Tsz Ho Chan, Angel Kumchev, Mate Wierdl
A classical additive basis question is Waring's problem. It has been extended to integer polynomial and non-integer power sequences. In this paper, we will consider a wider class of functions, namely functions from a Hardy field, and show that they are asymptotic bases.
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With the uniform positions we prove theorems of Landau and Hardy-Littlwood type for Goldbach, Chen, Lemoine-Levy and other binary partitions of positive integers. We also pose some new conjectures.
Let $b$ be a numeration base. A $b$-additive Ramanujan-Hardy number $N$ is an integer for which there exists at least an integer $M$, called additive multiplier, such that the product of $M$ and the sum of base $b$ digits of $N$, added to the reversal of the product, gives $N$. We show that for any $b$ there exists an infinity of $b$-additive Ramanujan-Hardy numbers and an infinity of additive multipliers. A $b$-multiplicative Ramanujan-Hardy number $N$ is an integer for which there exists at least an integer $M$, called multiplicative multiplier, such that the product of $M$ and the sum of ba…
Non-Linear Additive Twists of $\mathrm{GL}_{3}$ Hecke Eigenvalues
本文通过 DFI 圆法及降低导子机制界定了 $\mathrm{GL}_{3}$ Hecke 特征值的非线性加法扭曲，并应用于证明 Hardy 函数第一矩的非平凡界。
We continue our recent work on additive problems with prime summands: we already studied the \emph{average} number of representations of an integer as a sum of two primes, and also considered individual integers. Furthermore, we dealt with representations of integers as sums of powers of prime numbers. In this paper, we study a Cesàro weighted partial \emph{explicit} formula for generalised Hardy-Littlewood numbers (integers that can be written as a sum of a prime power and a square) thus extending and improving our earlier results.
The classical Dedekind sums $s(d, c)$ can be represented as sums over the partial quotients of the continued fraction expansion of the rational $\frac{d}{c}$. Hardy sums, the analog integer-valued sums arising in the transformation of the logarithms of $θ$-functions under a subgroup of the modular group, have been shown to satisfy many properties which mirror the properties of the classical Dedekind sums. The representation as sums of partial quotients has, however, been missing so far. We define non-classical continued fractions and prove that Hardy sums can be expressed as a sums of partial …
Hardy and Littlewood initiated the study of the $2k$-th moments of the Riemann zeta function on the critical line. In 1918 Hardy and Littlewood established an asymptotic formula for the second moment and in 1926 Ingham established an asymptotic formula for the fourth moment. Since then no other moments have been asymptotically evaluated. In this article we study the sixth moment of the zeta function on the critical line. We show that a conjectural formula for a certain family of ternary additive divisor sums implies an asymptotic formula with power savings error term for the sixth moment of th…
A subset $C$ of an abelian group $G$ is a minimal additive complement to $W \subseteq G$ if $C + W = G$ and if $C' + W \neq G$ for any proper subset $C' \subset C$. In this paper, we study which sets of integers arise as minimal additive complements. We confirm a conjecture of Kwon, showing that bounded-below sets with arbitrarily large gaps arise as minimal additive complements. Moreover, our construction shows that any such set belongs to a co-minimal pair, strengthening a result of Biswas and Saha for lacunary sequences. We bound the upper and lower Banach density of syndetic sets that aris…
The integral $R(t)=π^{-1}(lnζ(\frac{1}{2}+it)+i\vartheta (t))$ of the logarithmic derivative of the Hardy Z function $Z(t)=e^{i\vartheta (t)}ζ(\frac{1}{2}+it)$, where $\vartheta (t)$ is the Riemann-Siegel theta function, and $ζ(t)$ is the Riemann zeta function, is used as a basis for the construction of a pair of transcendental entire functions $ν(t)=-ν(1-t)=-{ΔR(\frac{i}{2}-it)}^{-1}=-G(\frac{i}{2}-it)$ where $G=-(ΔR(t))^{-1}$ is the derivative of the additive inverse of the reciprocal of the Laplacian of $R(t)$ and $χ(t)=-χ(1-t)=\dotν (t)=-iH(\frac{i}{2}-it)$ where $H(t)=\dot{G} (t)$ has roo…
Exponential sums weighted by additive functions
本文研究了由加性函数加权的指数和，证明了其紧界，并应用Hardy-Littlewood圆法解决了与总素因子个数相关的Goldbach-Vinogradov三元问题的渐进行为。
Around the Gauss circle problem: Hardy's conjecture and the distribution of lattice points near circles
本文围绕 Hardy 猜想，通过研究圆边界附近格点间的相互作用检验民间启发式方法，证明其自相关性由随机模型决定，并指出充分分离的格点表现独立，同时提出了关于格点对分布的猜想。
Additive triples of bijections, or the toroidal semiqueens problem
本文用改造的Hardy–Littlewood圆法证明了模n群上双射加法三元组数量的渐近公式，统一解决了环面半皇后、正交态射及循环拉丁方横截线等计数问题。
We define $\overlineψ$ to be the multiplicative arithemtic function that satisfies \[\overlineψ(p^α)=\begin{cases} p^{α-1}(p+1), & \mbox{if } p\neq 2; \\ p^{α-1}, & \mbox{if } p=2 \end{cases}\] for all primes $p$ and positive integers $α$. Let $λ(n)$ be the number of iterations of the function $\overlineψ$ needed for $n$ to reach $2$. It follows from a theorem due to White that $λ$ is additive. Following Shapiro's work on the iterated $φ$ function, we determine bounds for $λ$. We also use the function $λ$ to partition the set of positive integers into three sets $S_1,S_2,S_3$ and determine som…
We study sets of recurrence, in both measurable and topological settings, for actions of $(\mathbb{N},\times)$ and $(\mathbb{Q}^{>0},\times)$. In particular, we show that autocorrelation sequences of positive functions arising from multiplicative systems have positive additive averages. We also give criteria for when sets of the form $\{(an+b)^{\ell}/(cn+d)^{\ell}: n \in \mathbb{N}\}$ are sets of multiplicative recurrence, and consequently we recover two recent results in number theory regarding completely multiplicative functions and the Omega function.
Additive functionals of Harmonic samples: the conditioned Dickman regime
研究调和分布下可加算术函数的分布行为，发现其经适当标准化后收敛到条件Dick曼型泊松积分，与均匀采样下的高斯极限形成对比。
Algebraic Lattices Arising from Congruence Submodules in Subfields of $p$-th Cyclotomic Fields
本研究从p次分圆域子域的代数整数环中特定ℤ-模族出发构造代数格，计算其中心密度下界，得到2、3、5维最优填充密度的代数格，助力经典球体填充问题研究。
Continuation of Dirichlet series I
本文研究亚纯函数内积空间上线性泛函形式的Dirichlet级数，建立其边界不连续性与亚纯函数虚轴极点零点的关系，并应用Delange Tauber型定理条件性证明加性除数问题的猜想渐近式。
Weighted uniform distribution of subpolynomial functions along primes and applications
本文研究了 Hardy 域中次多项式函数沿自然数和素数序列的加权均匀分布，给出了充要条件，推广了已有结果并应用于均匀分布、遍历理论与加性组合学。
In the analytic study of trace functions of $\ell$-adic sheaves over finite fields, a crucial issue is to control the conductor of sheaves constructed in various ways. We consider cohomological transforms on the affine line over a finite field which have trace functions given by linear operators with an additive character of a rational function in two variables as a kernel. We prove that the conductor of such a transform is bounded in terms of the complexity of the input sheaf and of the rational function defining the kernel, and discuss applications of this result, including motivating exampl…
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