A short note on Jacobsthal's function
Fintan Costello, Paul Watts
函数g（n）表示最小数Q，使得每个Q个连续整数的序列都包含一个与n互质的整数。给出了该函数的一个新的明确的上界。
The function g(n) represents the smallest number Q such that every sequence of Q consecutive integers contains an integer coprime to n. We give a new and explicit upper bound on this function.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
A short note on the computation of the generalised Jacobsthal function for paired progressions
Jacobsthal函数最近被推广到成对级数的情况。证明了该函数的一个特定界足以证明哥德巴赫猜想和素数对猜想的正确性。我们扩展和适应算法描述的计算的共同Jacobsthal函数，并计算各自的函数值的配对Jacobsthal函数的素数的素数高达73。所有这些值均满足所述的特异性结合。除了本说明，我们还提供了算法方法的详细回顾和辅助文件中的完整计算结果。
Jacobsthal and Jacobsthal-Lucas numbers with Lehmar property
本文证明了 Jacobsthal 和 Jacobsthal-Lucas 数列中不存在具有 Lehmer 性质的数。
Modified third-order Jacobsthal sequence is defined in this study. Some properties involving this sequence, including the Binet-style formula and the generating function are also presented.
Jacobsthal's function and a short proof of the density of a set in the unit hypercube
我们使用Jacobsthal函数的上界来完成一个已知密度结果的证明。除了这里使用的Jacobsthal函数的界限，我们完成的证明只使用了初等方法和关于等差数列中素数无穷大的狄利克雷定理。
Dual third order Jacobsthal and dual third order Jacobsthal-Lucas numbers are defined. In this study, we work on these dual numbers and we obtain the properties e.g. some quadratic identities, summation, norm, negadual third order Jacobsthal identities, Binet formulas and relations of them. We also define new vectors which are called dual third order Jacobsthal vectors and dual third order Jacobsthal-Lucas vectors. We give properties of these vectors to exert in geometry of dual space.
A Short Note on Discrete Log Problem in $F_p$
设$p$是奇素数，使得2是有限域$F_p*$的本原元.在这篇短文中，我们提出了一个新的算法计算离散对数的$F_p*$。该算法是基于有限域的基本性质，是纯理论的。
In this short note we formalized the definition for the Ducci operator $D$ in the context of the $p$-adic field $\mathbb{Q}_p$ as a natural extension of the classical one. Moreover we will describe the behavior of the operator and will provide some simple results as a counterpart to the classical one.
A Short Note on Gaps between Powers of Consecutive Primes
设$α, β\geq 0$和$α+ β< 1$。本文证明了$\liminf_{n \to \infty} p_n^β(p_{n+1}^α- p_n^α) = 0$，其中$p_n$是$n$第素数.这说明了一个改进的结果Sándor和提供额外的证据对猜想的Andrica。这直接遵循最近的结果从梅纳德，陶，张素对。
Averages of short correlations: a note
我们给出了一个完整的初步研究的平均短相关的所谓的筛函数（一个相当一般的类算术函数）。
A short note on p-adic families of Hilbert Modular Forms
本文利用Buzzard的思想推广了作者以前的工作，给出了Hilbert模特征形的非普通$p$ -adic族的初等构造.
A note on Primes in Short Intervals
我们可以假设Hardy-Littlewood素数$k$ -元组猜想的平均形式，而不是强定量形式，并且仍然得到Montgomery和Soundarajan [1]关于$ψ(x+h) - ψ(x)$的相同分布结果.
A note on the Liouville function in short intervals
本文给出了一个简短而完备的证明，证明了对任何$δ> 0$，$\sum_{x \leq n \leq x+x^δ} λ(n) = o(x^δ)$对几乎所有的$x \in [X, 2X]$ .我们还勾画了这样的结果一般实值乘性函数的证明。这两个结果都是我们最近更复杂和冗长的预印本中结果的特殊情况。
A Short Note on the Bruinier-Kohnen Sign Equidistribution Conjecture and Halász' Theorem
本文利用Halász定理的一个结果改进了半整权模本征形关于自然密度的Bruinier-Kohnen符号等分布猜想.此外，应用塞尔的结果，我们删除所有未经证明的假设。
A short note on reduced residues
我们解决了一个问题，由于Recamán的最大可能的长度之间的所有等差数列的最小约化剩余系统模$n$，作为$n \to \infty$的下界行为。
Following the work of Hildebrand we improve the Po'lya- Vinogradov inequality in a specific range, we also give a general result that shows its dependency on Burgess bound and at last we improve the range of validity for a special case of Burgess' character sum estimate.
