Following Katz–Sarnak, Iwaniec–Luo–Sarnak and Rubinstein, we use the one- two-level densities to study distribution of low-lying zeros for one-parameter rational families elliptic curves over . Modulo standard conjectures, small support agree with Katz Sarnak's predictions. Further, confirm that curves' L-functions behave in a manner consistent having r at critical point, as predicted by Birch Swinnerton-Dyer conjecture. By studying some constant sign families, find first examples where can...

SUMMARY: Auditors are required to use analytical procedures identify the existence of unusual transactions, events, and trends. Benford's Law gives expected patterns digits in numerical data, has been advocated as a test for authenticity reliability transaction level accounting data. This paper describes new second-order that calculates digit frequencies differences between ordered (ranked) values data set. These approximate most sets. The is applied four sets transactional detected errors...

A beautiful theorem of Zeckendorf states that every positive integer can be uniquely decomposed as a sum non-consecutive Fibonacci numbers $\{F_n\}$, where $F_1 = 1$, $F_2 2$ and $F_{n+1} F_n + F_{n-1}$. For general recurrences $\{G_n\}$ with non-negative coefficients, there is notion legal decomposition which again leads to unique representation, the number summands in representations uniformly randomly chosen $m \in [G_n, G_{n+1})$ converges normal distribution $n \to \infty$. We consider...

The M&M Game was created to help young kids explore probability by modeling a response the question: \emph{If two people are born on same day, will they die day?} Each player starts with fixed number of M&M's and fair coin; turn consists players simultaneously tossing their coin eating an only if toss is head, person ``dying'' when have eaten stash. tie can naturally be written as infinite sum binomial products, reformulated into finite calculation using memoryless processes, recursion...

ADVERTISEMENT RETURN TO ISSUEPREVArticleNEXTAntibacterials. Synthesis and structure-activity studies of 3-aryl-2-oxooxazolidines. 2. The "A" groupWalter A. Gregory, David R. Brittelli, C. L. J. Wang, Hollis S. Kezar III, Randall K. Carlson, Chung Ho Park, Peter F. Corless, Steven Miller, P. Rajagopalan, Cite this: Med. Chem. 1990, 33, 9, 2569–2578Publication Date (Print):September 1, 1990Publication History Published online1 May 2002Published inissue 1 September...

We investigate the moments of a smooth counting function zeros near central point L-functions weight k cuspidal newforms prime level N. split by sign functional equations and show that for test functions whose Fourier transform is supported in (−1/n,1/n), as N→∞ first n centered are Gaussian. By extending support to (−1/(n−1),1/(n−1)), we see non-Gaussian behavior; particular, odd-centered nonzero such functions. If do not sign, obtain Gaussian behavior (−2/n,2/n) if 2k≥n. The nth-centered...

We investigate the large weight (, where they find evidence of orthogonal and symplectic symmetry, respectively. The present examples suggest a relation between symmetry type family that its twistings, which will be further studied in subsequent paper. Both GL(4) GL(6) families above have all even functional equations, neither is naturally split from an family. A folklore conjecture states such must symplectic, true for first but false second. Thus, theory low lying zeros more than just...

Recently Conrey, Farmer and Zirnbauer conjectured formulas for the averages over a family of ratios products shifted L-functions. Their L-functions Ratios Conjecture predicts both main lower order terms many problems, ranging from n-level correlations densities to mollifiers moments vanishing at central point. There are now results showing agreement between number theory random matrix theory; however, there very few families where known. These often depend on subtle arithmetic properties...

Let {ℱN} and {𝒢M} be families of primitive automorphic L-functions for GLn(𝔸ℚ) GLm(𝔸ℚ), respectively, such that, as N, M → ∞, the statistical behavior (1-level density) low-lying zeros in ℱN 𝒢M agrees with that eigenvalues near 1 matrices G1 G2, size tend to infinity, where each Gi is one classical compact groups (unitary U, symplectic Sp, or orthogonal O, SO(even), SO(odd)). Assuming convolved × are automorphic, we study their 1-level density. (We also form f a fixed f.) Under natural...

In this article, we discuss the remarkable connection between two very different fields, number theory and nuclear physics. We describe essential aspects of these quantities studied, how insights in one have been fruitfully applied other. The exciting branch modern mathematics, random matrix theory, provides fields. assume no detailed knowledge physics, or theory; all that is required some familiarity with linear algebra probability as well results from complex analysis. Our goal to provide...

Zeckendorf proved that every positive integer has a unique representation as sum of non-consecutive Fibonacci numbers. Once this been shown, it's natural to ask how many summands are needed. Using continued fraction approach, Lekkerkerker the average number such needed for integers in $[F_n, F_{n+1})$ is $n / (φ^2 + 1) O(1)$, where $φ= \frac{1+\sqrt{5}}2$ golden mean. Surprisingly, no one appears have investigated distribution summands; our main result converges Gaussian $n\to\infty$....

We study the $1$-level density of low-lying zeros Dirichlet $L$-functions in family all characters modulo $q$, with $Q/2 < q\leq Q$. For test functions whose Fourier transform is supported $(-3/2, 3/2)$, we calculate this quantity beyond square-root cancellation expansion arising from $L$-function Ratios Conjecture Conrey, Farmer and Zirnbauer. discover existence a new lower-order term which not predicted by powerful conjecture. This first where 1-level determined well enough to see...

We show the leading digits of a variety systems satisfying certain conditions follow Benford's Law.For each system proving this involves two main ingredients.One is structure theorem limiting distribution, specific to system.The other general technique applying Poisson Summation distribution.We distribution values L-functions near central line and (in some sense) iterates 3x + 1 Problem are Benford.