Representing an element in F_q[t] as the sum of two irreducibles

Representing multiples of $m$ in real quadratic fields as sums of squares

We study real quadratic fields $\mathbb{Q}(\sqrt{D})$ such that, for a given rational integer $m$, all $m$-multiples of totally positive integers are sums of squares. We prove quite sharp necessary and sufficient conditions for this to happen. Further, we give a fast algorithm that solves this question for specific $m$, $D$ and we give complete results for $m \leq 5000$.

Explicit upper bound for the average number of divisors of irreducible quadratic polynomials

Consider the divisor sum $\sum_{n\leq N}τ(n^2+2bn+c)$ for integers $b$ and $c$. We extract an asymptotic formula for the average divisor sum in a convenient form, and provide an explicit upper bound for this sum with the correct main term. As an application we give an improvement of the maximal possible number of $D(-1)$-quadruples.

Irreducible polynomials with prescribed sums of coefficients

Let $q$ be a power of a prime, let $\mathbb{F}_q$ be the finite field with $q$ elements and let $n \geq 2$. For a polynomial $h(x) \in \mathbb{F}_q[x]$ of degree $n \in \mathbb{N}$ and a subset $W \subseteq [0,n] := \{0, 1, \ldots, n\}$, we define the sum-of-digits function $$S_W(h) = \sum_{w \in W}[x^{w}] h(x)$$ to be the sum of all the coefficients of $x^w$ in $h(x)$ with $w \in W$. In the case when $q = 2$, we prove, except for a few genuine exceptions, that for any $c \in \mathbb{F}_2$ and any $W \subseteq [0,n]$ there exists an irreducible polynomial $P(x)$ of degree $n$ over $\mathbb{F}_…

Some sums over irreducible polynomials

We prove a number of conjectures due to Dinesh Thakur concerning sums of the form $\sum_P h(P)$ where the sum is over monic irreducible polynomials $P$ in $\mathbb{F}_q[T]$, the function $h$ is a rational function and the sum is considered in the $T^{-1}$-adic topology. As an example of our results, in $\mathbb{F}_2[T]$, the sum $\sum_P \tfrac{1}{P^k - 1}$ always converges to a rational function, and is $0$ for $k=1$.

将整数表示为素数混合次幂之和

Representing integers as sums of mixed powers of primes

该研究证明了每个足够大的奇整数可以表示为素数的混合次幂之和，其中包含平方、立方和更高次幂的组合。

Representing an integer as the sum of a prime and the product of two small factors

Let c > 0.55. Every large n can be written in the form p +ab, where p is prime, a and b are significantly smaller than x^1/2 and ab is less than n^c. This strengthens a result of Heath-Brown, which has the requirement c>3/4. We introduce the idea of 'intersecting two sieves' as a tool in the proof of this result.

Character sums over products of prime polynomials

We study sums of Dirichlet characters over polynomials in $\mathbb{F}_q[t]$ with a prescribed number of irreducible factors. Our main results are explicit formulae for these sums in terms of zeros of Dirichlet L-functions. We also exhibit new phenomena concerning Chebyshev-type biases of such sums when the number of irreducible factors is very large.

Representing Integers as the Sum of Two Squares in the Ring $\Z_n$

A classical theorem in number theory due to Euler states that a positive integer $z$ can be written as the sum of two squares if and only if all prime factors $q$ of $z$, with $q\equiv 3 \pmod{4}$, have even exponent in the prime factorization of $z$. One can consider a minor variation of this theorem by not allowing the use of zero as a summand in the representation of $z$ as the sum of two squares. Viewing each of these questions in $\Z_n$, the ring of integers modulo $n$, we give a characterization of all integers $n\ge 2$ such that every $z\in \Z_n$ can be written as the sum of two squares…

The sum of irreducible fractions with consecutive denominators is never an integer in a very weak arithmetic

Two theorems of elmentary arithmetic, one stating that the sum of the reciprocals of any number of consecutive positive integers is never an integer, and a generalization thereof by Trygve Nagell, are shown to be provable inside a very weak arithmetic, Richard Kaye's $PA^-$, in which there is no induction axiom whatsoever.

Irreducible polynomials with prescribed trace and restricted norm

Let GF(q), q=p^r, be a finite field with a primitive element g. In this paper we use exponential sums and Jacobi sums to compute the number of the irreducible polynomials of degree m over GF(q) with trace fixed and norm restricted to a coset of a subgroup <g^s>, s|(q-1). We give the number explicitly for s=2, 3, 4 when q=p, and for s|(p^e+1) when r=2en. Finally, we give explicit formulae for the number when both trace and norm are fixed, p=2 and m =< 30.

An effective analytic formula for the number of distinct irreducible factors of a polynomial

We obtain an effective analytic formula, with explicit constants, for the number of distinct irreducible factors of a polynomial $f \in \mathbb{Z}[x]$. We use an explicit version of Mertens' theorem for number fields to estimate a related sum over rational primes. For a given $f \in \mathbb{Z}[x]$, our result yields a finite list of primes that certifies the number of distinct irreducible factors of $f$.

