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Daodao Yang

Publications and source records attributed to Daodao Yang.

11 recordsLinked to original sources
arXiv2024

An almost sharp quantitative version of the Duffin-Schaeffer conjecture

We prove a quantitative version of the Duffin-Schaeffer conjecture with an almost sharp error term. Precisely, let $\psi:\mathbb{N}\to[0,1/2]$ be a function such that the series $\sum_{q=1}^\infty \varphi(q)\psi(q)/q$ diverges. In addition, given $\alpha\in\mathbb{R}$ and $Q\geqslant1$, let $N(\alpha;Q)$ be the number of coprime pairs $(a,q)\in\mathbb{Z}\times\mathbb{N}$ with $q\leqslant Q$ and $|\alpha-a/q|<\psi(q)/q$. Lastly, let $\Psi(Q)=\sum_{q\leqslant Q}2\varphi(q)\psi(q)/q$, which is the expected value of $N(\alpha;Q)$ when $\alpha$ is uniformly chosen from $[0, 1]$. We prove that $N(\alpha;Q)=\Psi(Q)+O_{\alpha,\varepsilon}(\Psi(Q)^{1/2+\varepsilon})$ for almost all $\alpha$ (in the Lebesgue sense) and for every fixed $\varepsilon>0$. This improves upon results of Koukoulopoulos-Maynard and of Aistleitner-Borda-Hauke.

math.NT↗
arXiv2023

Mean values of ratios of the Riemann zeta function

It is proved that $$\int_{T}^{2T} \left|\frac{\zeta\left(\frac{1}{2}+{\rm i} t\right)}{\zeta\left(1+2{\rm i} t\right)}\right|^2 {\rm d} t = \frac{1}{\zeta(2)} T \log T + \left( \frac{\log \frac{2}{\pi} + 2\gamma -1 }{\zeta(2)} -4 \,\frac{\zeta^{\prime}(2)}{\zeta^2(2)} \right) T + O\left(T\, \left(\log T\right)^{-2023} \right) , \quad \forall T \geqslant 100. $$ For given $a \in \mathbb N$, we also establish similar formulas for second moments of $|\zeta(\tfrac{1}{2} + {\rm i} t)/\zeta(1 + {\rm i} at)|.$ We have \begin{align*} \lim_{a \to \infty} \lim_{T \to \infty}\frac{1}{T \log T} \int_{T}^{2T} \left|\frac{\zeta\left(\frac{1}{2}+{\rm i} t\right)}{\zeta\left(1+{\rm i} at\right)}\right|^2 {\rm d} t = \frac{\zeta(2)}{\zeta(4)}. \end{align*}

math.NT↗
arXiv2023

Extreme values of Dirichlet polynomials with multiplicative coefficients

We study extreme values of Dirichlet polynomials with multiplicative coefficients, namely \[D_N(t) : = D_{f,\, N}(t)= \frac{1}{\sqrt{N}} \sum_{n\leqslant N} f(n) n^{it}, \] where $f$ is a completely multiplicative function with $|f(n)|=1$ for all $n\in\mathbb{N}$. We use Soundararajan's resonance method to produce large values of $\left|D_N(t)\right|$ uniformly for all such $f$. In particular, we improve a recent result of Benatar and Nishry, where they establish weaker lower bounds and only for almost all such $f$.

math.NT↗
arXiv2022

Extreme values of derivatives of zeta and $L$-functions

It is proved that as $T \to \infty$, uniformly for all positive integers $\ell \leqslant (\log_3 T) / (\log_4 T)$, we have \begin{equation*} \max_{T\leqslant t\leqslant 2T}\left|\zeta^{(\ell)}\Big(1+it\Big)\right| \geqslant \big(\mathbf Y_{\ell}+ o\left(1\right)\big)\left(\log_2 T \right)^{\ell+1} \,, \end{equation*} where $\mathbf Y_{\ell} = \int_0^{\infty} u^{\ell} \rho (u) du$. Here $\rho(u)$ is the Dickman function. We have $\mathbf Y_{\ell} > e^{\gamma}/(\ell + 1)$ and $ \log\, \mathbf Y_{\ell} = \left(1 + o\left(1\right) \right) \ell \log \ell$ when $ \ell \to \infty $, which significantly improves previous results in [17, 40]. Similar results are established for Dirichlet $L$-functions. On the other hand, when assuming the Riemann Hypothesis and the Generalized Riemann Hypothesis, we establish upper bounds for $ \left| \zeta^{(\ell)}\left(1+it\right)\right| $ and $\left|L^{(\ell)}(1, \chi) \right|$. Furthermore, when assuming the Granville-Soundararajan Conjecture is true, we establish the following asymptotic formulas $$\max_{ \substack{ \chi \neq \chi_0 \\ \chi(\text{mod}\, q)}} \left|L^{(\ell)}(1, \chi) \right| \sim \mathbf Y_{\ell}\left(\log_2 q\right)^{\ell+1},\,\, \quad \text{as}\,\quad q \to \infty,$$ where $q$ is prime and $\ell \in \mathbb{N}$ is given.

