Alessandro Gambini, Remis Tonon, Alessandro Zaccagnini
We investigate the distribution of the digits of quotients of randomly chosen positive integers taken from the interval $[1,T]$, improving the previously known error term for the counting function as $T\to+\infty$. We also resolve some natural variants of the problem concerning points with prime coordinates and points that are visible from the origin.
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Quotient sets have attracted the attention of mathematicians in the past three decades. The set of quotients of primes is dense in the positive real numbers and the set of all quotients of Gaussian primes is also dense in the complex plane. Sittinger has proved that the set of quotients of primes in an imaginary quadratic ring is dense in the complex plane and the set of quotients of primes in a real quadratic number ring is dense in R. An interesting open question is introduced by Sittinger: Is the set of quotients of Hurwitz primes dense in the quaternions? In this paper, we answer the quest…
It has been observed many times, both in the Monthly and elsewhere, that the set of all quotients of prime numbers is dense in the positive real numbers. In this short note we answer the related question: "Is the set of all quotients of Gaussian primes dense in the complex plane?"
It has been established on many occasions that the set of quotients of prime numbers is dense in the set of positive real numbers. More recently, it has been proved that the set of quotients of primes in the Gaussian integers is dense in the complex plane. In this article, we not only extend this result to any imaginary quadratic number ring, but also prove that the set of quotients of primes in any real quadratic number ring is dense in the set of real numbers. To conclude, we show how to extend these results to an arbitrary algebraic number ring.
It is known that all modular forms on SL_2(Z) can be expressed as a rational function in eta(z), eta(2z) and eta(4z). By utilizing known theorems, and calculating the order of vanishing, we can compute the eta-quotients for a given level. Using this count, knowing how many eta-quotients are linearly independent and using the dimension formula, we can figure out a subspace spanned by the eta-quotients. In this paper, we primarily focus on the case where N=p a prime.
Unexpected primes of good reduction in quotients of modular and Shimura curves
研究权重2、平方自由层数N且有固定阿特金 - 勒纳符号模式的新形式零维空间，借此对平方自由层数的模曲线和志村曲线阿特金 - 勒纳商中良好约化的意外素数进行分类。
We determine the conditions under which singular values of multiple $η$-quotients of square-free level, not necessarily prime to~6, yield class invariants, that is, algebraic numbers in ring class fields of imaginary-quadratic number fields. We show that the singular values lie in subfields of the ring class fields of index $2^{k' - 1}$ when $k' \geq 2$ primes dividing the level are ramified in the imaginary-quadratic field, which leads to faster computations of elliptic curves with prescribed complex multiplication. The result is generalised to singular values of modular functions on $X_0^+ (…
Wilson's theorem modulo higher prime powers I: Fermat and Wilson quotients
该研究利用Bell多项式与Newton恒等式，证明威尔逊定理及威尔逊商可由含费马商幂和项的任意更高素数幂模超同余式描述，还给出相关多元多项式的递推公式与更多性质，推进了数论中超同余问题的研究。
An integral power series is called lacunary modulo $M$ if almost all of its coefficients are divisible by $M$. Motivated by the parity problem for the partition function, $p(n)$, Gordon and Ono studied the generating functions for $t$-regular partitions, and determined conditions for when these functions are lacunary modulo powers of primes. We generalize their results in a number of ways by studying infinite products called Dedekind eta-quotients and generalized Dedekind eta-quotients. We then apply our results to the generating functions for the partition functions considered by Nekrasov, Ok…
From the Modularity Theorem proven by Wiles, Taylor, et al, we know that all elliptic curves are modular. It has been shown by Martin and Ono exactly which are represented by eta-quotients, and some examples of elliptic curves represented by modular forms that are linear combinations of eta-quotients have been given by Pathakjee, RosnBrick, and Yoong. In this paper, we first show that eta-quotients which are modular for any congruence subgroup of level $N$ coprime to $6$ can be viewed as modular for $Γ_0(N)$. We then categorize when even weight eta-quotients can exist in $M_k (Γ_1(p))$ and $M_…
Garcia, Kahoro, and Luca showed that the Bateman-Horn conjecture implies $ϕ(p-1) \geq ϕ(p+1)$ for a majority of twin-primes pairs $p,p+2$ and that the reverse inequality holds for a small positive proportion of the twin primes. That is, $p$ tends to have more primitive roots than does $p+2$. We prove that Dickson's conjecture, which is much weaker than Bateman-Horn, implies that the quotients $\frac{ϕ(p+1)}{ϕ(p-1)}$, as $p,p+2$ range over the twin primes, are dense in the positive reals. We also establish several Schinzel-type theorems, some of them unconditional, about the behavior of $\frac{…
We use the method of quadratic Chabauty on the quotients $X_0^+(N)$ of modular curves $X_0(N)$ by their Fricke involutions to provably compute all the rational points of these curves for prime levels $N$ of genus four, five, and six. We find that the only such curves with exceptional rational points are of levels $137$ and $311$. In particular there are no exceptional rational points on those curves of genus five and six. More precisely, we determine the rational points on the curves $X_0^+(N)$ for $N=137,173,199,251,311,157,181,227,263,163,197,211,223,269,271,359$.
