arXiv subjects

Haozhe Gou

Publications and source records attributed to Haozhe Gou.

5 recordsLinked to original sources

arXiv2026

Large fluctuations of extended Rademacher random multiplicative functions

Let $f$ be an extended Rademacher random multiplicative function (RMF). We show that, for every fixed deterministic function $V(x)$ tending to infinity, almost surely there are arbitrarily large $x$ for which \[ \sum_{n\leq x}f(n) \geq \frac{\sqrt{x}(\log\log x)^{1/4}}{V(x)}. \] The corresponding negative fluctuation holds as well. In particular, this gives an affirmative answer to Erdős Problem~\#1144. As a byproduct, our result has a direct corollary giving new almost sure lower bounds $\log\log x$ on the number of sign changes of partial sums up to $x$ for all sufficiently large $x$.

math.NT↗

arXiv2026

Explicit bounds for the graphicality of the prime gap sequence

We establish explicit unconditional results on the graphic properties of the prime gap sequence. Let $p_n$ denote the $n$-th prime number (with $p_0=1$) and $\mathrm{PD}_n = (p_\ell - p_{\ell-1})_{\ell=1}^n$ be the sequence of the first $n$ prime gaps. Building upon the recent work by Erdős et al, which proved the graphic nature of $\mathrm{PD}_n$ for large $n$ unconditionally, and for all $n$ under RH, we provide the first explicit unconditional threshold such that: (1) For all $n \geq \exp\exp(30.32)$, $\mathrm{PD}_n$ is graphic. (2) For all $n \geq \exp\exp(34.33)$, every realization $G_n$ of $\mathrm{PD}_n$ satisfies that $(G_n, p_{n+1}-p_n)$ is DPG-graphic. Our proofs utilize a more refined criterion for when a sequence is graphic, and better estimates for the first moment of large prime gaps proven through an explicit zero-free region and explicit zero-density estimate for the Riemann zeta function.

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arXiv2026

Moments of the number of representations as sums of two prime squares

We prove, for every fixed integer $k\ge 4$, the correct order of magnitude for the $k$th moments of the function that counts the number of representations of an integer as sums of two prime squares. The upper bound for $k=4$ was previously known up to $\log\log\log x$, and the lower bound for $k\ge 4$ was only known conditionally on a conjectural uniform version of the Green-Tao theorem on linear equations in primes by the work of Sabuncu \cite{Sabuncu2024}. As an application of our method, we give a simpler proof of the lower bounds for the moments of the shifted prime divisor function, thereby recovering the lower-bound part of Gabdullin's recent result on a conjecture of Fan and Pomerance.

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arXiv2026

Extremal Problems for GCDs and LCMs in Higher Dimensions

We study extremal problems for tuples of integers chosen from sets $A_i \subset [X_i,2X_i]$ for $1\le i\le k$, under large GCD and small LCM conditions. For the GCD problem, we extend the work of Green and Walker to higher dimensions. Specifically, for $k\ge 3$, if $\gcd(a_1,\dots,a_k)\ge D$ for at least a proportion $δ$ of the tuples in $\prod_{i=1}^k A_i$, then $$ \prod_{i=1}^k |A_i| \ll_{k,\varepsilon} δ^{-k/(k-1)-\varepsilon} \frac{\prod_{i=1}^k X_i}{D^k}. $$ The proof is based on a minimal counterexample argument and a new high-dimensional measure concentration lemma. We also establish a large sieve-type inequality to obtain a complementary estimate for the GCD problem. For the LCM problem, we use a quite different method to show that, for all $k\ge 2$, $$ \prod_{i=1}^k |A_i| \ll_{k,\varepsilon} δ^{-k/(k-1)} \frac{L^{k/(k-1)+\varepsilon}} {\bigl(\prod_{i=1}^k X_i\bigr)^{1/(k-1)}}, $$ whenever $\operatorname{lcm}(a_1,\dots,a_k)\le L$ for at least a proportion $δ$ of the $k$-tuples in $\prod_{i=1}^k A_i$. Finally, we show that these bounds are essentially best possible up to $\varepsilon$-losses in the exponent.

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arXiv2025

On the Second Moment of Twisted Higher Degree $L$-functions

Assuming the Ramanujan conjecture, the zero density estimate and some subconvexity type bound, we describe a general method to obtain the log-saving upper bound for the second moment of standard twisted higher degree $L$-function in the $q$-aspect. Specifically, let $L(s, F)$ be a standard $L$-function of degree $d\geq3$. Under these foundational hypotheses. the bound \[ \sideset{}{^*}{\sum}_{χ\pmod q}\Big|L\big(\frac{1}{2}, F\times χ\big)\Big |^2\ll_{F,η} \frac{q^{\frac{d}{2}}}{\log^ηq} \] holds for some small $η>0$

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