Pascal Stumpf
We solve a problem due to Recamán about the lower bound behavior of the maximum possible length among all arithmetic progressions in the least reduced residue system modulo $n$, as $n \to \infty$.
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Improving on some recent results of Matomäki and of Wright, we show that the number of Carmichael numbers to $X$ in a coprime residue class exceeds $X^{1/(6\log\log\log X)}$ for all sufficiently large $X$ depending on the modulus of the residue class.
In this note, we revisit a result of Benli's and Pollack's on the number of small prime $k^{th}$ power residues. The proof is based on their idea of using reciprocity laws, but the argument is simplified and we prove a slightly stronger bound.
In 1940 Paul Erdős made a conjecture about the distribution of reduced residues. Here we study the distribution of $k$-tuple of reduced residues.
Let $q$ be a large prime number, $a$ be any integer, $ε$ be a fixed small positive quantity. Friedlander and Shparlinksi \cite{FSh} have shown that there exists a positive integer $n\ll q^{5/2+ε}$ such that $ϕ(n)$ falls into the residue class $a \pmod q.$ Here, $ϕ(n)$ denotes Euler's function. In the present paper we improve this bound to $n\ll q^{2+ε}.$
Fix coprime natural numbers $a,q$. Assuming the Prime $k$-tuple Conjecture, we show that there exist arbitrarily long arithmetic progressions of Carmichael numbers, each of which lies in the reduced residue class $a$ mod $q$ and is a product of three distinct prime numbers.
In Sen's theory in the imperfect residue field case, Brinon defined a functor from the category of C_p-representations to the category of linear representations of certain Lie algebra. We give a comparison theorem between the continuous Galois cohomology of C_p-representations and the Lie algebra cohomology of the associated representations. The key ingredients of the proof are Hyodo's calculation of Galois cohomology and the effaceability of Lie algebra cohomology for solvable Lie algebras.
It is known that Hall's sextic residue sequence has some desirable features of pseudorandomness: an ideal two-level autocorrelation and linear complexity of the order of magnitude of its period $p$. Here we study its correlation measure of order $k$ and show that it is, up to a constant depending on $k$ and some logarithmic factor, of order of magnitude $p^{1/2}$, which is close to the expected value for a random sequence of length $p$. Moreover, we derive from this bound a lower bound on the $N$th maximum order complexity of order of magnitude $\log p$, which is the expected order of magnitud…
This note contains some asymptotic formulas for the sums of various residue classes of Euler's phi-function.
Given positive integers $a_1,\ldots,a_k$, we prove that the set of primes $p$ such that $p \not\equiv 1 \bmod{a_i}$ for $i=1,\ldots,k$ admits asymptotic density relative to the set of all primes which is at least $\prod_{i=1}^k \left(1-\frac{1}{φ(a_i)}\right)$, where $φ$ is the Euler's totient function. This result is similar to the one of Heilbronn and Rohrbach, which says that the set of positive integer $n$ such that $n \not\equiv 0 \bmod a_i$ for $i=1,\ldots,k$ admits asymptotic density which is at least $\prod_{i=1}^k \left(1-\frac{1}{a_i}\right)$.
We present a short, self-contained, and purely combinatorial proof of Linnik's theorem: for any $\varepsilon > 0$ there exists a constant $C_\varepsilon$ such that for any $N$, there are at most $C_\varepsilon$ primes $p \leqslant N$ such that the least positive quadratic non-residue modulo $p$ exceeds $N^\varepsilon$.
Given an odd prime p, we prove a congruence on the p-residue of the Fermat quotient q_p(a) in base a with 0<a<p, which arises from a generalization of the Voronoi congruences and from some other congruences on sums and weighted sums of divided Bernoulli numbers. As an application in the base 2 case, we retrieve a congruence for the first generalized harmonic number of order (p-1)/2, a generalization originally due to Z-H. Sun of a classical congruence known since long for the harmonic number of order (p-1)/2 as a special case of the Lerch formula. We find a sharpening of the Voronoi congruence…
A note on the second supplementary law of rational power residue symbols
本文针对具有特殊和式形式的素数 $p$，利用负指标多对数函数给出 $q$ 次幂剩余符号 $(q/p)_q=1$ 的显式条件，从而推广第二补律。
Problems 66 and 67 on sums of residue classes and primes of András Sárközy's collection of unsolved problems
本文讨论András Sárközy 2001年未解决问题集中的第66、67题，证明了素数稀疏子集和集的最优结果。
For positive integers $q$, Dirichlet's theorem states that there are infinitely many primes in each reduced residue class modulo $q$. A stronger form of the theorem states that the primes are equidistributed among the $φ(q)$ reduced residue classes modulo $q$. This paper considers patterns of sequences of consecutive primes $(p_n, p_{n+1}, \ldots, p_{n+k})$ modulo $q$. Numerical evidence suggests a preference for certain prime patterns. For example, computed frequencies of the pattern $(a,a)$ modulo $q$ up to $x$ are much less than the expected frequency $π(x)/φ(q)^2$. We begin to rigorously c…
In this short note we will use the residue theorem to establish a formula for Euler's constant. In particular, we offer a slightly generalized version of an interesting infinite series due to Flajolet, Gourdon, and Dumas.