math.NT↗
arXiv2022

A note on log-type GCD sums and derivatives of the Riemann zeta function

In [Yan22a], we defined so-called ``log-type" GCD sums and proved the lower bounds $\Gamma^{(\ell)}_1(N) \gg_{\ell} \left(\log\log N\right)^{2+2\ell}$. We will establish the upper bounds $\Gamma^{(\ell)}_1(N)\ll_{\ell} \left(\log \log N\right)^{2+2\ell}$ in this note, which generalizes G\'{a}l's theorem on GCD sums (corresponding to the case $\ell = 0$). This result will be proved by two different methods. The first method is unconditional. We establish sharp upper bounds for spectral norms along $\alpha-$lines when $\alpha$ tends to $1$ with certain fast rates. As a corollary, we obtain upper bounds for log-type GCD sums. The second method is conditional. We prove that lower bounds for log-type GCD sums $\Gamma^{(\ell)}_1(N)$ can produce lower bounds for large values of derivatives of the Riemann zeta function on the 1-line. So from conditional upper bound for $\left| \zeta^{(\ell)}\left(1+ i t\right)\right|$, we obtain upper bounds for log-type GCD sums.

math.NT↗
arXiv2021

Extreme values of derivatives of the Riemann zeta function

It is proved that if $T$ is sufficiently large, then uniformly for all positive integers $\ell \leqslant (\log T) / (\log_2 T)$, we have \begin{equation*} \max_{T\leqslant t\leqslant 2T}\left|ζ^{(\ell)}\Big(1+it\Big)\right| \geqslant e^γ\cdot \ell^{\ell}\cdot (\ell+1)^{ -(\ell+1)}\cdot\Big(\log_2 T - \log_3 T + O(1)\Big)^{\ell+1} \,, \end{equation*} where $γ$ is the Euler constant. We also establish lower bounds for maximum of $\big|ζ^{(\ell)}(σ+it)\big|$ when $\ell \in \mathbb N $ and $σ\in [1/2, \,1)$ are fixed.

math.NT↗
arXiv2020

On a variant of Pillai's problem with transcendental numbers

In this paper, we study the asymptotic behaviour of the number of solutions $(m, n)\in \mathbb{N}^2$ to the inequality $ | \alpha^n - \beta^m | \leq x $ when $x$ tends to infinity. Here $\alpha, \beta$ are given multiplicatively independent complex numbers with $|\alpha| > 1$ and $|\beta|>1$.

math.NT↗
arXiv2020

Integers representable as differences of linear recurrence sequences

Let $\{U_n\}_{n \geq 0}$ and $\{V_m\}_{m \geq 0}$ be two linear recurrence sequences. We establish an asymptotic formula for the number of integers $c$ in the range $[-x, x]$ which can be represented as differences $ U_n - V_m$. In particular, the density of such integers is $0$.

math.NT↗
arXiv2020

Integers representable as differences of linear recurrence sequences

Let $\{U_n\}_{n \geqslant 0}$ and $\{G_m\}_{m \geqslant 0}$ be two linear recurrence sequences defined over the integers. We establish an asymptotic formula for the number of integers $c$ in the range $[-x, x]$ which can be represented as differences $ U_n - G_m$, when $x$ goes to infinity. In particular, the density of such integers is $0$.

math.NT↗
arXiv2016

Picard-Lefschetz Monodromy Groups of Quadratic Hypersurfaces

We study the topology of the space of affine hyperplanes $L \subset \CC^n$ which are in general position with respect to a given generic quadratic hypersurface $A$, and calculate the monodromy action of the fundamental group of this space on the relative homology groups $H_*(\CC^n, A \cup L)$ associated with such hyperplanes.

math.AG↗