The Fermat quotient $q_p(a):=(a^{p-1}-1)/p$, for prime $p\nmid a$, and the Wilson quotient $w_p:=((p-1)!+1)/p$ are integers. If $p\mid w_p,$ then $p$ is a Wilson prime. For odd $p,$ Lerch proved that $(\sum_{a=1}^{p-1} q_p(a) - w_p)/p$ is also an integer; we call it the Lerch quotient $\ell_p.$ If $p\mid\ell_p$ we say $p$ is a Lerch prime. A simple Bernoulli-number test for Lerch primes is proven. There are four Lerch primes 3, 103, 839, 2237 up to $3\times10^6$; we relate them to the known Wilson primes 5, 13, 563. Generalizations are suggested. Next, if $p$ is a non-Wilson prime, then $q_p(w…
Metric properties of continued fractions with large prime partial quotients
研究连分数展开中存在至少两个同时大且素数的部分商的数集，建立其勒贝格测度的零一定律并确定其Hausdorff维度。
We show that for every positive integer $k$, there exist $k$ consecutive primes having the property that if any digit of any one of the primes, including any of the infinitely many leading zero digits, is changed, then that prime becomes composite.
Tao has shown that in any fixed base, a positive proportion of prime numbers cannot have any digit changed and remain prime. In other words, most primes are "digitally delicate". We strengthen this result in a manner suggested by Tao: A positive proportion of primes become composite under any change of a single digit and any insertion a fixed number of arbitrary digits at the beginning or end.
This paper explores some previously-unrecognized consequences of Lerch's 1905 formula for the Fermat quotient, with special attention to the sums which he introduced in this context. A generalization of his result is proved, and a new proof given of a sharpened result by Skula (2008). We also sharpen the criteria given by Emma Lehmer in 1938 for a Wieferich prime to be simultaneously a Mirimanoff prime.
Wilson's theorem modulo higher prime powers II: Bernoulli numbers and polynomials
该研究将作者此前用费马商幂和超同余描述威尔逊定理与威尔逊商的成果，转化为幂和与伯努利数的同余式，给出更简洁的证明，还算出了模p⁴的威尔逊商等结果并可推广到更高素数幂。
We improve an estimate of A.Granville (1987) on the number of vanishing Fermat quotients $q_p(\ell)$ modulo a prime $p$ when $\ell$ runs through primes $\ell \le N$. We use this bound to obtain an unconditional improvement of the conditional (under the Generalised Riemann Hypothesis) estimate of Y. Ihara (2006) on a certain sum, related to vanishing Fermat quotients. In turn this sum appears in the study of the index of certain subfields of of cyclotomic fields $\Q(\exp(2 πi/p^2))$.
We survey some different results on the digits of prime numbers, giving a simplified proof of weak forms of a result of Maynard and Mauduit-Rivat.