In this note, we establish a duality result under the residue paring between certain two-dimensional adelic spaces, which are associated to a closed point on an arithmetic surface.
We show that for any $\varepsilon > 0$ and a sufficiently large cube-free $q$, any reduced residue class modulo $q$ can be represented as a product of $14$ integers from the interval $[1, q^{1/4e^{1/2} + \varepsilon}]$. The length of the interval is at the lower limit of what is possible before the Burgess bound on the smallest quadratic nonresidue is improved. We also consider several variations of this result and give applications to Fermat quotients.
For a wide range of $x$ and $y$ we show that ${\Cal S}(x,y)$, the set of integers below $x$ composed only of prime factors below $y$, is equidistributed in the reduced residue classes $\pmod q$ for all $q<y^{4\sqrt{e}-ε}$. This improves earlier work of Granville; any improvement of this range of $q$ would have interesting consequences for Vinogradov's conjecture on the least quadratic non-residue. For larger ranges of $q$ we prove the existence of a large subgroup of the group of reduced residues such that ${\Cal S}(x,y)$ is equidistributed within cosets of that subgroup.
Let $q$ be a prime. We classify the odd primes $p\neq q$ such that the equation $x^2\equiv q\pmod{p}$ has a solution, concretely, we find a subgroup $\mathbb{L}_{4q}$ of the multiplicative group $\mathbb{U}_{4q}$ of integers relatively prime with $4q$ (modulo $4q$) such that $x^2\equiv q\pmod{p}$ has a solution iff $p\equiv c\pmod{4q}$ for some $c\in\mathbb{L}_{4q}$. Moreover, $\mathbb{L}_{4q}$ is the only subgroup of $\mathbb{U}_{4q}$ of half order containing $-1$. Considering the ring $\mathbb{Z}[\sqrt{2}]$, for any odd prime $p$ it is known that the equation $x^2\equiv 2\pmod{p}$ has a solu…
Products of two elements of Granville's random model in residue classes
本文在Granville随机素数模型中无条件证明了Erdős-Odlyzko-Sárközy猜想关于模q既约剩余类包含两个素数乘积的类似版本，适用范围扩展到q^Θ（Θ>3/4），并指出模型选择的关键性。
In this work we introduce the numerical constant, LOPI, N LOPI is congruent to LOPI mod 18, equal to the lowest odd partition identity in conjunction with the reduced residue system Modulo 18, a complete disjoint covering residue system when considered in its whole set of residues from 0 to 17. By convolution of specific LOPI Dirichlet linear progressions for each LOPI superset, the non-prime elements of the 6 reduced residue congruence classes mod 18 are generated sequentially and uniquely. Through set generation and complete composite number subset generation based on the pattern of sequenti…
A Note on the Walsh Spectrum of Power Residue S-Boxes
本文界定了用于零知识友好型哈希函数的幂剩余S-盒的Walsh谱绝对值，并特别证明了Polocolo S-Box猜想的相关性。
While the sequence of primes is very well distributed in the reduced residue classes (mod $q$), the distribution of pairs of consecutive primes among the permissible $ϕ(q)^2$ pairs of reduced residue classes (mod $q$) is surprisingly erratic. This paper proposes a conjectural explanation for this phenomenon, based on the Hardy-Littlewood conjectures. The conjectures are then compared to numerical data, and the observed fit is very good.
