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Jul 20, 2013
07/13

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Steven J Miller

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We derive an identity for certain linear combinations of polylogarithm functions with negative exponents, which implies relations for linear combinations of Eulerian numbers. The coefficients of our linear combinations are related to expanding moments of Satake parameters of holomorphic cuspidal newforms in terms of the moments of the corresponding Fourier coefficients, which has applications in analyzing lower order terms in the behavior of zeros of L-functions near the central point.

Source: http://arxiv.org/abs/0804.3611v1

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Sep 18, 2013
09/13

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Steven J. Miller

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

Source: http://arxiv.org/abs/0704.0927v5

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Sep 21, 2013
09/13

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Steven J. Miller

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We explore the effect of zeros at the central point on nearby zeros of elliptic curve L-functions, especially for one-parameter families of rank r over Q. By the Birch and Swinnerton Dyer Conjecture and Silverman's Specialization Theorem, for t sufficiently large the L-function of each curve E_t in the family has r zeros (called the family zeros) at the central point. We observe experimentally a repulsion of the zeros near the central point, and the repulsion increases with r. There is greater...

Source: http://arxiv.org/abs/math/0508150v3

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Sep 18, 2013
09/13

by
Steven J. Miller

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We investigate a one-parameter family of probability densities (related to the Pareto distribution, which describes many natural phenomena) where the Cramer-Rao inequality provides no information.

Source: http://arxiv.org/abs/0704.0923v1

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Sep 22, 2013
09/13

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Steven J. Miller

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Following Katz-Sarnak, Iwaniec-Luo-Sarnak, and Rubinstein, we use the 1- and 2-level densities to study the distribution of low lying zeros for one-parameter rational families of elliptic curves over Q(t). Modulo standard conjectures, for small support the densities agree with Katz and Sarnak's predictions. Further, the densities confirm that the curves' L-functions behave in a manner consistent with having r zeros at the critical point, as predicted by the Birch and Swinnerton-Dyer conjecture....

Source: http://arxiv.org/abs/math/0310159v1

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Sep 20, 2013
09/13

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Steven J. Miller

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Michel proved that for a one-parameter family of elliptic curves over Q(T) with non-constant j(T) that the second moment of the number of solutions modulo p is p^2 + O(p^{3/2}). We show this bound is sharp by studying y^2 = x^3 + Tx^2 + 1. Lower order terms for such moments in a family are related to lower order terms in the n-level densities of Katz and Sarnak, which describe the behavior of the zeros near the central point of the associated L-functions. We conclude by investigating similar...

Source: http://arxiv.org/abs/math/0506461v2

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Sep 21, 2013
09/13

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Steven J. Miller

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Using mostly elementary results and functions from probability, we prove Wallis's formula for pi: pi/2 = prod_n (2n * 2n) / ((2n-1) * (2n+1)). The proof involves normalization constants and the Gamma function, Standard normal, and the Student t-Distribution.

Source: http://arxiv.org/abs/0709.2181v2

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Sep 22, 2013
09/13

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Steven J. Miller

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We test the predictions of the L-functions Ratios Conjecture for the family of cuspidal newforms of weight k and level N, with either k fixed and N --> oo through the primes or N=1 and k --> oo. We study the main and lower order terms in the 1-level density. We provide evidence for the Ratios Conjecture by computing and confirming its predictions up to a power savings in the family's cardinality, at least for test functions whose Fourier transforms are supported in (-2, 2). We do this...

Source: http://arxiv.org/abs/0805.4208v1

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Sep 18, 2013
09/13

by
Steven J. Miller

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The Katz-Sarnak density conjecture states that, in the limit as the conductors tend to infinity, the behavior of normalized zeros near the central point of families of L-functions agree with the N -> oo scaling limits of eigenvalues near 1 of subgroups of U(N). Evidence for this has been found for many families by studying the n-level densities; for suitably restricted test functions the main terms agree with random matrix theory. In particular, all one-parameter families of elliptic curves...

Source: http://arxiv.org/abs/0704.0924v4

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Sep 18, 2013
09/13

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Steven J. Miller

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It has been noted that in many professional sports leagues a good predictor of a team's won-loss percentage is Bill James' Pythagorean Formula RSobs^c / (RSobs^c + RAobs^c), where RSobs (resp. RAobs) is the observed average number of runs scored (allowed) per game and c is a constant for the league; for baseball the best agreement is when c is about 1.82. We provide a theoretical justification for this formula and value of c by modelling the number of runs scored and allowed in baseball games...

