A comparison of the Carlitz and digit derivatives bases in function field arithmetic

Arithmetic of characteristic p special L-values (with an appendix by V. Bosser)

Recently the second author has associated a finite $\F_q[T]$-module $H$ to the Carlitz module over a finite extension of $\F_q(T)$. This module is an analogue of the ideal class group of a number field. In this paper we study the Galois module structure of this module $H$ for `cyclotomic' extensions of $\F_q(T)$. We obtain function field analogues of some classical results on cyclotomic number fields, such as the $p$-adic class number formula, and a theorem of Mazur and Wiles about the Fitting ideal of ideal class groups. We also relate the Galois module $H$ to Anderson's module of circular un…

Fermat-Wilson Supercongruences, arithmetic derivatives and strange factorizations

In [Tha15], we looked at two (`multiplicative' and `Carlitz-Drinfeld additive') analogs each, for the well-known basic congruences of Fermat and Wilson, in the case of polynomials over finite fields. When we look at them modulo higher powers of primes, i.e. at `supercongruences', we find interesting relations linking them together, as well as linking them with arithmetic derivatives and zeta values. In the current work, we expand on the first analog and connections with arithmetic derivatives more systematically, giving many more equivalent conditions linking the two, now using `mixed derivati…

一阶Drinfeld模在有理函数域上的安德森生成函数

Anderson generating function of rank-one Drinfeld Module over rational function fields

本文研究了一阶Drinfeld模在有理函数域上的安德森生成函数，通过构造生成函数并将其与Carlitz周期关联，解决了无限地方度数大于1的理论空白，提出新的方法论进展。

The sum-of-digits function on arithmetic progressions

Let $s_2$ be the sum-of-digits function in base $2$, which returns the number of non-zero binary digits of a nonnegative integer $n$. We study $s_2$ alon g arithmetic subsequences and show that --- up to a shift --- the set of $m$-tuples of integers that appear as an arithmetic subsequence of $s_2$ has full complexity.

Quasi-Holonomic Modules in Positive Characteristic

We study modules over the Carlitz ring, a counterpart of the Weyl algebra in analysis over local fields of positive characteristic. It is shown that some basic objects of function field arithmetic, like the Carlitz module, Thakur's hypergeometric polynomials, and analogs of binomial coefficients arising in the function field version of umbral calculus, generate quasi-holonomic modules. This class of modules is, in many respects, similar to the class of holonomic modules in the characteristic zero theory.

Evolution Equations and Functions of Hypergeometric Type over Fields of Positive Characteristic

We consider a class of partial differential equations with Carlitz derivatives over a local field of positive characteristic, for which an analog of the Cauchy problem is well-posed. Equations of such type correspond to quasi-holonomic modules over the ring of differential operators with Carlitz derivatives. The above class of equations includes some equations of hypergeometric type. Building on the work of Thakur, we develop his notion of the hypergeometric function of the first kind (whose parameters belonged initially to $\mathbb Z$) in such a way that it becomes fully an object of the func…

Multiplicative relations between coefficients of logarithmic derivatives of $\mathbb{F}_q$-linear functions and applications

We prove some interesting multiplicative relations which hold between the coefficients of the logarithmic derivatives obtained in a few simple ways from $\mathbb{F}_q$-linear formal power series. Since the logarithmic derivatives connect power sums to elementary symmetric functions via the Newton identities, we establish, as applications, new identities between important quantities of function field arithmetic, such as the Bernoulli-Carlitz fractions and power sums as well as their multi-variable generalizations. Resulting understanding of their factorizations has arithmetic significance, as w…

缺失一位数字的素数：算术级数中的分布及在筛法理论中的应用

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。

Arithmetics in number systems with cubic base

This paper focuses on greedy expansions, one possible representation of numbers, and on arithmetical operations with them. Performing addition or multiplication some additional digits can appear. We study bounds on the number of such digits assuming the finiteness of the expansion of the considered sum or product, especially for the case of cubic Pisot units.

On the Polynomial Ramanujan Sums over Finite Fields

The polynomial Ramanujan sum was first introduced by Carlitz [7], and a generalized version by Cohen [10]. In this paper, we study the arithmetical and analytic properties of these sums, derive various fundamental identities, such as H older formula, reciprocity formula, orthogonality relation and Davenport{Hasse type formula. In particular, we show that the special Dirichlet series involving the polynomial Ramanujan sums are, indeed, the entire functions on the whole complex plane, we also give a square mean values estimation. The main results of this paper are new appearance to us, which ind…

数字和函数在等差数列中的分布水平

The level of distribution of the sum-of-digits function in arithmetic progressions

本文将 Thue--Morse 序列分布水平为 1 的结论推广至基于数字和函数的复指数序列，并在上界中给出明确的指数。

Carryless Arithmetic Mod 10

We investigate what arithmetic would look like if carry digits into other digit position were ignored, so that 9 + 4 = 3, 5 + 5 = 0, 9 X 4 = 6, 5 X 4 = 0, and so on. For example, the primes are now 21, 23, 25, 27, 29, 41, 43, 45, 47, ... .