Residue maps, Azumaya algebras, and buildings
本文利用扭转上闭链给出矩阵代数扭转留数的显式公式，并结合仿射建筑上的定点定理，快速证明了用Azumaya代数刻画非分歧代数的经典结论。
We generalize and solve the $\roman{mod}\,q$ analogue of a problem of Littlewood and Offord, raised by Vaughan and Wooley, concerning the distribution of the $2^n$ sums of the form $\sum_{i=1}^n\varepsilon_ia_i$, where each $\varepsilon_i$ is $0$ or $1$. For all $q$, $n$, $k$ we determine the maximum, over all reduced residues $a_i$ and all sets $P$ consisting of $k$ arbitrary residues, of the number of these sums that belong to $P$.
Bounds for the residue of Dedekind zeta functions under GRH and applications
该研究在GRH下给出戴德金ζ函数留数的显式上下界，改进了此前结果，其下界可检测类数≤2000的虚二次域，上界对固定非平凡域扩张可显著缩小。
The least quadratic residue and integers represented by quadratic forms
该研究针对模n的最小非平凡既约二次剩余ℓ(n)，从n的大小与素因子数维度建立近乎最优界，构造出ℓ(n)异常大的模，并将结果应用于推导有界判别式二元二次型表示不超过N的正整数的速率界。
For A,epsilon>0 and any sufficiently large odd n we show that for almost all k up to n^{1/5-epsilon} there exists a representation n=p1+p2+p3 with primes in residue classes b1,b2,b3 mod k for almost all admissible triplets b1,b2,b3 of reduced residues mod k.
We investigate, using the weighted linear sieve, the distribution of almost-primes among the residue classes (mod p) that generate the multiplicative group of reduced residue classes. We are concerned with finding an upper bound for the least prime or almost-prime primitive root (mod p) that holds uniformly for all p, analogous to Linnik's Theorem on a uniform upper bound for the least prime in a single arithmetic progression (mod p).
We show that for every fixed $A>0$ and $θ>0$ there is a $\vartheta=\vartheta(A,θ)>0$ with the following property. Let $n$ be odd and sufficiently large, and let $Q_{1}=Q_{2}:=n^{\h}(\log n)^{-\vartheta}$ and $Q_{3}:=(\log n)^θ$. Then for all $q_{3}\leq Q_{3}$, all reduced residues $a_{3}$ mod $q_{3}$, almost all $q_{2}\leq Q_{2}$, all admissible residues $a_{2}$ mod $q_{2}$, almost all $q_{1}\leq Q_{1}$ and all admissible residues $a_{1}$ mod $q_{1}$, there exists a representation $n=p_{1}+p_{2}+p_{3}$ with primes $p_{i}\equiv a_{i} (q_{i})$, $i=1,2,3$.
Binary Cubic Forms and Rational Cube Sum Problem
本文利用整二元三次型研究有理数立方和问题，无条件证明了在特定剩余类中存在无穷多个素数可表示为两个有理数的立方和。
A note on $t$-designs in isodual codes
本文利用扩展二元二次剩余码及其对偶码构造了3-设计，为组合设计理论提供了新的构造方法。
Notes on the LVP and CVP in $p$-adic Fields
本文提出在$p$-adic域中通过多项式时间算法解决LVP和CVP问题，利用$p$-adic范数的非阿基米德性质，构造$p$-adic晶格的正交基。
Products of primes in arithmetic progressions
本文证明了Erdős素数乘积猜想的三元版本，通过建立素数特征和的乘性稠密模型定理，证明了任意充分大无立方因子整数模数的简化剩余类可表示为三个不超过该整数的素数之积。
The problem of constructing a perfect Euler cuboid is reduced to a single Diophantine equation of the degree 12.
We show, for any $q\ge 3$ and distinct reduced residues $a,b \pmod q$, the existence of certain hypothetical sets of zeros of Dirichlet $L$-functions lying off the critical line implies that $π(x;q,a)<π(x;q,b)$ for a set of real $x$ of asymptotic density 1.
In this note we investigate the problem of existence of a torsor structure for Galois covers of (formal) schemes over a complete discrete valuation ring of residue characteristic $p>0$ in the case of abelian Galois groups of type (p,p,...,p).
In this note we, first, recall that the sets of all representatives of some special ordinary residue classes become $\left( m,n\right) $-rings. Second, we introduce a possible $p$-adic analog of the residue class modulo a $p$-adic integer. Then, we find the relations which determine, when the representatives form a $\left( m,n\right) $-ring. At the very short spacetime scales such rings could lead to new symmetries of modern particle models.
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