Let C denote the Fermat curve over Q of prime exponent l. The Jacobian Jac(C) of C splits over Q as the product of Jacobians Jac(C_k), 0< k < l-1, where C_k are curves obtained as quotients of C by certain subgroups of automorphisms of C. It is well known that Jac(C_k) is the power of an absolutely simple abelian variety B_k with complex multiplication. We call degenerate those pairs (l,k) for which B_k has degenerate CM type. For a non-degenerate pair (l,k), we compute the Sato-Tate group of Jac(C_k), prove the generalized Sato-Tate Conjecture for it, and give an explicit method to compute th…
On continued fraction partial quotients of square roots of primes
本文证明素数平方根连分数中特定部分商出现次数的有限性，给出周期长度可被4整除时部分商1的存在性，并建立周期长度上界及多项式分解，回答Miska和Ulas的若干问题。
We prove a generalization of the Davenport-Heilbronn theorem to quotients of ideal class groups of quadratic fields by the primes lying above a fixed set of rational primes $S$. Additionally, we obtain average sizes for the relaxed Selmer group $\mathrm{Sel}_3^S(K)$ and for $\mathcal{O}_{K,S}^\times/(\mathcal{O}_{K,S}^\times)^3$ as $K$ varies among quadratic fields with a fixed signature ordered by discriminant.
Let $\mathcal{A}'$ be the set of integers missing any three fixed digits from their decimal expansion. We produce primes in a thin sequence by proving an asymptotic formula for counting primes of the form $p = m^2 + \ell^2$, with $\ell \in \mathcal{A}'$. The proof draws on ideas from the work of Friedlander-Iwaniec on primes of the form $p = x^2+y^4$, as well as ideas from the work of Maynard on primes with restricted digits.
Continued fractions with large prime partial quotients
本文研究部分商为大素数的连分数实数集合，通过证明几乎素数zeta函数尾端的新渐近结果，确定了相应集合的勒贝格测度与豪斯多夫维数，推广并涵盖了多个已有经典与近期定理。
Making use of covering systems and a theorem of D. Shiu, the first and second authors showed that for every positive integer $k$, there exist $k$ consecutive widely digitally delicate primes. They also noted that for every positive integer $k$, there exist $k$ consecutive primes which are Brier numbers. We show that for every positive integer $k$, there exist $k$ consecutive primes that are both widely digitally delicate and Brier numbers.
The Prime Digit Distribution Conjecture: A Formal Proof of Average Digit Equidistribution in the Prime Numbers
研究素数数字分布，结合素数定理等方法证明素数中数字出现概率渐近为\(\frac{1}{10}\)，建立内部数字定量等分布，阐明与其他相关问题区别，得出素数数字汇总后渐近等分布的结论。
Benford's Law describes the prevalence of small numbers as the leading digits of numbers in many sets of integers. We prove a variant of Benford's law for many positive-density subsets of the primes. This follows from a more general result over number fields.
Let $S$ be a string of $l$ decimal digits. We give an explicit upper bound on some prime $p$ whose decimal representation contains the string $S$. We also show, as a corollary of the Green-Tao theorem, that there are arbitrarily long arithmetic progressions of prime numbers all of whose decimal representations contain $S$.
Let $b \ge 2$ be an integer. Not much is known on the representation in base $b$ of prime numbers or of numbers whose prime factors belong to a given, finite set. Among other results, we establish that any sufficiently large integer which is not a multiple of $b$ and has only small (in a suitable sense) prime factors has at least four nonzero digits in its representation in base $b$.
The index of a certain quotient of the Hecke algebra in its normalization
本文研究了Hecke代数在正规化下的商的指数，通过分析数域中素因子与特征形式共轭的关系，揭示了指数与素数间的关系。
We obtain the expected asymptotic formula for the number of primes $p<N=2^n$ with $r$ prescribed (arbitrarly placed) binary digits, provided $r<cn$ for a suitable constant $c>0$. This result improves on our earlier result where $r$ was assumed to satisfy $r<c(\frac n{\log n})^{4/7}$.
