Timothy Foo
In this note, we prove that under some conditions, certain products of integers related to Gauss factorials are always quadratic residues.
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In this paper we initiate a study on Gauss factorials of polynomials over finite fields, which are the analogues of Gauss factorials of positive integers.
In this paper we establish a connection between the Gauss factorials and Iwasawa's cyclotomic $λ$-invariant for an imaginary quadratic field $K$. As a result, we will explain a corespondance between the 1-exceptional primes of Cosgrave and Dilcher for $m = 3$ and $m = 4$, and the primes for which the $λ$-invariants for $K = \mathbb{Q}(\sqrt{-3})$ and $K = \mathbb{Q}(i)$ is greater than one, respectively. We refer to the latter primes as ``non-trivial'' for their respective fields. We will also see that similar correspondences are true for $K = \mathbb{Q}(\sqrt{-d})$ when $d = 2,5$ and $6$. As …
We obtain a lower bound on the multiplicative order of Gauss periods which generate normal bases over finite fields. This bound improves the previous bound of J. von zur Gathen and I. E. Shparlinski.
We obtain explicit lower bounds on multiplicative order of elements that have more general form than finite field Gauss period. In a partial case of Gauss period this bound improves the previous bound of O.Ahmadi, I.E.Shparlinski and J.F.Voloch
We obtain divisibility conditions on the multiplicative orders of elements of the form $ζ+ ζ^{-1}$ in a finite field by exploiting a link to the arithmetic of real quadratic fields.
We describe the points of diff erentiability of the multiplicative autocorrelation function of the "fractional part" function. In connection with this question, we study series involving the fi rst Bernoulli function, the arithmetical function "number of divisors", and the Gauss map from the theory of continued fractions. A key role is played by a function defi ned in 1933 by J. R. Wilton, similar to the Brjuno function of dynamical systems theory. A unifying theme of our exposition is the use of functional equations involving the Gauss map, allowing us to reprove and re fine a theorem of Wilt…
We prove a structure theorem for multiplicative functions on the Gaussian integers, showing that every bounded multiplicative function on the Gaussian integers can be decomposed into a term which is approximately periodic and another which has a small U^{3}-Gowers uniformity norm. We apply this to prove partition regularity results over the Gaussian integers for certain equations involving quadratic forms in three variables. For example, we show that for any finite coloring of the Gaussian integers, there exist distinct nonzero elements x and y of the same color such that x^{2}-y^{2}=n^{2} for…
We prove new lower bounds for the upper tail probabilities of suprema of Gaussian processes. Unlike many existing bounds, our results are not asymptotic, but supply strong information when one is only a little into the upper tail. We present an extended application to a Gaussian version of a random process studied by Halasz. This leads to much improved lower bound results for the sum of a random multiplicative function. We further illustrate our methods by improving lower bounds for some classical constants from extreme value theory, the Pickands constants $H_α$, as $α\rightarrow 0$.
We prove that if $ω$ is uniformly distributed on $[0,1]$, then as $T\to\infty$, $t\mapsto ζ(iωT+it+1/2)$ converges to a non-trivial random generalized function, which in turn is identified as a product of a very well behaved random smooth function and a random generalized function known as a complex Gaussian multiplicative chaos distribution. This demonstrates a novel rigorous connection between number theory and the theory of multiplicative chaos -- the latter is known to be connected to many other areas of mathematics. We also investigate the statistical behavior of the zeta function on the …
We prove convergence of a stochastic approximation of powers of the Riemann $ζ$ function to a non-Gaussian multiplicative chaos measure, and prove that this measure is a non-trivial multifractal random measure. The results cover both the subcritical and critical chaos. A basic ingredient of the proof is a 'good' Gaussian approximation of the induced random fields that is potentially of independent interest.
On averages of completely multiplicative functions over co-prime integer pairs
本文研究了完全乘法函数在互质整数对上的平均值行为，推导了高斯整数上的Wirsing定理，并在多线性平均方面取得新进展。
The Fourier coefficients of the critical holomorphic multiplicative chaos
本文研究临界全纯乘性混沌（HMC）的傅里叶系数，将HMC傅里叶系数的分布收敛性从L²相（β>4）和L¹相扩展到临界情形β=2，并证明连续傅里叶系数的联合收敛性及久期系数的分布收敛性。
The distribution of partial sums of random multiplicative functions with a large prime factor
本文研究 Steinhaus 随机可乘函数在限制最大素因子大于 $\sqrt{x}$ 时的归一化部分和，证明其依分布收敛，并刻画极限分布为临界高斯可乘混沌积分平方根与独立标准复正态变量的乘积。
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