arXiv subjects

Yangbo Ye

Publications and source records attributed to Yangbo Ye.

10 recordsLinked to original sources

arXiv2026

Probing automorphic induction through zero correlations and secondary arithmetic terms

Zero correlations offer a way to test the cuspidal structure of an automorphic L-function, but their leading self statistics retain only coarse degree and multiplicity data. We therefore adjoin a specified cuspidal L-function as a probe and study the resulting raw mixed correlations. Assuming a factorization into cuspidal L-functions over Q, we show that a mixed pair statistic detects the probe's occurrence and recovers its multiplicity. A suitable Gevrey-order-two test eliminates the smooth-density terms and gives a subexponential error. For arbitrary smooth plateau tests, the prime number theorem of Harcos and Thorner gives an error smaller than every fixed inverse power of log T, without a self-contragredience assumption. The stronger prime number theorem of Liu and Ye gives a subexponential remainder under its stated structural assumptions, with the possible norm-twist terms made explicit. To obtain arithmetic information beyond this overlap, we compute the correlation expansion at every fixed level and inverse-logarithmic order within the original strict small-support range. We treat the pair and higher correlations separately, using Jiang's rational-field fixed-data estimates to justify the required prime-power convergence. The pair calculation identifies the mixed arithmetic finite parts and gives detailed error estimates, including the contribution of norm twists. The higher-level calculation retains connected prime-power invariants, whose first cubic contribution appears at order (log T)^{-3}. Explicit third- and fourth-level formulas and examples explain the arithmetic content of these coefficients and the limitations of self-correlations. Varying powers of the probe separates their mixed components. These formulas, together with the distinct-index sieve, provide a collection of arithmetic probe responses toward the further problem of recovering automorphic induction structure.

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arXiv2026

Functoriality probes over number fields by mixed zero correlations

We use an automorphic L-function over one number field as a probe for the functorial structure of another automorphic L-function over a possibly different field. No automorphic induction is assumed. We first determine the mixed Euler factors arising from the explicit formula, including the residue-degree multiplicities that occur when local traces are grouped by absolute norm. This local analysis gives four probe constructions. Rational-field probes with known base change detect constituent multiplicities and, in equal rank, exact Satake equalities. Probes over a common Galois field detect conjugacy and stabilizers. Dirichlet-character probes give congruence-weighted combinations of Satake traces even when the leading overlap vanishes. A tensor-product construction realizes general different-field probes whenever the required base changes exist. We then separate the pair calculation from the higher connected calculations and work out the cubic and quartic arithmetic explicitly. In an explicit small Fourier box these expressions yield all finite secondary orders of the centered statistics; the required pair prime number theorems also give raw and distinct-index correlations. The main analytic tools are an independent fixed-data proof of Hypothesis H over number fields, a positive-margin estimate for rationally grouped coefficients, and the theorem of Harcos-Thorner for the realized Rankin-Selberg factors. This expanded version includes the full computations, local examples, and an appendix explaining the character-extension issue avoided by the proof of Hypothesis H. The resulting probes give concrete tests of functorial relationships, while recovery of automorphic induction remains a longer-term goal.

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arXiv2026

Zero density in short strips and short-interval bounds toward Hypothesis S

We deduce zero-density estimates in short vertical strips from short-interval moments of globally tempered automorphic $L$-functions, by adapting the large-value estimate of Guth and Maynard and the Type II zero argument of Maynard and Pratt to the framework of Ye and Zhang. For the Riemann zeta function, we obtain $N_ζ(σ;T,T+T^a)\ll T^{15a(1-σ)/(3+5σ)+\varepsilon}$ for $2/3\le a\le1$ and $7/10\leσ\le4/5$, improving Ye--Zhang's exponent in the interior of this range. Combining the localized density bound with the recent quasi-Riemann theorems, we estimate smoothed prime-power sums with phase $\exp(2πiαn^β)$ on intervals of length $K^δ$. For $0<β<13/20$, we obtain a power saving whenever $\max\{17/30,1-β\}<δ<1$; in particular the square-root phase admits every $δ>17/30$. A revised transform estimate retains both the stationary spread and the Fourier width $K/L$. For example, the bound at $β=1/2$, $δ=3/5$ is $K^{15/26+\varepsilon}$, uniformly for frequencies $mα$ with $1\le m\le K^{1/10}$. We also retain power savings for certain self-dual $GL(2)$ twists and self-dual $GL(3)$ cusp forms, without assuming Ramanujan. All three applications have explicit frequency dependence.

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arXiv2026

Nonlinear twists of Rankin-Selberg coefficients in the eigenvalue aspect

Let $g$ be a fixed primitive Hecke--Maass cusp form, and let $f$ vary with spectral parameter $k_f=K$, at a fixed level and with trivial nebentypus. We study smooth sums of $λ_f(n)λ_g(n)e(αn^β)$ for fixed real $α\ne0$. The crucial input is the smooth spectral second moment of Lau, Liu, and Ye, for which the window length can be taken to be $L=K^{1/3+a}$ for arbitrarily small fixed $a>0$. We show that, for every $κ>0$, there is $β_0>0$ such that the sum is $O(K^{2/3+κ}X^{1/2})$ for $1\le X\le K^2$ and $0<β\leβ_0$. We give a detailed Baire category argument which makes the dependence on the smooth weight uniform through one finite derivative norm. A second argument keeps the phase explicit, applies Voronoi summation and shifted convolution estimates, and gives the numerical bound $O(K^{423/224+(5/7)β+\varepsilon})$ at $X=K^2$ for $0<β<5/32$.

