We examine two counting problems that seem very group-theoretic on the surface but, on closer examination, turn out to concern integers with restrictions on their prime factors.
First, given an odd prime $q$ and a finite abelian $q$-group $H$, we consider the set of integers $n\le x$ such that the Sylow $q$-subgroup of the multiplicative group $(\mathbb Z/n\mathbb Z)^\times$ is isomorphic to $H$. We show that the counting function of this set of integers is asymptotic to $K x(\log\log x)^\ell/(\log x)^{1/(q-1)}$ for explicit constants $K$ and $\ell$ depending on $q$ and $H$.
Second, we consider the set of integers $n\le x$ such that the multiplicative group $(\mathbb Z/n\mathbb Z)^\times$ is "maximally non-cyclic", that is, such that all of its prime-power subgroups are elementary groups. We show that the counting function of this set of integers is asymptotic to $A x/(\log x)^{1-ξ}$ for an explicit constant $A$, where $ξ$ is Artin's constant.
As it turns out, both of these group-theoretic problems can be reduced to problems of counting integers with restrictions on their prime factors, allowing them to be addressed by classical techniques of analytic number theory.
We examine how closely a multiplicative function resembles an additive function. Given a multiplicative function $g$ and an additive function $f$, we examine the size of the quantity $E(f,g;x)=\# \{n\leq x:f(n)=g(n)\}$. We establish a lower bound for $E(Ω,g,x)$.
A famous conjecture of Littlewood (c. 1930) concerns approximating two real numbers by rationals of the same denominator, multiplying the errors. In a lesser-known paper, Wang and Yu (1981) established an asymptotic formula for the number of such approximations, valid almost always. Using the quantitative Koukoulopoulos--Maynard theorem of Aistleitner--Borda--Hauke, together with bounds arising from the theory of Bohr sets, we deduce lower bounds of the expected order of magnitude for inhomogeneous and fibre refinements of the problem.
The cardinality of the set of $D\leqslant x$ for which the fundamental solution of the Pell equation $t^2-Du^2=1$ is less than $D^{\frac{1}{2}+α}$ with $α\in[\frac{1}{2},1]$ is studied and certain lower bounds are obtained, improving previous results of Fouvry by introducing the $q$-analogue of van der Corput method to algebraic exponential sums with smooth moduli.
We continue our study on counting irreducible polynomials over a finite field with prescribed coefficients. We set up a general combinatorial framework using generating functions with coefficients from a group algebra which is generated by equivalent classes of polynomials with prescribed coefficients. Simplified expressions are derived for some special cases. Our results extend some earlier results.
A set S of integers is said to be multiplicative if for every pair m and n of coprime integers we have that mn is in S iff both m and n are in S. Both Landau and Ramanujan gave approximations to S(x), the number of n<=x that are in S, for specific choices of S. The asymptotical precision of their respective approaches are being compared and related to Euler-Kronecker constants, a generalization of Euler's constant gamma=0.57721566.... This paper claims little originality, its aim is to give a survey on the literature related to this theme with an emphasis on the contributions of the author (an…
Harron and Snowden counted the number of elliptic curves over $\mathbb{Q}$ up to height $X$ with torsion group $G$ for each possible torsion group $G$ over $\mathbb{Q}$. In this paper we generalize their result to all number fields and all level structures $G$ such that the corresponding modular curve $X_G$ is a weighted projective line $\mathbb{P}(w_0,w_1)$ and the morphism $X_G\to X(1)$ satisfies a certain condition. In particular, this includes all modular curves $X_1(m,n)$ with coarse moduli space of genus $0$. We prove our results by defining a size function on $\mathbb{P}(w_0,w_1)$ follo…
Let $λ_1(n)$ denote the least invariant factor in the invariant factor decomposition of the multiplicative group $M_n = (\mathbb Z/n\mathbb Z)^\times$. We give an asymptotic formula, with order of magnitude $x/\sqrt{\log x}$, for the counting function of those integers $n$ for which $λ_1(n)\ne2$. We also give an asymptotic formula, for any even $q\ge4$, for the counting function of those integers $n$ for which $λ_1(n)=q$. These results require a version of the Selberg-Delange method whose dependence on certain parameters is made explicit, which we provide in an appendix. As an application, we …
Mazur's theorem states that there are exactly 15 possibilities for the torsion subgroup of an elliptic curve over the rational numbers. We determine how often each of these groups actually occurs. Precisely, if $G$ is one of these 15 groups, we show that the number of elliptic curves up to height $X$ whose torsion subgroup is isomorphic to $G$ is on the order of $X^{1/d}$, for some number $d=d(G)$ which we compute.