Source: http://arxiv.org/abs/math/0509698v4

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Sep 19, 2013
09/13

by
Steven J. Miller; David Montague

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We generalize Tennenbaum's geometric proof of the irrationality of sqrt(2) to sqrt(n) for n = 3, 5, 6 and 10.

Source: http://arxiv.org/abs/0909.4913v2

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Jun 30, 2018
06/18

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Victor Luo; Steven J. Miller

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Bill James invented the Pythagorean expectation in the late 70's to predict a baseball team's winning percentage knowing just their runs scored and allowed. His original formula estimates a winning percentage of ${\rm RS}^2/({\rm RS}^2+{\rm RA}^2)$, where ${\rm RS}$ stands for runs scored and ${\rm RA}$ for runs allowed; later versions found better agreement with data by replacing the exponent 2 with numbers near 1.83. Miller and his colleagues provided a theoretical justification by modeling...

Topics: Mathematics, Applications, History and Overview, Statistics

Source: http://arxiv.org/abs/1406.3402

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Sep 21, 2013
09/13

by
Levent Alpoge; Steven J. Miller

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The Katz-Sarnak density conjecture states that the scaling limits of the distributions of zeroes of families of automorphic L-functions agree with the scaling limits of eigenvalue distributions of classical subgroups of the unitary groups U(N). This conjecture is often tested by way of computing particular statistics, such as the one-level density, which evaluates a test function with compactly supported Fourier transform at normalized zeroes near the central point. Iwaniec, Luo, and Sarnak...

Source: http://arxiv.org/abs/1301.5702v1

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Sep 21, 2013
09/13

by
Steven J. Miller; Ralph Morrison

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A number of mathematical methods have been shown to model the zeroes of $L$-functions with remarkable success, including the Ratios Conjecture and Random Matrix Theory. In order to understand the structure of convolutions of families of $L$-functions, we investigate how well these methods model the zeros of such functions. Our primary focus is the convolution of the $L$-function associated to Ramanujan's tau function with the family of quadratic Dirichlet $L$-functions, for which J.B. Conrey...

Source: http://arxiv.org/abs/1011.0229v1

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Sep 21, 2013
09/13

by
Virginia Hogan; Steven J. Miller

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We study the relationship between the number of minus signs in a generalized sumset, $A+...+A-...-A$, and its cardinality; without loss of generality we may assume there are at least as many positive signs as negative signs. As addition is commutative and subtraction is not, we expect that for most $A$ a combination with more minus signs has more elements than one with fewer; however, recently Iyer, Lazarev, Miller and Zhang proved that a positive percentage of the time the combination with...

Source: http://arxiv.org/abs/1301.5703v1

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Sep 21, 2013
09/13

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Steven J. Miller; Yinghui Wang

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A beautiful theorem of Zeckendorf states that every integer can be written uniquely as a sum of non-consecutive Fibonacci numbers $\{F_n\}_{n=1}^{\infty}$. Lekkerkerker proved that the average number of summands for integers in $[F_n, F_{n+1})$ is $n/(\phi^2 + 1)$, with $\phi$ the golden mean. This has been generalized to the following: given nonnegative integers $c_1,c_2,...,c_L$ with $c_1,c_L>0$ and recursive sequence $\{H_n\}_{n=1}^{\infty}$ with $H_1=1$, $H_{n+1}...

Source: http://arxiv.org/abs/1008.3202v3

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Sep 20, 2013
09/13

by
Eduardo Duenez; Steven J. Miller

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We investigate the large weight (k --> oo) limiting statistics for the low lying zeros of a GL(4) and a GL(6) family of L-functions, {L(s,phi x f): f in H_k(1)} and {L(s,phi times sym^2 f): f in H_k(1)}; here phi is a fixed even Hecke-Maass cusp form and H_k(1) is a Hecke eigenbasis for the space H_k(1) of holomorphic cusp forms of weight k for the full modular group. Katz and Sarnak conjecture that the behavior of zeros near the central point should be well modeled by the behavior of...

Source: http://arxiv.org/abs/math/0506462v2

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Sep 23, 2013
09/13

by
Daniel Fiorilli; Steven J. Miller

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We study the distribution of the zeros near the central point for weighted and unweighted families of Dirichlet $L$-functions. As the conductors tend to infinity, the main term of the 1-level densities agrees with the scaling limit of unitary matrices for even $C^2$ test functions whose Fourier transforms are supported in $(-2,2)$, supporting the Katz-Sarnak conjecture. The lower order terms agree with the prediction from the $L$-function Ratios Conjecture in the regime where both can be...

