Doychin Tolev
We cosider the number of r-tuples of squarefree numbers in a short interval. We prove that it cannot be much bigger than the expected value and we also estabish an asymptotic formula if the interval is not very short.
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Squarefree numbers in short intervals
该数论研究通过引入计数曲线附近受限格点的新技术，结合Green和Tao的定量定理，将短区间含无平方因子数的区间长度指数从1/5改进至1/5 - η。
Squarefree numbers in short intervals: explicit and formalized
作者将短区间无平方因子数的相关结果显式化并在Lean 4中形式化，给出了特定参数下的误差界，阐述了指数显式化及形式化的相关工作。
We calculate the mean and variance of sums of the Möbius function and the indicator function of the squarefrees, in both short intervals and arithmetic progressions, in the context of the ring of polynomials over a finite field of $q$ elements, in the limit $q\to \infty$. We do this by relating the sums in question to certain matrix integrals over the unitary group, using recent equidistribution results due to N. Katz, and then by evaluating these integrals. In many cases our results mirror what is either known or conjectured for the corresponding problems involving sums over the integers, whi…
On the variance of squarefree integers in short intervals and arithmetic progressions
本文渐近计算了短区间和等差数列中无平方因子整数数量的方差，并在假设 Lindelöf 假设及其推广下改进了适用范围，同时证明在完整范围内获得精确界等价于 Riemann 假设。
Squarefrees are Gaussian in short intervals
本文证明了在特定短区间内无平方因子数的计数趋于高斯分布，回答了Hall在1989年提出的问题，并将其推广至B-free整数，展示了其缩放后趋向分数布朗运动。
On differences of perfect powers and prime powers
本研究提出一种结合Frey-Hellegouarch曲线伽罗瓦表示、Thue-Mahler方程有效求解及改进对数下界的方法，显式确定完全幂与素数幂的差的无平方因子部分对应元组，并求解了指定参数范围的情形。
New small gaps between squarefree numbers
本文通过建立位移间渐近关系、推广两种经典差分并引入新差分，证明充分大的 x 附近长度为 Cx^{5/26} 的区间内必有一个无平方因子数。
Instead of a strong quantitative form of the Hardy-Littlewood prime $k$-tuple conjecture, one can assume an average form of it and still obtains the same distribution result on $ψ(x+h) - ψ(x)$ by Montgomery and Soundararajan [1].
A problem of D. H. Lehmer in short intervals. I
研究与模q乘法逆元奇偶性不同的整数个数问题，利用算术指数对和素变量不完全克洛斯特曼和估计，在特定条件下得到短区间内此类整数个数渐近公式并证明素变量版本，改进了前人结果。
Prime numbers have attracted the attention of mathematiciansand enthusiasts for millenniums due to their simple definition and remarkable properties. In this paper, we study primorial numbers (the product of the first prime numbers) to define primorial sets, primorial intervals, primorial tables, and primorial totative numbers. We establish relationships between prime numbers and primorial totative numbers and between admissible k-tuples of prime numbers and admissible k-tuples of primorial totative. Finally, we study the Goldbach conjecture and derive four Goldbach conjectures using primordia…
In this paper we establish a function field analogue of a conjecture in number theory which is a combination of several famous conjectures, including the Hardy-Littlewood prime tuple conjecture, conjectures on the number of primes in arithmetic progressions and in short intervals, and the Goldbach conjecture. We prove an asymptotic formula for the number of simultaneous prime values of $n$ linear functions, in the limit of a large finite field.
Given a positive integer $n$ the $k$-fold divisor function $d_k(n)$ equals the number of ordered $k$-tuples of positive integers whose product equals $n$. In this article we study the variance of sums of $d_k(n)$ in short intervals and establish asymptotic formulas for the variance of sums of $d_k(n)$ in short intervals of certain lengths for $k=3$ and for $k \ge 4$ under the assumption of the Lindelöf hypothesis.
We discuss in some detail the general problem of computing averages of convergent Euler products, and apply this to examples arising from singular series for the $k$-tuple conjecture and more general problems of polynomial representation of primes. We show that the singular series for the $k$-tuple conjecture have a limiting distribution when taken over $k$-tuples with (distinct) entries of growing size, and observe that its moments have a curious symmetry property. We also give conditional arguments that would imply that the number of twin primes (or more general polynomial prime patterns) in…
Short Interval Results For Powerfree Polynomials Over Finite Fields
该研究将有限域上无平方因子多项式分布的相关定理推广到所有k≥2的无k次幂情形，发展了对应经典数论技巧的多项式版本，得到若干经典定理的多项式域类似结果，补充了中等次数下的分布结论。
Let $η>0$ be a fixed positive number, let $N$ be a sufficiently large number. In this paper, we study the second moment of the sum of Hecke eigenvalues over primes in short intervals (whose length is $η\log N$) on average (with some weights) over the family of weight $k$ holomorphic Hecke cusp forms. We also generalize the above result to Hecke-Maass cusp forms for $SL(2,\mathbb{Z})$ and $SL(3,\mathbb{Z}).$ By applying the Hardy-Littlewood prime 2-tuples conjecture, we calculate the exact values of the mean values.
