\'ark\"ozy's theorem for shifted primes in $\mathbb{Z}$ using the van der Corput property, we show that \[ |A| \ll q^{(N+1)(11/12+o(1))}, \] improving upon the bound $O\big(q^{(1-c/\log N)(N+1)}\big)$ due to L\^{e} and Spencer. An important distinction between Green's argument and ours lies in the properties of exponential sums over function fields, which differ in several interesting ways from their number-field counterparts.

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arXiv2025

The maximal order of the shifted-prime divisor function

For each positive integer $n$, we denote by $\omega^*(n)$ the number of shifted-prime divisors $p-1$ of $n$, i.e., \[\omega^*(n):=\sum_{p-1\mid n}1.\] First introduced by Prachar in 1955, this function has interesting applications in primality testing and bears a strong connection with counting Carmichael numbers. Prachar showed that for a certain constant $c_0 > 0$, \[\omega^*(n)>\exp\left(c_0\frac{\log n}{(\log\log n)^2}\right)\] for infinitely many $n$. This result was later improved by Adleman, Pomerance and Rumely, who established an inequality of the same shape with $(\log\log n)^2$ replaced by $\log\log n$. Assuming the Generalized Riemann Hypothesis for Dirichlet $L$-functions, Prachar also proved the stronger inequality \[\omega^*(n)>\exp\left(\left(\frac{1}{2}\log2+o(1)\right)\frac{\log n}{\log\log n}\right)\] for infinitely many $n$. By refining the arguments of Prachar and of Adleman, Pomerance and Rumely, we improve on their results by establishing \begin{align*} \omega^*(n)&>\exp\left(0.6736\log 2\cdot\frac{\log n}{\log\log n}\right) \quad\text{(unconditionally)},\\ \omega^*(n)&>\exp\left(\left(\log\left(\frac{1+\sqrt{5}}{2}\right)+o(1)\right)\frac{\log n}{\log\log n}\right) \quad\text{(under GRH)}, \end{align*} for infinitely many $n$.

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arXiv2025

The Hardy--Ramanujan inequality for sifted sets and its applications

The well-known Hardy--Ramanujan inequality states that if $\omega(n)$ denotes the number of distinct prime factors of a positive integer $n$, then there is an absolute constant $C>0$ such that uniformly for $x\ge2$ and $k\in\mathbb{N}$, \[\#\{n\le x\colon\omega(n)=k\}\ll\frac{x(\log\log x+C)^{k-1}}{(k-1)!\log x}.\] A myriad of generalizations and variations of this inequality have been discovered. In this paper, we establish a weighted version of this inequality for sifted sets, which generalizes an earlier result of Hal\'asz and implies Timofeev's theorems on shifted primes. We then explore its applications to a variety of intriguing problems, such as large deviations of $\omega$ on subsets of integers, the Erd\H{o}s multiplication table problem, divisors of shifted primes, and the image of the Carmichael $\lambda$-function. Building on the same circle of ideas, we also generalize Troupe's result on the normal order of $\omega(s(n))$ for the sum-of-proper-divisors function $s(n)$, confirming for the first time the weighted version of a special case of a 1992 conjecture by Erd\H{o}s, Granville, Pomerance, and Spiro.

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arXiv2025

Counting primes with a given primitive root, uniformly

The celebrated Artin conjecture on primitive roots asserts that given any integer $g$ which is neither $-1$ nor a perfect square, there is an explicit constant $A(g)>0$ such that the number $\Pi(x;g)$ of primes $p\le x$ for which $g$ is a primitive root is asymptotically $A(g)\pi(x)$ as $x\to\infty$, where $\pi(x)$ counts the number of primes not exceeding $x$. Artin's conjecture has remained unsolved since its formulation in 1927. Nevertheless, Hooley demonstrated in 1967 that Artin's conjecture is a consequence of the Generalized Riemann Hypothesis (GRH) for Dedekind zeta functions of certain cyclotomic-Kummer extensions over $\mathbb{Q}$. In this paper, we use GRH to establish a uniform version of the Artin--Hooley asymptotic formula. Specifically, we prove that $\Pi(x;g) \sim A(g) x/\log{x}$ whenever $\log{x}/\log\log{2|g|} \to \infty$, i.e., whenever $x$ tends to infinity faster than any power of $\log{(2|g|)}$. Under GRH, we also show that the least prime $p_g$ possessing $g$ as a primitive root satisfies the upper bound $p_g=O(\log^{19}(2|g|))$ uniformly for all non-square $g\ne-1$. We conclude with an application to the average value of $p_g$ and a discussion of an analogue concerning the least "almost-primitive'' root.

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arXiv2025

Extremal elasticity of quadratic orders

We study how large and small elasticity can be for orders belonging to a fixed quadratic field, in terms of the corresponding conductors. For example, we show that if $K$ is an imaginary quadratic field, then the order of conductor $f$ in $K$ has elasticity exceeding $(\log{f})^{c_1 \log\log\log{f}}$ for all $f$ that are sufficiently large. On the other hand, this elasticity is smaller than $(\log{f})^{c_2\log\log\log{f}}$ for infinitely many $f$. Here $c_1, c_2$ are universal positive constants. The proofs borrow methods from analytic number theory previously employed to study statistics of the multiplicative groups $(\mathbb{Z}/m\mathbb{Z})^{\times}$.