Arithmetic Progressions with Restricted Digits

For an integer $b \geqslant 2$ and a set $S\subset \{0,\cdots,b-1\}$, we define the Kempner set $\mathcal{K}(S,b)$ to be the set of all non-negative integers whose base-$b$ digital expansions contain only digits from $S$. These well-studied sparse sets provide a rich setting for additive number theory, and in this paper we study various questions relating to the appearance of arithmetic progressions in these sets. In particular, for all $b$ we determine exactly the maximal length of an arithmetic progression that omits a base-$b$ digit.

A local Benford Law for a class of arithmetic sequences

It is well-known that sequences such as the Fibonacci numbers and the factorials satisfy Benford's Law, that is, leading digits in these sequences occur with frequencies given by $P(d)=\log_{10}(1+1/d)$, $d=1,2,\dots,9$. In this paper, we investigate leading digit distributions of arithmetic sequences from a local point of view. We call a sequence locally Benford distributed of order $k$ if, roughly speaking, $k$-tuples of consecutive leading digits behave like $k$ independent Benford-distributed digits. This notion refines that of a Benford distributed sequence, and it provides a way to quant…

Large deviation principle for harmonic/geometric/arithmetic mean of digits in backward continued fraction expansion

We establish the (level-1) large deviation principles for three kinds of means associated with the backward continued fraction expansion. We show that: for the harmonic and geometric means, the rate functions vanish exactly at one point; for the arithmetic mean, it is completely degenerate, vanishing at every point in its effective domain. Our method of proof employs the thermodynamic formalism for finite Markov shifts, and a multifractal analysis for the Rényi map generating the backward continued fraction digits. We completely determine the class of unbounded arithmetic functions for which t…

Arithmetic Properties of Integers in Chains and Reflections of $g$-ary Expansions

Recently, there has been a sharp rise of interest in properties of digits primes. Here we study yet another question of this kind. Namely, we fix an integer base $g \ge 2$ and then for every infinite sequence $${\mathcal D} = \{d_i\}_{i=0}^\infty \in \{0, \ldots, g-1\}^\infty $$ of $g$-ary digits we consider the counting function $\varpi_{{\mathcal D},g}(N)$ of integers $n \le N$ for which $\sum_{i=0}^{n-1} d_i g^i$ is prime. We construct sequences ${\mathcal D}$ for which $\varpi_{{\mathcal D},g}(N)$ grows fast enough, and show that for some constant $\vartheta_g< g$ there are at most $O(\var…

Some normal numbers generated by arithmetic functions

Let $g \geq 2$. A real number is said to be g-normal if its base g expansion contains every finite sequence of digits with the expected limiting frequency. Let ϕdenote Euler's totient function, let σbe the sum-of-divisors function, and let λbe Carmichael's lambda-function. We show that if f is any function formed by composing ϕ, σ, or λ, then the number \[ 0. f(1) f(2) f(3) \dots \] obtained by concatenating the base g digits of successive f-values is g-normal. We also prove the same result if the inputs 1, 2, 3, \dots are replaced with the primes 2, 3, 5, \dots. The proof is an adaptation of …

On the constant in the Mertens product for arithmetic progressions. II. Numerical values

We give explicit numerical values with 100 decimal digits for the constant in the Mertens product over primes in the arithmetic progressions $a \bmod q$, for $q \in \{3$, ..., $100\}$ and $(a, q) = 1$.

The sum of digits of polynomial values in arithmetic progressions

Let $q, m\geq 2$ be integers with $(m,q-1)=1$. Denote by $s_q(n)$ the sum of digits of $n$ in the $q$-ary digital expansion. Further let $p(x)\in mathbb{Z}[x]$ be a polynomial of degree $h\geq 3$ with $p(\mathbb{N})\subset \mathbb{N}$. We show that there exist $C=C(q,m,p)>0$ and $N_0=N_0(q,m,p)\geq 1$, such that for all $g\in\mathbb{Z}$ and all $N\geq N_0$, $$#\{0\leq n< N: \quad s_q(p(n))\equiv g \bmod m\}\geq C N^{4/(3h+1)}.$$ This is an improvement over the general lower bound given by Dartyge and Tenenbaum (2006), which is $C N^{2/h!}$.

指数序列的自引用首位数：算术结构与认证搜索

Self-Referential Leading Digits of Exponential Sequences: Arithmetic Structure and Certified Search

研究指数序列自引用首位数，推导收缩目标准则等，对\(c\geq2\)得出相关结论，给出特定情况下计数公式等，还给出插值连分数定位器复杂度及\((2,10)\)的认证实例，其无穷性待解。

Dismal Arithmetic

Dismal arithmetic is just like the arithmetic you learned in school, only simpler: there are no carries, when you add digits you just take the largest, and when you multiply digits you take the smallest. This paper studies basic number theory in this world, including analogues of the primes, number of divisors, sum of divisors, and the partition function.