Source: http://arxiv.org/abs/1111.3896v2

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Sep 17, 2013
09/13

by
Steven J. Miller; Ryan Peckner

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One of the most important statistics in studying the zeros of L-functions is the 1-level density, which measures the concentration of zeros near the central point. Fouvry and Iwaniec [FI] proved that the 1-level density for L-functions attached to imaginary quadratic fields agrees with results predicted by random matrix theory. In this paper, we show a similar agreement with random matrix theory occurring in more general sequences of number fields. We first show that the main term agrees with...

Source: http://arxiv.org/abs/1003.5336v1

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Sep 18, 2013
09/13

by
Peter Hegarty; Steven J. Miller

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We investigate the relationship between the sizes of the sum and difference sets attached to a subset of {0,1,...,N}, chosen randomly according to a binomial model with parameter p(N), with N^{-1} = o(p(N)). We show that the random subset is almost surely difference dominated, as N --> oo, for any choice of p(N) tending to zero, thus confirming a conjecture of Martin and O'Bryant. The proofs use recent strong concentration results. Furthermore, we exhibit a threshold phenomenon regarding the...

Source: http://arxiv.org/abs/0707.3417v5

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Jun 28, 2018
06/18

by
Corey Manack; Steven J. Miller

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We determine the leading digit laws for the matrix components of a linear Lie group $G$. These laws generalize the observations that the normalized Haar measure of the Lie group $\mathbb{R}^+$ is $dx/x$ and that the scale invariance of $dx/x$ implies the distribution of the digits follow Benford's law, which is the probability of observing a significand base $B$ of at most $s$ is $\log_B(s)$; thus the first digit is $d$ with probability $\log_B(1 + 1/d)$). Viewing this scale invariance as left...

Topics: Metric Geometry, Number Theory, Mathematics, Probability

Source: http://arxiv.org/abs/1507.01605

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Jul 20, 2013
07/13

by
Steven J. Miller; Yinghui Wang

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A beautiful theorem of Zeckendorf states that every integer can be written uniquely as a sum of non-consecutive Fibonacci numbers $\{F_n\}_{n=1}^{\infty}$; Lekkerkerker proved that the average number of summands for integers in $[F_n, F_{n+1})$ is $n/(\phi^2 + 1)$, with $\phi$ the golden mean. Interestingly, the higher moments seem to have been ignored. We discuss the proof that the distribution of the number of summands converges to a Gaussian as $n \to \infty$, and comment on generalizations...

Source: http://arxiv.org/abs/1107.2718v1

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Sep 23, 2013
09/13

by
Christopher Hammond; Steven J. Miller

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Consider the ensemble of Real Symmetric Toeplitz Matrices, each entry iidrv from a fixed probability distribution p of mean 0, variance 1, and finite higher moments. The limiting spectral measure (the density of normalized eigenvalues) converges weakly to a new universal distribution with unbounded support, independent of p. This distribution's moments are almost those of the Gaussian's; the deficit may be interpreted in terms of Diophantine obstructions. With a little more work, we obtain...

Source: http://arxiv.org/abs/math/0312215v1

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Sep 20, 2013
09/13

by
Eduardo Duenez; Steven J. Miller

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L-functions for GL_n(A_Q) and GL_m(A_Q), respectively, such that, as N,M --> oo, the statistical behavior (1-level density) of the low-lying zeros of L-functions in F_N (resp., G_M) agrees with that of the eigenvalues near 1 of matrices in G_1 (resp., G_2) as the size of the matrices tend to infinity, where each G_i is one of the classical compact groups (unitary, symplectic or orthogonal). Assuming that the convolved families of L-functions F_N x G_M are automorphic, we study their 1-level...

Source: http://arxiv.org/abs/math/0607688v2

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Jun 29, 2018
06/18

by
Ray Li; Steven J. Miller

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Zeckendorf proved that every integer can be written uniquely as a sum of non-adjacent Fibonacci numbers $\{1,2,3,5,\dots\}$. This has been extended to many other recurrence relations $\{G_n\}$ (with their own notion of a legal decomposition) and to proving that the distribution of the number of summands of an $M \in [G_n, G_{n+1})$ converges to a Gaussian as $n\to\infty$. We prove that for any non-negative integer $g$ the average number of gaps of size $g$ in many generalized Zeckendorf...

Topics: Number Theory, Mathematics

Source: http://arxiv.org/abs/1606.08110

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Jun 28, 2018
06/18

by
Jesse Freeman; Steven J. Miller

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Given an $L$-function, one of the most important questions concerns its vanishing at the central point; for example, the Birch and Swinnerton-Dyer conjecture states that the order of vanishing there of an elliptic curve $L$-function equals the rank of the Mordell-Weil group. The Katz and Sarnak Density Conjecture states that this and other behavior is well-modeled by random matrix ensembles. This correspondence is known for many families when the test functions are suitably restricted. For...