Sárközy's theorem for shifted primes with restricted digits
该研究针对带受限数位的移位素数，通过构造关联指数和的局部逼近函数证明其具有van der Corput性质，进而得出正上Banach密度的自然数集A中存在满足a₁+p-1=a₂的相关元素及素数p的结论。
It is known that all modular forms on $SL_2(Z)$ can be expressed as a rational function in $η(z)$, $η(2z)$ and $η(4z)$. By using a theorem by Gordon, Hughes, and Newman, and calculating the order of vanishing, we can compute the $η$-quotients for a given level. Using this count, knowing how many $η$-quotients are linearly independent and using the dimension formula, we can figure out how the $η$-quotients span higher levels. In this paper, we primarily focus on the case where $N=p$ a prime, and some discussion for non-prime indicies.
Let $q$ be a sufficiently large integer, and $a_0\in\{0,\dots,q-1\}$. We show there are infinitely many prime numbers which do not have the digit $a_0$ in their base $q$ expansion. Similar results are obtained for values of a polynomial (satisfying the necessary local conditions) and if multiple digits are excluded. Our proof is based on the Hardy-Littlewood circle method and Fourier analysis of the set of integers with no digit equal to $a_0$ in base $q$.
Let $$γ^*=\frac{8}{9}+\frac{2}{3}\:\frac{\log(10/9)}{\log 10}\:(\approx 0.919\ldots)\:.$$ Let $γ^*<γ_0\leq 1$, $c_0=1/γ_0$ be fixed. Let also $a_0\in\{0,1,\ldots, 9\}$.\\ We prove on assumption of the Generalized Riemann Hypothesis that each sufficiently large odd integer $N_0$ can be represented in the form $$N_0=p_1+p_2+p_3\:,$$ where the $p_i$ are of the form $p_i=[n_i^{c_0}]$, $n_i\in\mathbb{N}$, for $i=1,2$ and the decimal expansion of $p_3$ does not contain the digit $a_0$.\\ The proof merges methods of J. Maynard from his paper on the infinitude of primes with restricted digits, results…
Atkin--Lehner Quotients of Modular Curves at Primes of Bad Reduction
本文构造了模曲线 $X_0(N)$ 的 Atkin--Lehner 商在坏约化素数处的半稳定模型，并给出特殊纤维的显式描述与亏格公式。
The double sum sum_(s >= 1) sum_p 1/(p^s log p^s) = 2.00666645... over the inverse of the product of prime powers p^s and their logarithms, is computed to 24 decimal digits. The sum covers all primes p and all integer exponents s>=1. The calculational strategy is adopted from Cohen's work which basically looks at the fraction as the underivative of the Prime Zeta Function, and then evaluates the integral by numerical methods.
We present a new result on counting primes $p<N=2^n$ for which $r$ (arbitrarily placed) digits in the binary expansion of $p$ are specified. Compared with earlier work of Harman and Katai, the restriction on $r$ is relaxed to $r< c\Big(\frac n{\log n}\Big)^{4/7}$. This condition results from the estimates of Gallagher and Iwaniec on zero-free regions of $L$-functions with `powerful' conductor.
Digital functions along Squares of Prime Numbers
本文将Gelfond猜想中沿素数与沿平方的分布问题相结合，证明沿平方的q-可乘函数素数定理，并由此得到沿素数平方的良好分布结果。
Motivated by Euler's observation that the polynomial $x^{2} + x + 41$ takes on prime values for $0 \leq x \leq 39$, we search for large values of $x$ for which $N = x^{2} + x + 41$ is prime. To apply classical primality proving results based on the factorization of $N-1$, we choose $x$ to have the form $g(y)$, chosen so that $g(y)^{2} + g(y) + 40$ is reducible. Our main result is an explicit, 60,000 digit prime number of the form $x^{2} + x + 41$.