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arXiv2024

The typical elasticity of a quadratic order

For an atomic domain $D$, the $elasticity$ $\rho(D)$ of $D$ is defined as $\sup\{r/s: \pi_1\cdots \pi_r = \rho_1 \cdots \rho_s,~ \text{where each $\pi_i, \rho_j$ is irreducible}\}$; the elasticity provides a concrete measure of the failure of unique factorization in $D$. Fix a quadratic number field $K$ with discriminant $\Delta_K$, and for each positive integer $f$, let $\mathcal{O}_f = \mathbb{Z} + f\mathcal{O}_K$ denote the order of conductor $f$ in $K$. Results of Halter-Koch imply that $\mathcal{O}_f$ has finite elasticity precisely when $f$ is $\textit{split-free}$, meaning not divisible by any rational prime $p$ with $(\Delta_K/p)=1$. When $K$ is imaginary, we show that for almost all split-free $f$, \[ \rho(\mathcal{O}_f) = f/(\log{f})^{\frac{1}{2}\log\log\log{f} + \frac{1}{2}C_K+o(1)}, \] for a constant $C_K$ depending on $K$. When $K$ is real, we prove under the assumption of the Generalized Riemann Hypothesis that \[ \rho(\mathcal{O}_f)= (\log{f})^{\frac12 +o(1)} \] for almost all split-free $f$. Underlying these estimates are new statistical theorems about class groups of orders in quadratic fields, whose proofs borrow ideas from investigations of Erd\H{o}s, Hooley, Li, Pomerance, Schmutz, and others into the multiplicative groups $(\mathbb{Z}/m\mathbb{Z})^\times$. One novelty of the argument is the development of a weighted version of the Tur\'{a}n--Kubilius inequality to handle a variety of sums over split-free integers.

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arXiv2024

The shifted prime-divisor function over shifted primes

Let $a,b\in\mathbb{Z}\setminus\{0\}$. For every $n\in\mathbb{N}$, denote by $\omega_a^*(n)$ the number of shifted-prime divisors $p-a$ of $n$, where $p>a$ is prime. In this paper, we study the moments of $\omega_a^*$ over shifted primes $p-b$. Specifically, we prove an asymptotic formula for the first moment and upper and lower bounds of the correct order of magnitude for the second moment. These results suggest that the average behavior of $\omega^*_a$ on shifted primes is similar to its average behavior on natural numbers. We shall also prove upper bounds for the mean values of sub-multiplicative functions in a nice class over the least common multiples of the shifted primes $p-a$ and $q-b$. Such upper bounds are intimately related to the second moments of $\omega^*_a$ over natural numbers and over shifted primes. Finally, we propose a new conjecture on the second moment of $\omega_1^*$ over natural numbers and provide a heuristic argument in support of this conjecture.

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arXiv2024

Shifted-prime divisors

Let $\omega^*(n)$ denote the number of divisors of $n$ that are shifted primes, that is, the number of divisors of $n$ of the form $p-1$, with $p$ prime. Studied by Prachar in an influential paper from 70 years ago, the higher moments of $\omega^*(n)$ are still somewhat a mystery. This paper addresses these higher moments and considers other related problems.

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arXiv2023

Weighted Erd\H{o}s-Kac Theorems via Computing Moments

By adapting the moment method developed by Granville and Soundararajan [17], Khan, Milinovich and Subedi [24] recently obtained a weighted version of the Erd\H{o}s--Kac theorem for $\omega(n)$ with multiplicative weight $d_k(n)$, where $\omega(n)$ denotes the number of distinct prime divisors of a positive integer $n$, and $d_k(n)$ is the $k$-fold divisor function with $k\in\mathbb{N}$. In this paper, we generalize their method to study the distribution of additive functions $f(n)$ weighted by nonnegative multiplicative functions $\alpha(n)$ in a wide class. In particular, we establish uniform asymptotic formulas for the moments of $f(n)$ with suitable growth rates. We also prove a qualitative result on the moments which extends a theorem of Delange and Halberstam [8]. As a consequence, we obtain a weighted analogue of the Kubilius--Shapiro theorem.

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arXiv2023

Numerically explicit estimates for the distribution of rough numbers

For $x\ge y>1$ and $u:= \log x/\log y$, let $\Phi(x,y)$ denote the number of positive integers up to $x$ free of prime divisors less than or equal to $y$. In 1950 de Bruijn [1] studied the approximation of $\Phi(x,y)$ by the quantity \[\mu_y(u)e^{\gamma}x\log y\prod_{p\leq y}\left(1-\frac{1}{p}\right),\] where $\gamma=0.5772156...$ is Euler's constant and \[\mu_y(u):=\int_{1}^{u}y^{t-u}\omega(t)\,dt.\] He showed that the asymptotic formula \[\Phi(x,y)=\mu_y(u)e^{\gamma}x\log y\prod_{p\leq y}\left(1-\frac{1}{p}\right)+O\left(\frac{xR(y)}{\log y}\right)\] holds uniformly for all $x\ge y\ge2$, where $R(y)$ is a positive decreasing function related to the error estimates in the Prime Number Theorem. In this paper we obtain numerically explicit versions of de Bruijn's result.

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