Topics: Functional Analysis, Number Theory, Mathematics

Source: http://arxiv.org/abs/1507.03598

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Sep 20, 2013
09/13

by
Steven J. Miller; Mark. J. Nigrini

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Fix a base B and let zeta have the standard exponential distribution; the distribution of digits of zeta base B is known to be very close to Benford's Law. If there exists a C such that the distribution of digits of C times the elements of some set is the same as that of zeta, we say that set exhibits shifted exponential behavior base B (with a shift of log_B C \bmod 1). Let X_1, >..., X_N be independent identically distributed random variables. If the X_i's are drawn from the uniform...

Source: http://arxiv.org/abs/math/0601344v5

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Sep 23, 2013
09/13

by
Steven J. Miller; David A. Thompson

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The purpose of this note is to discuss the real analogue of the Schwarz lemma from complex analysis.

Source: http://arxiv.org/abs/1012.0585v1

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Jun 26, 2018
06/18

by
Gene S. Kopp; Steven J. Miller

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The spherical orthogonal, unitary, and symplectic ensembles (SOE/SUE/SSE) $S_\beta(N,r)$ consist of $N \times N$ real symmetric, complex hermitian, and quaternionic self-adjoint matrices of Frobenius norm $r$, made into a probability space with the uniform measure on the sphere. For each of these ensembles, we determine the joint eigenvalue distribution for each $N$, and we prove the empirical spectral measures rapidly converge to the semicircular distribution as $N \to \infty$. In the unitary...

Topics: Probability, Mathematics, Mathematical Physics

Source: http://arxiv.org/abs/1501.01848

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Sep 24, 2013
09/13

by
Alex V. Kontorovich; Steven J. Miller

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

Source: http://arxiv.org/abs/math/0412003v2

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Sep 23, 2013
09/13

by
Steven J. Miller; Cesar E. Silva

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One of the greatest difficulties encountered by all in their first proof intensive class is subtly assuming an unproven fact in a proof. The purpose of this note is to describe a specific instance where this can occur, namely in results related to unique factorization and the concept of the greatest common divisor.

Source: http://arxiv.org/abs/1012.5866v1

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Sep 19, 2013
09/13

by
C. P. Hughes; Steven J. Miller

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We investigate the moments of a smooth counting function of the zeros near the central point of L-functions of weight k cuspidal newforms of prime level N. We split by the sign of the functional equations and show that for test functions whose Fourier transform is supported in (-1/n, 1/n), as N --> oo the first n centered moments are Gaussian. By extending the support to (-1/n-1, 1/n-1), we see non-Gaussian behavior; in particular the odd centered moments are non-zero for such test...

Source: http://arxiv.org/abs/math/0507450v3

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Jun 30, 2018
06/18

by
Iddo Ben-Ari; Steven J. Miller

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Generalized Zeckendorf decompositions are expansions of integers as sums of elements of solutions to recurrence relations. The simplest cases are base-$b$ expansions, and the standard Zeckendorf decomposition uses the Fibonacci sequence. The expansions are finite sequences of nonnegative integer coefficients (satisfying certain technical conditions to guarantee uniqueness of the decomposition) and which can be viewed as analogs of sequences of variable-length words made from some fixed...

Topics: Probability, Mathematics, Number Theory

Source: http://arxiv.org/abs/1405.2379

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Sep 22, 2013
09/13

by
Steven J. Miller; M. Ram Murty

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Extending recent work of others, we provide effective bounds on the family of all elliptic curves and one-parameter families of elliptic curves modulo p (for p prime tending to infinity) obeying the Sato-Tate Law. We present two methods of proof. Both use the framework of Murty-Sinha; the first involves only knowledge of the moments of the Fourier coefficients of the L-functions and combinatorics, and saves a logarithm, while the second requires a Sato-Tate law. Our purpose is to illustrate how...

Source: http://arxiv.org/abs/1004.2753v2

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Sep 20, 2013
09/13

by
Steven J. Miller; Mark J. Nigrini

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We derive a necessary and sufficient condition for the sum of M independent continuous random variables modulo 1 to converge to the uniform distribution in L^1([0,1]), and discuss generalizations to discrete random variables. A consequence is that if X_1, ..., X_M are independent continuous random variables with densities f_1, ..., f_M, for any base B as M \to \infty for many choices of the densities the distribution of the digits of X_1 * ... * X_M converges to Benford's law base B. The rate...