Let $a_0\in\{0,\dots,9\}$. We show there are infinitely many prime numbers which do not have the digit $a_0$ in their decimal expansion. The proof is an application of the Hardy-Littlewood circle method to a binary problem, and rests on obtaining suitable `Type I' and `Type II' arithmetic information for use in Harman's sieve to control the minor arcs. This is obtained by decorrelating Diophantine conditions which dictate when the Fourier transform of the primes is large from digital conditions which dictate when the Fourier transform of numbers with restricted digits is large. These estimates…
We prove a generalization of a result of Bhargava regarding the average size $\mathrm{Cl}(K)[2]$ as $K$ varies among cubic fields. For a fixed set of rational primes $S$, we obtain a formula for the average size of $\mathrm{Cl}(K)/\langle S \rangle[2]$ as $K$ varies among cubic fields with a fixed signature, where $\langle S \rangle$ is the subgroup of $\mathrm{Cl}(K)$ generated by the classes of primes of $K$ above primes in $S$. As a consequence, we are able to calculate the average sizes of $K_{2n}(\mathcal{O}_K)[2]$ for $n > 0$ and for the relaxed Selmer group $\mathrm{Sel}_2^S(K)$ as $K$ …
Let $$γ^*:=\frac{8}{9}+\frac{2}{3}\:\frac{\log(10/9)}{\log 10}\:(\approx 0.919\ldots)\:,\ γ^*<\frac{1}{c_0}\leq 1\:.$$ Let $γ^*<γ_0\leq 1$, $c_0=1/γ_0$ be fixed. Let also $a_0\in\{0,1,\ldots, 9\}$. In [23] we proved on assumption of the Generalized Riemann Hypothesis (GRH), that each sufficiently large odd integer $N_0$ can be represented in the form $$N_0=p_1+p_2+p_3\:,$$ where for $i=2, 3$ the primes $p_i$ are Piatetski-Shapiro primes - primes of the form $p_i=[n_i^{c_0}]$, $n_i\in\mathbb{N}$ - whereas the decimal expansion of $p_1$ does not contain the digit $a_0$. In this paper we replace …
Prime-Interval Algebras
研究从正整数\(n\)出发，在无先验信息时构造多项式商环，通过模指数运算恢复\((n,2n]\)内素数，仅用前一个素数就能得下一个素数，还发展了环结构、扩展结果并给出实现。
Arithmetic progressions of integers that are relatively prime to their digital sums
本文研究了$b$-反尼文数等差数列的最大长度，探讨了其在不同基底下的性质和特征。
The Position-wise Prime Digit Distribution Theorem: A Formal Proof of Position-wise Digit Equidistribution in the Prime Numbers
该论文形式证明了素数的位置数字等分布定理，指出内部数字极限概率为1/10、首位数字为1/9，明确了两种机制并给出误差界。
Wilson's theorem modulo higher prime powers III: The cases modulo $p^6$ and $p^7$
本研究拓展前期工作，算出更高素数幂下的威尔逊商及对应阶乘同余，确定费马商幂和模性质，揭示相关p进系数规律并验证原始威尔逊同余的适配性。
We prove a prime geodesic theorem for compact quotients of affine buildings and apply it to get class number asymptotics for global fields of positive characteristic.