Let $p\ge5$ be a prime and $T$ a Kodaira type of the special fiber of an elliptic curve. We estimate the number of elliptic curves over $\mathbb Q$ up to height $X$ with Kodaira type $T$ at $p$. This enables us find the proportion of elliptic curves over $\mathbb Q$, when ordered by height, with Kodaira type $T$ at a prime $p\ge5$ inside the set of all elliptic curves. This proportion is a rational function in $p$. For instance, we show that $\displaystyle\frac{p^8(p-1)}{p^9-1}$ of all elliptic curves with bad reduction at $p$ are of multiplicative reduction. Furthermore, we prove that the pri…
In this paper, we give some counting results on integer polynomials of fixed degree and bounded height whose distinct non-zero roots are multiplicatively dependent. These include sharp lower bounds, upper bounds and asymptotic formulas for various cases, although in general there is a logarithmic gap between lower and upper bounds.
Building on the concept of pretentious multiplicative functions, we give a new and largely elementary proof of the best result known on the counting function of primes in arithmetic progressions.
Following Wigert, various authors, including Ramanujan, Gronwall, Erdős, Ivić, Schwarz, Wirsing, and Shiu, determined the maximal order of several multiplicative functions, generalizing Wigert's result $$\max_{n\leq x} \log d(n) = \frac{\log x}{\log \log x} (\log 2 + o(1)). $$ On the contrary, for many multiplicative functions, the maximal order of iterations of the functions remains widely open. The case of the iterated divisor function was only solved recently, answering a question of Ramanujan from 1915. Here we determine the maximal order of $\log f(f(n))$ for a class of multiplicative fun…
Let F_q be the finite field of q elements. Let H be a multiplicative subgroup of F_q^*. For a positive integer k and element b\in F_q, we give a sharp estimate for the number of k-element subsets of H which sum to b.
Consider a strongly $b$-multiplicative sequence and a prime $p$. Studying its $p$-rarefaction consists in characterizing the asymptotic behaviour of the sums of the first terms indexed by the multiples of $p$. The integer values of the "norm" $3$-variate polynomial $\mathcal N_{p,i_1,i_2}(Y_0,Y_1,Y_2)\!:=\!\prod_{j=1}^{p-1}\left(Y_0{+}ζ_p^{i_1j}Y_1{+}ζ_p^{i_2j}Y_2\right),$ where $ζ_p$ is a primitive $p$-th root of unity, and $i_1,i_2{\in}\{1,2,\dots,p{-}1\},$ determine this asymptotic behaviour. It will be shown that a combinatorial method can be applied to $\mathcal N_{p,i_1,i_2}(Y_0,Y_1,Y_2)…
We consider the problem of counting the number of varieties in a family over $\mathbb{Q}$ with a rational point. We obtain lower bounds for this counting problem for some families over $\mathbb{P}^1$, even if the Hasse principle fails. We also obtain sharp results for some multinorm equations and for specialisations of certain Brauer group elements on higher dimensional projective spaces, where we answer some cases of a question of Serre. Our techniques come from arithmetic geometry and additive combinatorics.
For any prime p we consider the density of elements in the multiplicative group of the finite field F_p having order, respectively index, congruent to a(mod d). We compute these densities on average, where the average is taken over all finite fields of prime order. Some connections between the two densities are established. It is also shown how to compute these densities with high numerical accuracy.
In this paper we introduce and study a family $Φ_k$ of arithmetic functions generalizing Euler's totient function. These functions are given by the number of solutions to the equation $\gcd(x_1^2+\ldots +x_k^2, n)=1$ with $x_1,\ldots,x_k \in {\mathbb{Z}}/n{\mathbb{Z}}$ which, for $k=2,4$ and $8$ coincide, respectively, with the number of units in the rings of Gaussian integers, quaternions and octonions over ${\mathbb{Z}}/n{\mathbb{Z}}$. We prove that $Φ_k$ is multiplicative for every $k$, we obtain an explicit formula for $Φ_k(n)$ in terms of the prime-power decomposition of $n$ and derive an…
We give a precise estimate for the number of lattice points in certain bounded subsets of $\mathbb{R}^{n}$ that involve `hyperbolic spikes' and occur naturally in multiplicative Diophantine approximation. We use Wilkie's o-minimal structure $\mathbb{R}_{\exp}$ and expansions thereof to formulate our counting result in a general setting. We give two different applications of our counting result. The first one establishes nearly sharp upper bounds for sums of reciprocals of fractional parts, and thereby sheds light on a question raised by Lê and Vaaler, extending previous work of Widmer and of t…
For any positive integers $n\geq 3, r\geq 1$ we present formulae for the number of irreducible polynomials of degree $n$ over the finite field $\mathbb{F}_{2^r}$ where the coefficients of $x^{n-1}$, $x^{n-2}$ and $x^{n-3}$ are zero. Our proofs involve counting the number of points on certain algebraic curves over finite fields, a technique which arose from Fourier-analysing the known formulae for the $\mathbb{F}_2$ base field cases, reverse-engineering an economical new proof and then extending it. This approach gives rise to fibre products of supersingular curves and makes explicit why the fo…
We use Massey products and their relations to unipotent representations to parametrize and find an explicit formula for the number of Galois extensions of a given local field with the prescribed Galois group ${\mathbb U}_4({\mathbb F}_p)$ consisting of unipotent four by four matrices over ${\mathbb F}_p$. Further applications of this method involve the counting of certain Galois extensions with restricted ramifications, and counting the numbers of Galois ${\mathbb U}_4({\mathbb F}_p)$-extensions of some other fields. For each Demushkin pro-$p$-group, we find a very simple version of the condit…
In this paper, we present the problem of counting magic squares and we focus on the case of multiplicative magic squares of order 4. We give the exact number of normal multiplicative magic squares of order 4 with an original and complete proof, pointing out the role of the action of the symmetric group. Moreover, we provide a new representation for magic squares of order 4. Such representation allows the construction of magic squares in a very simple way, using essentially only five particular 4X4 matrices.