Consider the interval of integers $I_{m,n} = \{m, m+1, m+2,\ldots, m+n-1 \}$. For fixed integers $h,k,m$, and $c$, let $Φ_{h,k,m}^{(c)}(n)$ denote the number of solutions of the equation $(a_1+\cdots + a_h)- (a_{h+1} + \cdots + a_{h+k})=c$ with $a_i \in I_{m,n}$ for all $i=1,\ldots, h+k$. This is a polynomial in $n$ for all sufficiently large $n$, and the growth polynomial is constructed explicitly.
Biases in the distribution of primes in short intervals
在Hardy-Littlewood素数元组假设下，研究短区间内素数计数分布的二级渐近，发现算术修正导致比Cramér模型更强的向均值偏差，并通过有限筛论证和数值比较验证。
Products of consecutive integers with unusual anatomy
本文研究了连续自然数区间中连乘积的性质，定义了bad、very bad和type F3区间，并给出了关于这些区间数量的渐进行为及相关方程的解的近似数量。
We use short divisor sums to approximate prime tuples and moments for primes in short intervals. By connecting these results to classical moment problems we are able to prove that a positive proportion of consecutive primes are within a quarter of the average spacing between primes.
Our purpose is to give an account of the $r$-tuple problem on the increasing sequence of reduced fractions having denominators bounded by a certain size and belonging to a fixed real interval. We show that when the size grows to infinity, the proportion of the $r$-tuples of consecutive denominators with components in certain apriori fixed arithmetic progressions with the same ratio approaches a limit, which is independent on the interval. The limit is given explicitly and it is completely described in a few particular instances.
Large sieve inequality for sums of Legendre symbols over short intervals
本文利用Burgess界和Selberg筛法，研究Legendre符号在短区间上的二次矩界，并在特定条件下获得对h的幂次改进。
A Diophantine $m$-tuple is a set $A$ of $m$ positive integers such that $ab+1$ is a perfect square for every pair $a,b$ of distinct elements of $A$. We derive an asymptotic formula for the number of Diophantine quadruples whose elements are bounded by $x$. In doing so, we extend two existing tools in ways that might be of independent interest. The Erd\H os-Turán inequality bounds the discrepancy between the number of elements of a sequence that lie in a particular interval modulo 1 and the expected number; we establish a version of this inequality where the interval is allowed to vary. We also…
Répartition conjointe de trois nombres premiers et applications
本文研究三个素数的联合分布，证明相关奇异级数的渐近结果，并在哈代-利特尔伍德素数三元组猜想的定量假设下推导渐近公式，改进并补全已有结果，进而给出短区间素数分布的猜想性应用。
Sums of singular series with large sets and the tail of the distribution of primes
本文研究k相对于h较大时区间内集合的奇异级数常数平均值，将其应用于素数分布尾部，在Hardy--Littlewood猜想下得出含至少k个素数的对数长度区间数量的上界估计。
We investigate function field analogs of the distribution of primes, and prime $k$-tuples, in "very short intervals" of the form $I(f) := \{ f(x) + a : a \in \mathbb{F}_p \}$ for $f(x) \in \mathbb{F}_p[x]$ and $p$ prime, as well as cancellation in sums of function field analogs of the Möbius $μ$ function and its correlations (similar to sums appearing in Chowla's conjecture). For generic $f$, i.e., for $f$ a Morse polynomial, the error terms are roughly of size $O(\sqrt{p})$ (with typical main terms of order $p$). For non-generic $f$ we prove that independence still holds for "generic" set of …
Odd moments and adding fractions
论文研究了短区间内互质余数分布的奇 moment 上界，并验证了Montgomery 和 Vaughan 的猜想，同时证明了 Hardy-Littlewood 猜想中奇数 k-tuples 素数数量的平均值上界。
A central limit theorem for Hilbert modular forms
本文研究全实数域上 GL$_2$ 尖点自守表示的 Satake 角分布，在权向量分量快速增长等条件下证明落入固定区间的角数量满足中心极限定理。
In this paper we study the problem of counting Salem numbers of fixed degree. Given a set of disjoint intervals $I_1,\ldots, I_{k}\subset \left[0;π\right]$, $1\leq k\leq m$ let $Sal_{m,k}(Q,I_1,\ldots,I_{k})$ denote the set of ordered $(k+1)$-tuples $\left(α_0,\ldots,α_{k}\right)$ of conjugate algebraic integers, such that $α_0$ is a Salem numbers of degree $2m+2$ satisfying $α\leq Q$ for some positive real number $Q$ and $\argα_i\in I_i$. We derive the following asymptotic approximation \[ \# Sal_{m,k}(Q,I_1,\ldots,I_{k})=ω_m\,Q^{m+1}\,\int\limits_{I_1}\ldots\int\limits_{I_{k}}ρ_{m,k}(\boldsy…
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