Andrews' $(k, i)$-singular overpartition function $\overline{C}_{k, i}(n)$ counts the number of overpartitions of $n$ in which no part is divisible by $k$ and only parts $\equiv \pm i\pmod{k}$ may be overlined. In recent times, divisibility of $\overline{C}_{3\ell, \ell}(n)$, $\overline{C}_{4\ell, \ell}(n)$ and $\overline{C}_{6\ell, \ell}(n)$ by $2$ and $3$ are studied for certain values of $\ell$. In this article, we study divisibility of $\overline{C}_{3\ell, \ell}(n)$, $\overline{C}_{4\ell, \ell}(n)$ and $\overline{C}_{6\ell, \ell}(n)$ by primes $p\geq 5$. For all positive integer $\ell$ an…
The problem of N-digit sets all permutations of which give primes is discussed. Such sets may include only digits 1, 3, 7 and 9, and none of 0, 2, 5, 4, 6, 8. Direct calculations show that such full-permutation digit sets occur at N = 1, 2, 3 and are absent in the 4 - 10 interval of N. On the other hand the formal full permutation at N = 19, 23, 317 and 1031 (as well at N = 2) cases is provided by repunits (integers all digits of which are 1). The existence/nonexistence of other (not repunits) full-permutation digit sets for arbitrary large N is an open question with probable negative answer. …
We show that, for any fixed $\varepsilon > 0$ and almost all primes $p$, the $g$-ary expansion of any fraction $m/p$ with $\gcd(m,p) = 1$ contains almost all $g$-ary strings of length $k < (5/24 - \varepsilon) \log_g p$. This complements a result of J. Bourgain, S. V. Konyagin, and I. E. Shparlinski that asserts that, for almost all primes, all $g$-ary strings of length $k < (41/504 -\varepsilon) \log_g p$ occur in the $g$-ary expansion of $m/p$.
Primes with a missing digit: distribution in arithmetic progressions and an application in sieve theory
本文利用圆法证明了缺失一位数字的素数的Bombieri-Vinogradov型定理，并结合半线性筛，给出了形如p=1+m^2+n^2的此类素数数量的正确数量级上下界。
Let s_q(n) denote the base q sum of digits function, which for n<x, is centered around (q-1)/2 log_q x. In Drmota, Mauduit and Rivat's 2009 paper, they look at sum of digits of prime numbers, and provide asymptotics for the size of the set {p<x, p prime : s_q(p)=alpha(q-1)log_q x} where alpha lies in a tight range around 1/2. In this paper, we examine the tails of this distribution, and provide the lower bound |{p < x, p prime : s_q(p)>alpha(q-1)log_q x}| >>x^{2(1-alpha)}e^{-c(log x)^{1/2+epsilon}} for 1/2<alpha<0.7375. To attain this lower bound, we note that the multinomial distribution is s…
Unramified Iwasawa module of $\mathbb{Z}_2$-extension of certain quadratic fields with a bounded quotient
针对判别式含三个奇素因子且 2 分裂的实二次域，本文证明其 $\mathbb{Z}_2$-扩张的非分歧 Iwasawa 模具有有界商，从而验证 Greenberg 猜想并细化模结构。
Distributions of left prime truncations
研究了整数和多项式左素截断的分布特性，分析了其在所有ℓ位数或多项式中的比例、方差及最大比例。
Recently Oliver and Soundararajan made conjectures based on computational enumerations about the frequency of occurrence of pairs of last digits for consecutive primes. By studying Eratosthenes sieve, we have identified discrete dynamic systems that exactly model the populations of gaps across stages of Eratosthenes sieve. Our models provide some insight into the observed biases in the occurrences of last digits in consecutive primes, and the models suggest that the biases will ultimately be reversed for large enough primes. The exact model for populations of gaps across stages of Eratosthenes…
Vinogradov's theorem for primes with restricted digits
本文在基 g 展开中不含某固定数字的整数集合上，证明了每个充分大的奇数都能表示为三个此类素数之和，从而将维诺格拉多夫三素数定理推广到受限数字的素数情形。
An Explicit Surjectivity Threshold for Digit Sums of Primes
基于圆法框架，给出素数数字和满射性的显式数值阈值，并应用于证明OEIS A070027的无穷性。
Missing digits and sums of two prime squares
本文研究了在特定基数下省略固定数字且可表示为两个素数平方之和的整数，利用圆方法和筛法分析了其计数行为并给出了非平凡下界。
By a sphere-packing argument, we show that there are infinitely many pairs of primes that are close to each other for some metrics on the integers. In particular, for any numeration basis $q$, we show that there are infinitely many pairs of primes the base $q$ expansion of which differ in at most two digits. Likewise, for any fixed integer $t,$ there are infinitely many pairs of primes, the first $t$ digits of which are the same. In another direction, we show that, there is a constant $c$ depending on $q$ such that for infinitely many integers $m$ there are at least $c\log \log m$ primes which…
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