Carries and the arithmetic progression structure of sets

If we want to represent integers in base $m$, we need a set $A$ of digits, which needs to be a complete set of residues modulo $m$. When adding two integers with last digits $a_1, a_2 \in A$, we find the unique $a \in A$ such that $a_1 + a_2 \equiv a$ mod $m$, and call $(a_1 + a_2 -a)/m$ the carry. Carries occur also when addition is done modulo $m^2$, with $A$ chosen as a set of coset representatives for the cyclic group $\mathbb{Z}/m \mathbb{Z} \subseteq \mathbb{Z}/m^2\mathbb{Z}$. It is a natural to look for sets $A$ which minimize the number of different carries. In a recent paper, Diaconis…

The Digit Principle

A number of constructions in function field arithmetic involve extensions from linear objects using digit expansions. This technique is described here as a method of constructing orthonormal bases in spaces of continuous functions. We illustrate several examples of orthonormal bases from this viewpoint, and we also obtain a concrete model for the continuous functions on the integers of a local field as a quotient of a Tate algebra in countably many variables.

Arithmetic functions at consecutive shifted primes

For each of the functions $f \in \{ϕ, σ, ω, τ\}$ and every natural number $k$, we show that there are infinitely many solutions to the inequalities $f(p_n-1) < f(p_{n+1}-1) < \dots < f(p_{n+k}-1)$, and similarly for $f(p_n-1) > f(p_{n+1}-1) > \dots > f(p_{n+k}-1)$. We also answer some questions of Sierpiński on the digit sums of consecutive primes. The arguments make essential use of Maynard and Tao's method for producing many primes in intervals of bounded length.

关于Erdős-Borwein常数的二进制数字

On the binary digits of the Erdős-Borwein constant

通过结合Erdős同余构造和Alford-Granville-Pomerance素数计数估计，证明了Erdős-Borwein常数E的二进制展开中字符串11无限次出现。

Periods of Drinfeld modules and local shtukas with complex multiplication

Colmez conjectured a product formula for periods of abelian varieties over number fields with complex multiplication and proved it in some cases. His conjecture is equivalent to a formula for the Faltings height of CM abelian varieties in terms of the logarithmic derivatives at $s=0$ of certain Artin $L$-functions. In a series of articles we investigate the analog of Colmez's theory in the arithmetic of function fields. There abelian varieties are replaced by Drinfeld modules and their higher dimensional generalizations, so-called $A$-motives. In the present article we prove the product formul…

Memory Efficient Arithmetic

In this paper we give an algorithm for computing the mth base-b digit (m=1 is the least significant digit) of an integer n (actually, it finds sharp approximations to n/b^m mod 1), where n is defined as the last number in a sequence of integers s1,s2,...,sL=n, where s1=0, s2=1, and each successive si is either the sum, product, or difference of two previous sj's in the sequence. In many cases, the algorithm will find this mth digit using far less memory than it takes to write down all the base-b digits of n, while the number of bit operations will grow only slighly worse than linear in the num…

Large Deviation Principle for arithmetic functions in continued fraction expansion

Khinchin proved that the arithmetic mean of continued fraction digits of Lebesgue almost every irrational number in $(0,1)$ diverges to infinity. Hence, none of the classical limit theorems such as the weak and strong laws of large numbers or central limit theorems hold. Nevertheless, we prove the existence of a large deviations rate function which estimates exponential probabilities with which the arithmetic mean of digits stays away from infinity.

The arithmetic-geometric scaling spectrum for continued fractions

To compare continued fraction digits with the denominators of the corresponding approximants we introduce the arithmetic-geometric scaling. We will completely determine its multifractal spectrum by means of a number theoretical free energy function and show that the Hausdorff dimension of sets consisting of irrationals with the same scaling exponent coincides with the Legendre transform of this free energy function. Furthermore, we identify the asymptotic of the local behaviour of the spectrum at the right boundary point and discuss a connection to the set of irrationals with continued fractio…

Digit sums and generating functions

We connect a primitive operation from arithmetic -- summing the digits of a base-$B$ integer -- to $q$-series and product generating functions analogous to those in partition theory. We find digit sum generating functions to be intertwined with distinctive classes of "$B$-ary" Lambert series, which themselves enjoy nice transformations. We also consider digit sum Dirichlet series.

