Fernando Chamizo, Jorge Jiménez Urroz
We estimate the number of primes represented by a general quadratic polynomial with discriminant $Δ$, assuming that the corresponding real character is exceptional.
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The goal of this note is to provide an analysis of the positive integers that are represented everywhere locally, but not globally, by each of the 29 spinor regular positive definite integral ternary quadratic forms that are not regular.
We describe a method for bounding the set of exceptional integers not represented by a given additive form in terms of the exceptional set corresponding to a subform. Illustrating our ideas with examples stemming from Waring's problem for cubes, we show, in particular, that the number of positive integers not exceeding N, that fail to have a representation as the sum of six cubes of natural numbers, is O(N^{3/7}).
We give an account of the arguments that lead from the assumption of the existence of exceptional characters to the asymptotics in related ranges for the counting function of twin primes.
In this paper we continue our study, begun in part I, of the exceptional set of integers, not restricted by elementary congruence conditions, which cannot be represented as sums of three or four squares of primes. We correct a serious oversight in our first paper, but make further progress on the exponential sums estimates needed, together with an embellishment of the previous sieve technique employed. This leads to an improvement in our bounds for the maximal size of the exceptional sets.
Exceptional characters and prime numbers in sparse sets
本文在假设有无穷多个例外特征的前提下提出素数下界筛法，以更少算术信息获得非平凡下界，并据此证明形如 $a^2+b^8$ 的素数有无穷多个。
Let $ψ$ be a real primitive character modulo $D$. If the $L$-function $L(s,ψ)$ has a real zero close to $s=1$, known as a Landau-Siegel zero, then we say the character $ψ$ is exceptional. Under the hypothesis that such exceptional characters exist, we prove that at least fifty percent of the central values $L(1/2,χ)$ of the Dirichlet $L$-functions $L(s,χ)$ are nonzero, where $χ$ ranges over primitive characters modulo $q$ and $q$ is a large prime of size $D^{O(1)}$. Under the same hypothesis we also show that, for almost all $χ$, the function $L(s,χ)$ has at most a simple zero at $s = 1/2$.
We prove that the existence of exceptional real zeroes of Dirichlet $L$-functions would lead to cancellations in the sum $\sum_{p\leq x} \Kl(1, p)$ of Kloosterman sums over primes, and also to sign changes of $\Kl(1, n)$, where $n$ runs over integers with exactly two prime factors. Our arguments involve a variant of Bombieri's sieve, bounds for twisted sums of Kloosterman sums, and work of Fouvry and Michel on sums of $\left| \Kl(1, n)\right|$.
This paper improves the upper bound for the exceptional zeroes of Dirichlet L-functions with even characters. The result is obtained by improving on explicit estimate for $L'(σ;χ)$ for $σ$ close to unity, using a result on the average of Dirichlet characters, and on the lower bound for $L(1;χ)$, with computational aid.
Hybrid estimation of single exponential sums, exceptional characters and primes in short intervals
本文提出一种结合Van der Corput、Huxley和Bourgain结果的单一指数和混合估计，并利用该估计在存在例外Dirichlet特征假设下，将短区间素数问题的可容许范围从0.4937改进至0.4923。
A note on exceptional characters and non-vanishing of Dirichlet $L$-functions
本文在存在例外实本原 Dirichlet 特征的假设下，证明大范围素数模上几乎所有 Dirichlet 特征的 $L$-函数在中心点非零。
Iwasawa invariants and exceptional pairs of certain abelian fields
本文针对Q(√d,ζₚ)研究岩泽不变量与例外对，确定岩泽模的Λ同构类及例外对结构差异，在指定范围内发现15个新例外对，包括λ不变量为2的特殊情形。
Bernoulli-Carlitz numbers were introduced by L. Carlitz in 1935, they are the analogues in positive characteristic of Bernoulli numbers. We prove a conjecture formulated by F. Pellarin and the first author on the non-vanishing modulo a given prime of families of Bernoulli-Carlitz numbers. We then show that the "exceptional zeros" of certain $L$-series are intimately connected to the Bernoulli-Carlitz numbers.
We produce a new family of polynomials f(x) over fields K of characteristic 2 which are exceptional, in the sense that f(x)-f(y) has no absolutely irreducible factors in K[x,y] besides the scalar multiples of x-y; when K is finite, this condition is equivalent to saying there are infinitely many finite extensions L/K for which the map c --> f(c) is bijective on L. Our polynomials have degree (2^e-1)*2^(e-1), where e is odd. Combined with our previous paper arxiv:0707.1835, this completes the classification of indecomposable exceptional polynomials of degree not a power of the characteristic. T…
Planar functions are special functions from a finite field to itself that give rise to finite projective planes and other combinatorial objects. We consider polynomials over a finite field $K$ that induce planar functions on infinitely many extensions of $K$; we call such polynomials exceptional planar. Exceptional planar monomials have been recently classified. In this paper we establish a partial classification of exceptional planar polynomials. This includes results for the classical planar functions on finite fields of odd characteristic and for the recently proposed planar functions on fi…
There has recently been interest in relating properties of matrices drawn at random from the classical compact groups to statistical characteristics of number-theoretical L-functions. One example is the relationship conjectured to hold between the value distributions of the characteristic polynomials of such matrices and value distributions within families of L-functions. These connections are here extended to non-classical groups. We focus on an explicit example: the exceptional Lie group G_2. The value distributions for characteristic polynomials associated with the 7- and 14-dimensional rep…
Let $q$ be an odd prime power and $n$ be a positive integer. Let $\ell\in \mathbb F_{q^n}[x]$ be a $q$-linearised $t$-scattered polynomial of linearized degree $r$. Let $d=\max\{t,r\}$ be an odd prime number. In this paper we show that under these assumptions it follows that $\ell=x$. Our technique involves a Galois theoretical characterization of $t$-scattered polynomials combined with the classification of transitive subgroups of the general linear group over the finite field $\mathbb F_q$.
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