Extremal invariant polynomials not satisfying the Riemann hypothesis

Construction of divisible formal weight enumerators and extremal polynomials not satisfying the Riemann hypothesis

The formal weight enumerators were first introduced by M. Ozeki. They form a ring of invariant polynomials which is similar to that of the weight enumerators of Type II codes. Later, the zeta functions for linear codes were discovered and their theory was developed by I. Duursma. It was generalized to certain invariant polynomials including Ozeki's formal weight enumerators by the present author. One of the famous and important problems is whether extremal weight enumerators satisfy the Riemann hypothesis. In this paper, first we formulate the notion of divisible formal weight enumerators and …

An abundance of invariant polynomials satisfying the Riemann hypothesis

In 1999, Iwan Duursma defined the zeta function for a linear code as a generating function of its Hamming weight enumerator. It can also be defined for other homogeneous polynomials not corresponding to existing codes. If the homogeneous polynomial is invariant under the MacWilliams transform, then its zeta function satisfies a functional equation and we can formulate an analogue of the Riemann hypothesis. As far as existing codes are concerned, the Riemann hypothesis is believed to be closely related to the extremal property. In this article, we show there are abundant polynomials invariant b…

Bounding $S(t)$ and $S_1(t)$ on the Riemann hypothesis

Let $πS(t)$ denote the argument of the Riemann zeta-function, $ζ(s)$, at the point $s=\frac{1}{2}+it$. Assuming the Riemann hypothesis, we present two proofs of the bound $$ |S(t)| \leq \left(\tfrac{1}{4} + o(1) \right)\tfrac{\log t}{\log \log t} $$ for large $t$. This improves a result of Goldston and Gonek by a factor of 2. The first method consists in bounding the auxiliary function $S_1(t) = \int_0^{t} S(u) du$ using extremal functions constructed by Carneiro, Littmann and Vaaler. We then relate the size of $S(t)$ to the size of the functions $S_1(t\pm h)-S_1(t)$ when $h\asymp 1/\log\log t…

Hilbert spaces and the pair correlation of zeros of the Riemann zeta-function

Montgomery's pair correlation conjecture predicts the asymptotic behavior of the function $N(T,β)$ defined to be the number of pairs $γ$ and $γ'$ of ordinates of nontrivial zeros of the Riemann zeta-function satisfying $0<γ,γ'\leq T$ and $0 < γ'-γ\leq 2πβ/\log T$ as $T\to \infty$. In this paper, assuming the Riemann hypothesis, we prove upper and lower bounds for $N(T,β)$, for all $β>0$, using Montgomery's formula and some extremal functions of exponential type. These functions are optimal in the sense that they majorize and minorize the characteristic function of the interval $[-β, β]$ in a w…

线性化多项式的分解与奇特征下的极值曲线

Factorizations of linearized polynomials and extremal curves in odd characteristic

本文给出奇特征有限域上构造极值van der Geer–van der Vlugt曲线的完整方法，输入数据为基础域的非零元素和满足迹条件的线性子空间，是作者先前在特征2下结果的类比。

Difference independence of the Riemann zeta function

It is proved that the Riemann zeta function does not satisfy any nontrivial algebraic difference equation whose coefficients are meromorphic functions $ϕ$ with Nevanlinna characteristic satisfying $T(r, ϕ)=o(r)$ as $r\to \infty$