Use of difference tables at studying of properties of binomial coefficients

Some properties of diagonal binomial coefficients were studied in respect to frequency of their units digits. An approach was formulated that led to use of difference tables to predict if certain units digits can appear in the values of binomial coefficients at quadratic terms of the binomial theorem. Frequency distributions of units digits of binomial coefficients contain gaps (zero frequency) under most common numbering systems with supposed exclusion to systems with 2^n bases. Binomial coefficient arithmetics may be used to model cell population dynamics in a multicellular organism, which i…

短算术项表示有限域上魏尔斯特拉斯标准形式椭圆曲线的基数

Short arithmetic terms for the cardinality of elliptic curves over rings of remainder classes and over finite fields

该研究改进了椭圆曲线计数的算术项构造方法，降低了中间整数位数，对素模p≥17时大幅减少了运算次数，提升了有限域上魏尔斯特拉斯标准形式椭圆曲线基数的计算效率。

广义数字映射：周期性、Prouhet-Tarry-Escott 解与求和恒等式

A Generalized Digit Map: Periodicity, Prouhet-Tarry-Escott Solutions, and Summation Identities

本文研究基-M到基-N移位映射所得序列b(n)的算术性质，证明其最终周期性条件并界定最小周期，在条件不满足时为Prouhet-Tarry-Escott问题提供新解，并引入有限差分恒等式扩展经典数字和恒等式。

任意基下的 Zsiflaw--Legeis 定理

The Zsiflaw--Legeis theorem for arbitrary bases

本文推广了素数指数和的结果，证明了任意基下素数数字反转的 Dirichlet 定理与 Siegel--Walfisz 定理的类似结论。

Arithmetics in numeration systems with negative quadratic base

We consider positional numeration system with negative base $-β$, as introduced by Ito and Sadahiro. In particular, we focus on arithmetical properties of such systems when $β$ is a quadratic Pisot number. We study a class of roots $β>1$ of polynomials $x^2-mx-n$, $m\geq n\geq 1$, and show that in this case the set ${\rm Fin}(-β)$ of finite $(-β)$-expansions is closed under addition, although it is not closed under subtraction. A particular example is $β=τ=\frac12(1+\sqrt5)$, the golden ratio. For such $β$, we determine the exact bound on the number of fractional digits appearing in arithmetic…

Continued fraction digit averages an Maclaurin's inequalities

A classical result of Khinchin says that for almost all real numbers $α$, the geometric mean of the first $n$ digits $a_i(α)$ in the continued fraction expansion of $α$ converges to a number $K = 2.6854520\ldots$ (Khinchin's constant) as $n \to \infty$. On the other hand, for almost all $α$, the arithmetic mean of the first $n$ continued fraction digits $a_i(α)$ approaches infinity as $n \to \infty$. There is a sequence of refinements of the AM-GM inequality, Maclaurin's inequalities, relating the $1/k$-th powers of the $k$-th elementary symmetric means of $n$ numbers for $1 \leq k \leq n$. On…

Arithmetics in number systems with negative base

We study the numeration system with negative basis, introduced by Ito and Sadahiro. We focus on arithmetic operations in the set ${\rm Fin}(-β)$ and $\Z_{-β}$ of numbers having finite resp. integer $(-β)$-expansions. We show that ${\rm Fin}(-β)$ is trivial if $β$ is smaller than the golden ratio $\frac12(1+\sqrt5)$. For $β\geq\frac12(1+\sqrt5)$ we prove that ${\rm Fin}(-β)$ is a ring, only if $β$ is a Pisot or Salem number with no negative conjugates. We prove the conjecture of Ito and Sadahiro that ${\rm Fin}(-β)$ is a ring if $β$ is a quadratic Pisot number with positive conjugate. For quadr…

受限数字差素数

Primes with Restricted-Digit Differences

本文研究差或公差属于受限数字集的素数对及三项算术级数，提出局部傅里叶准则并借助传递算子论证，在显式参数范围内验证了渐近公式的有效性。

Multi-Base Representations of Integers: Asymptotic Enumeration and Central Limit Theorems

In a multi-base representation of an integer (in contrast to, for example, the binary or decimal representation) the base (or radix) is replaced by products of powers of single bases. The resulting numeral system has desirable properties for fast arithmetic. It is usually redundant, which means that each integer can have multiple different digit expansions, so the natural question for the number of representations arises. In this paper, we provide a general asymptotic formula for the number of such multi-base representations of a positive integer $n$. Moreover, we prove central limit theorems …

Polynomial Expressions of Carries in p-ary Arithmetics

It is known that any $n$-variable function on a finite prime field of characteristic $p$ can be expressed as a polynomial over the same field with at most $p^n$ monomials. However, it is not obvious to determine the polynomial for a given concrete function. In this paper, we study the concrete polynomial expressions of the carries in addition and multiplication of $p$-ary integers. For the case of addition, our result gives a new family of symmetric polynomials, which generalizes the known result for the binary case $p = 2$ where the carries are given by elementary symmetric polynomials. On th…