Kloosterman sums, elliptic curves, and irreducible polynomials with prescribed trace and norm

Let $\F_q$ ($q=p^r$) be a finite field. In this paper the number of irreducible polynomials of degree $m$ in $\F_q[x]$ with prescribed trace and norm coefficients is calculated in certain special cases and a general bound for that number is obtained improving the bound by Wan if $m$ is small compared to $q$. As a corollary, sharp bounds are obtained for the number of elements in $\F_{q^3}$ with prescribed trace and norm over $\F_q$ improving the estimates by Katz in this special case. Moreover, a characterization of Kloosterman sums over $\F_{2^r}$ divisible by three is given generalizing the …

Visibly irreducible polynomials over finite fields

H. Lenstra has pointed out that a cubic polynomial of the form (x-a)(x-b)(x-c) + r(x-d)(x-e), where {a,b,c,d,e} is some permutation of {0,1,2,3,4}, is irreducible modulo 5 because every possible linear factor divides one summand but not the other. We classify polynomials over finite fields that admit an irreducibility proof with this structure.

关于 $\so(5)$ 的不可约表示数目

On the number of irreducible representations of $\so(5)$

本文研究李代数 $\mathfrak{so}(5)$ 的不可约表示计数问题，通过其维数公式 $jk(j+k)(j+2k)/6$ 建立计数函数，并给出求和函数 $\sum_{1\leq n \leq x}\varrho_{\mathfrak{so}(5)}(n)$ 的渐近公式。

不可约二次多项式上Hecke特征值和的抵消现象与质量等分布

Cancellation in Sums of Hecke Eigenvalues Over Quadratic Polynomials and Mass Equidistribution

研究短区间不可约二次多项式上Hecke特征值和的抵消现象，通过Hecke形式基平均得到平方根抵消，结合Kloosterman和界及Nelson工作，证明几乎所有模形式满足质量等分布猜想且收敛速度为k^(-δ)（δ<1/2）。

On divisors of sums of polynomials

Let $\mathcal{A}$ and $\mathcal{B}$ be sets of polynomials of degree $n$ over a finite field. We show, that if $\mathcal{A}$ and $\mathcal{B}$ are large enough, then $A+B$ has an irreducible divisor of large degree for some $A\in\mathcal{A}$ and $B\in \mathcal{B}$.

On Quadratic Fields Generated by Discriminants of Irreducible Trinomials

A. Mukhopadhyay, M. R. Murty and K. Srinivas (http://arxiv.org/abs/0808.0418) have recently studied various arithmetic properties of the discriminant $Δ_n(a,b)$ of the trinomial $f_{n,a,b}(t) = t^n + at + b$, where $n \ge 5$ is a fixed integer. In particular, it is shown that, under the $abc$-conjecture, for every $n \equiv 1 \pmod 4$, the quadratic fields $\Q(\sqrt{Δ_n(a,b)})$ are pairwise distinct for a positive proportion of such discriminants with integers $a$ and $b$ such that $f_{n,a,b}$ is irreducible over $\Q$ and $|Δ_n(a,b)|\le X$, as $X\to \infty$. We use the square-sieve and bounds …

Partial Franel和

Partial Franel sums

该研究针对特定N推导了Farey序列中不可约分数的位置解析表达式与更优误差下界的渐近行为，定义了部分Franel和并得出其增长阶，还探索了其他部分Franel和，为黎曼假设相关研究提供了新的数论结果。

Plane curves in boxes and equal sums of two powers

Given an absolutely irreducible ternary form $F$, the purpose of this paper is to produce better upper bounds for the number of integer solutions to the equation F=0, that are restricted to lie in very lopsided boxes. As an application of the main result, a new paucity estimate is obtained for equal sums of two like powers.

Archimedean Asai 和外平方 L-因子的有限和实现

Finite-Sum Realization of Archimedean Asai and Exterior-Square $L$-Factors

该研究在数论领域，将 Archimedean Asai L-因子与外平方 L-因子分别表示为特定局部 ζ 积分的有限和，完善了 L-因子的有限和实现相关理论。

Non-unique factorization of polynomials over residue class rings of the integers

We investigate non-unique factorization of polynomials in Z_{p^n}[x] into irreducibles. As a Noetherian ring whose zero-divisors are contained in the Jacobson radical, Z_{p^n}[x] is atomic. We reduce the question of factoring arbitrary non-zero polynomials into irreducibles to the problem of factoring monic polynomials into monic irreducibles. The multiplicative monoid of monic polynomials of Z_{p^n}[x] is a direct sum of monoids corresponding to irreducible polynomials in Z_p[x], and we show that each of these monoids has infinite elasticity. Moreover, for every positive integer m, there exis…