A short note on a conjecture of Okounkov about a q-analogue of multiple zeta values
在[OK] Okounkov研究了一个特定的$q$ -模拟多个zeta值，并作出了一些说明其代数结构。本文将Okounkovs $q$ -类似物与文献[BK 1]中定义的多因子和的生成函数进行比较。我们还提出了一个关于它们的维数的猜想，补充了Okounkov的维数公式，并给出了一些数值证据。
Short Note: Every Large Set of Integers Contains a Three Term Arithmetic Progression
我表明，一个平凡的修改标准证明罗斯定理的三倍在等差数列将导致以下定理：如果A是一个“大集合”，即它的元素是单调递增的整数和倒数的总和，其元素发散，然后序列包含一个等差数列的长度为3。
Note On Prime Gaps And Very Short Intervals
本文在黎曼假设的前提下，讨论了一个新的初等论证，它似乎证明了一个有限序列的素数p_1，p_2，.的最大素数间隔，p_n <= x，满足max {p_（n+1）- p_n：p_n <= x} <= c1（（logx）^2）/loglogx，c1 > 0常数。等价地，它表明非常短的区间（x，x + y）包含素数，对于所有y > c2（（logx）^2）/loglogx，c2 > 0常数，并且x足够大> 0。
In this note, we are interested in obtaining uniform upper bounds for the number of powerful numbers in short intervals $(x, x + y]$. We obtain unconditional upper bounds $O(\frac{y}{\log y})$ and $O(y^{11/12})$ for all powerful numbers and $y^{1/2}$-smooth powerful numbers respectively. Conditional on the $abc$-conjecture, we prove the bound $O(\frac{y}{\log^{1+ε} y})$ for squarefull numbers and the bound $O(y^{(2 + ε)/k})$ for $k$-full numbers when $k \ge 3$. They are related to Roth's theorem on arithmetic progressions and the conjecture on non-existence of three consecutive squarefull numb…
For any integer $l\geq 1$, let $p_1, p_2, \ldots, p_{l+2}$ be distinct prime numbers $\geq 5.$ For all real numbers $X>1,$ we let $N_{3,l}(X)$ denote the number of real quadratic fields $K$ whose absolute discriminant $d_K\leq X$ and $d_K$ is divisible by $(p_1\ldots p_{l+2})$ together with the class number $h_K$ of $K$ divisible by $2^{l}\cdot 3.$ Then, in this short note, by following the method in \cite{Byeonkoh}, we prove that $N_{3,l}(X) \gg X^\frac{7}{8}$ for all large enough $X$'s.
Notes on Universality in Short Intervals and Exponential Shifts
本文改进了Antanas Laurinčikas关于Riemann zeta-function在短区间内的普遍性定理，涉及区间长度，并证明平移可具有指数增长，该研究源于Laurinčikas提出的两个问题。
In this work, we define a new type of Eisenstein-like series by using Pell-Lucas numbers and call them the Pell-Lucas-Eisenstein Series. Firstly, we show that the Pell-Lucas-Eisenstein series are convergent on their domain. Afterwards we prove that they satisfy some certain functional equations. Proofs follows from some on calculations on Pell-Lucas numbers.
Classical mock modular and quantum modular forms are known to have an intimate relationship with Mordell integrals thanks to Zwegers' groundbreaking PhD thesis. More recently, generalisations of mock/quantum modular forms to so-called "higher depth" versions have been intensively studied. In essence, a mock/quantum modular form of depth $d$ is such that the error of modularity transforms as another mock/quantum modular form of depth $d - 1$. In this short note we use techniques of Bringmann, Kaszian, and Milas to show that the double Eichler integrals of a family of depth two quantum modular f…
A short note on sign changes
本文给出了满模群$SL_2(\mathbb{Z})$的正规化Hecke特征尖点形式的傅里叶系数序列$\{a(n^j)\}_{n\ge 1}, j=2,3,4$的符号变化数的一个定量结果.对于同余子群$N$为无平方数的$Γ_0(N)$，我们还证明了某些广义模函数的$q$ -指数$c(p) (p {vary over primes})$符号变化数的类似定量结果.
We give a short proof of Manin-Mumford in the multiplicative group based on the pigeon-hole principle and the so-called structure theorem for anomalous subvarieties. The arguments appear to be new and perhaps applicable in other situations.
We obtain an upper bound for the multiplicative energy of the spectrum of an arbitrary set from $\mathbb{F}_p$, which is the best possible up to the results on exponential sums over subgroups.
Let $f(z)=q+\sum_{n\geq 2}a(n)q^n$ be a weight $k$ normalized newform with integer coefficients and trivial residual mod 2 Galois representation. We extend the results of Amir and Hong in \cite{AH} for $k=2$ by ruling out or locating all odd prime values $|\ell|<100$ of their Fourier coefficients $a(n)$ when $n$ satisfies some congruences. We also study the case of odd weights $k\geq 1$ newforms where the nebentypus is given by a real quadratic Dirichlet character.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。