Define {\em the Liouville function for $A$}, a subset of the primes $P$, by $λ_{A}(n) =(-1)^{Ω_A(n)}$ where $Ω_A(n)$ is the number of prime factors of $n$ coming from $A$ counting multiplicity. For the traditional Liouville function, $A$ is the set of all primes. Denote $$L_A(n):=\sum_{k\leq n}λ_A(n)\quad{and}\quad R_A:=\lim_{n\to\infty}\frac{L_A(n)}{n}.$$ We show that for every $α\in[0,1]$ there is an $A\subset P$ such that $R_A=α$. Given certain restrictions on $A$, asymptotic estimates for $\sum_{k\leq n}λ_A(k)$ are also given. With further restrictions, more can be said. For {\em character…
Counting the number of prime numbers up to a certain natural number and describing the asymptotic behavior of such a counting function has been studied by famous mathematicians like Gauss, Legendre, Dirichlet, and Euler. The prime number theorem determines that such asymptotic behavior is similar to the asymptotic behavior of the number divided by its natural logarithm. In this paper, we take advantage of a multiplicative representation of a number and the properties of the logarithm function to express the prime number theorem in terms of primorial numbers, and n-primorial totative numbers. A…
In this paper we derive a formula for the number of $N$-free elements over a finite field $\mathbb{F}_q$ with prescribed trace, in particular trace zero, in terms of Gaussian periods. As a consequence, we derive a simple explicit formula for the number of primitive elements, in quartic extensions of Mersenne prime fields, having absolute trace zero. We also give a simple formula in the case when $Q = (q^m-1)/(q-1)$ is prime. More generally, for a positive integer $N$ whose prime factors divide $Q$ and satisfy the so called semi-primitive condition, we give an explicit formula for the number of…
We study Tamagawa numbers of elliptic curves with torsion $\mathbb{Z}/2\mathbb{Z}\oplus \mathbb{Z}/14\mathbb{Z}$ over cubic fields and of elliptic curves with an $n-$isogeny over $\mathbb{Q}$, for $n\in\{6,8,10,12,14,16,17,18,19,37,43,67,163\}$. Bruin and Najman proved that every elliptic curve with torsion $\mathbb{Z}/2\mathbb{Z}\oplus \mathbb{Z}/14\mathbb{Z}$ over a cubic field is a base change of an elliptic curve defined over $\mathbb{Q}$. We find that Tamagawa numbers of elliptic curves defined over $\mathbb{Q}$ with torsion $\mathbb{Z}/2\mathbb{Z}\oplus \mathbb{Z}/14\mathbb{Z}$ over a cu…
We prove prime geodesic theorems counting primitive closed geodesics on a compact hyperbolic 3-manifold with length and holonomy in prescribed intervals, which are allowed to shrink. Our results imply effective equidistribution of holonomy and have both the rate of shrinking and the strength of the error term fully symmetric in length and holonomy.
Random Multiplicative Functions and Making Squares from Polynomial Values
Université Paris Cité（巴黎西岱大学）；Sorbonne Université（索邦大学）；CNRS UMR 7586 Institut Universitaire de France（法国国家科学研究中心联合研究实验室7586 法国高等科学研究院）；Institut de Mathématiques de Jussieu-Paris Rive Gauche（儒勒·昂里希数学研究所（巴黎左岸））；Institute of Mathematics, Academia Sinica（中央研究院数学研究所）；Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）
We prove by using simple number-theoretic arguments formulae concerning the number of elements of a fixed order and the number of cyclic subgroups of a direct product of several finite cyclic groups. We point out that certain multiplicative properties of related counting functions for finite Abelian groups are immediate consequences of these formulae.