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arXiv2024

Distribution of neighboring values of the Liouville and Möbius functions

Let $λ(n)$ and $μ(n)$ denote the Liouville function and the Möbius function, respectively. In this study, relationships between the values of $λ(n)$ and $λ(n+h)$ up to $n\leq10^8$ for $1\leq h\leq1,000$ are explored. Chowla's conjecture predicts that the conditional expectation of $λ(n+h)$ given $λ(n)=1$ for $1\leq n\leq X$ converges to the conditional expectation of $λ(n+h)$ given $λ(n)=-1$ for $1\leq n\leq X$ as $X\rightarrow\infty$. However, for finite $X$, these conditional expectations are different. The observed difference, together with the significant difference in $χ^2$ tests of independence, reveals hidden additive properties among the values of the Liouville function. Similarly, such additive structures for $μ(n)$ for square-free $n$'s are identified. These findings pave the way for developing possible, and hopefully efficient, additive algorithms for these functions. The potential existence of fast, additive algorithms for $λ(n)$ and $μ(n)$ may eventually provide scientific evidence supporting the belief that prime factorization of large integers should not be too difficult. For $1\leq h\leq1,000$, the study also tested the convergence speeds of Chowla's conjecture and found no relation on $h$.

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arXiv2022

Double square moments and bounds for resonance sums of cusp forms

Let $f$ and $g$ be holomorphic cusp forms for the modular group $SL_2(\mathbb Z)$ of weight $k_1$ and $k_2$ with Fourier coefficients $λ_f(n)$ and $λ_g(n)$, respectively. For real $α\neq0$ and $0<β\leq1$, consider a smooth resonance sum $S_X(f,g;α,β)$ of $λ_f(n)λ_g(n)$ against $e(αn^β)$ over $X\leq n\leq2X$. Double square moments of $S_X(f,g;α,β)$ over both $f$ and $g$ are nontrivially bounded when their weights $k_1$ and $k_2$ tend to infinity together. By allowing both $f$ and $g$ to move, these double moments are indeed square moments associated with automorphic forms for $GL(4)$. By taking out a small exceptional set of $f$ and $g$, bounds for individual $S_X(f,g;α,β)$ will then be proved. These individual bounds break the resonance barrier of $X^\frac58$ for $\frac16<β<1$ and achieve a square-root cancellation for $\frac13<β<1$ for almost all $f$ and $g$ as an evidence for Hypothesis S for cusp forms over integers. The methods used in this study include Petersson's formula, Poisson's summation formula, and stationary phase integrals.

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arXiv2017

Improved subconvexity bounds for GL(2)xGL(3) and GL(3) L-functions by weighted stationary phase

Let $f$ be a fixed self-contragradient Hecke-Maass form for $SL(3,\mathbb Z)$, and $u$ an even Hecke-Maass form for $SL(2,\mathbb Z)$ with Laplace eigenvalue $1/4+k^2$, $k>0$. A subconvexity bound $O\big(k^{4/3+\varepsilon}\big)$ in the eigenvalue aspect is proved for the central value at $s=1/2$ of the Rankin-Selberg $L$-function $L(s,f\times u)$. Meanwhile, a subconvexity bound $O\big((1+|t|)^{2/3+\varepsilon}\big)$ in the $t$ aspect is proved for $L(1/2+it,f)$. These bounds improved corresponding subconvexity bounds proved by Xiaoqing Li (Annals of Mathematics, 2011). The main technique in the proof, other than those used by Li, is an $n$th-order asymptotic expansion of a weighted stationary phase integral, for arbitrary $n\geq1$. This asymptotic expansion sharpened the classical result for $n=1$ by Huxley.

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arXiv2016

Weighted stationary phase of higher orders

An $n$th-order first derivative test for oscillatoric integrals is established. When the phase has a single stationary point, an $n$th-order asymptotic expansion of a weighted stationary phase integral is proved for arbitrary $n\geq1$. This asymptotic expansion sharpened the classical result for $n=1$ by Huxley. Possible applications include analysis and analytic number theory.

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arXiv2015

Asymptotics for cuspidal representations by functoriality from GL(2)

Let $π$ be a unitary automorphic cuspidal representation of $GL_2(\mathbb{Q}_\mathbb{A})$ with Fourier coefficients $λ_π(n)$. Asymptotic expansions of certain sums of $λ_π(n)$ are proved using known functorial liftings from $GL_2$, including symmetric powers, isobaric sums, exterior square from $GL_4$ and base change. These asymptotic expansions are manifestation of the underlying functoriality and reflect value distribution of $λ_π(n)$ on integers, squares, cubes and fourth powers.

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arXiv2014

Resonance and rapid decay of exponential sums of Fourier coefficients of a Maass form for $GL_m(\mathbb Z)$

Let $f$ be a full-level cusp form for $GL_m(\mathbb Z)$ with Fourier coefficients $A_f(n_1,...,n_{m-1})$. In this paper an asymptotic expansion of Voronoi's summation formula for $f$ is established. As applications of this formula, a smoothly weighted average of $A_f(n,1,...,1)$ against $e(α|n|^β)$ is proved to be rapidly decayed when $0<β<1/m$. When $β=1/m$ and $α$ equals or approaches $\pm mq^{1/m}$ for a positive integer $q$, this smooth average has a main term of the size of $|A_f(1,...,1,q)+A_f(1,...,1,-q)|X^{1/(2m)+1/2}$, which is a manifestation of resonance of oscillation exhibited by the Fourier coefficients $A_f(n,1,...,1)$. Similar estimate is also proved for a sharp-cut sum.

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