Source: http://arxiv.org/abs/math/0607686v2

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Sep 21, 2013
09/13

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Steven J. Miller; Matthew Schiffman; Ben Wieland

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Using the fact that the irrationality measure of zeta(2) = pi^2/6 is finite, one can deduce explicit lower bounds for the number of primes at most x. The best estimate this method yields is (basically) a lower bound of loglog(x) / logloglog(x) for infinitely many x, almost as good as Euclid's argument. Unfortunately, the standard proofs of the finiteness of the irrationality measure of zeta(2) use the prime number theorem to estimate lcm(1,...,n)! By a careful analysis of the irrationality...

Source: http://arxiv.org/abs/0709.2184v3

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Sep 19, 2013
09/13

by
Frank W. K. Firk; Steven J. Miller

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In this article, we discuss the remarkable connection between two very different fields, number theory and nuclear physics. We describe the essential aspects of these fields, the quantities studied, and how insights in one have been fruitfully applied in the other. The exciting branch of modern mathematics, random matrix theory, provides the connection between the two fields. We assume no detailed knowledge of number theory, nuclear physics, or random matrix theory; all that is required is some...

Source: http://arxiv.org/abs/0909.4914v1

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Sep 17, 2013
09/13

by
Steven Jackson; Steven J. Miller; Thuy Pham

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Consider the ensemble of real symmetric Toeplitz matrices whose entries are i.i.d random variables chosen from a fixed probability distribution p of mean 0, variance 1 and finite higher moments. Previous work [BDJ,HM] showed that the limiting spectral measures (the density of normalized eigenvalues) converge weakly and almost surely to a universal distribution almost that of the Gaussian, independent of p. The deficit from the Gaussian distribution is due to obstructions to solutions of...

Source: http://arxiv.org/abs/1003.2010v2

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Jun 30, 2018
06/18

by
Victoria Cuff; Allison Lewis; Steven J. Miller

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Benford's law states that many data sets have a bias towards lower leading digits (about $30\%$ are 1s). There are numerous applications, from designing efficient computers to detecting tax, voter and image fraud. It's important to know which common probability distributions are almost Benford. We show the Weibull distribution, for many values of its parameters, is close to Benford's law, quantifying the deviations. As the Weibull distribution arises in many problems, especially survival...

Topics: Probability, Mathematics

Source: http://arxiv.org/abs/1402.5854

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Sep 23, 2013
09/13

by
Oleg Lazarev; Steven J. Miller; Kevin O'Bryant

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For any finite set of integers X, define its sumset X+X to be {x+y: x, y in X}. In a recent paper, Martin and O'Bryant investigated the distribution of |A+A| given the uniform distribution on subsets A of {0, 1, ..., n-1}. They also conjectured the existence of a limiting distribution for |A+A| and showed that the expectation of |A+A| is 2n - 11 + O((3/4)^{n/2}). Zhao proved that the limits m(k) := lim_{n --> oo} Prob(2n-1-|A+A|=k) exist, and that sum_{k >= 0} m(k)=1. We continue this...

Source: http://arxiv.org/abs/1109.4700v3

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Sep 23, 2013
09/13

by
Steven J. Miller; Sean Pegado; Luc Robinson

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A More Sums Than Differences (MSTD) set is a set of integers A contained in {0, ..., n-1} whose sumset A+A is larger than its difference set A-A. While it is known that as n tends to infinity a positive percentage of subsets of {0, ..., n-1} are MSTD sets, the methods to prove this are probabilistic and do not yield nice, explicit constructions. Recently Miller, Orosz and Scheinerman gave explicit constructions of a large family of MSTD sets; though their density is less than a positive...

Source: http://arxiv.org/abs/1303.0605v1

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Sep 18, 2013
09/13

by
Steven J. Miller; Tim Novikoff; Anthony Sabelli

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Recently Friedman proved Alon's conjecture for many families of d-regular graphs, namely that given any epsilon > 0 `most' graphs have their largest non-trivial eigenvalue at most 2 sqrt{d-1}+epsilon in absolute value; if the absolute value of the largest non-trivial eigenvalue is at most 2 sqrt{d-1} then the graph is said to be Ramanujan. These graphs have important applications in communication network theory, allowing the construction of superconcentrators and nonblocking networks, coding...

Source: http://arxiv.org/abs/math/0611649v2

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Sep 18, 2013
09/13

by
Carlos Dominguez; Steven J. Miller; Siman Wong

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For any integer $k\ge 1$, we show that there are infinitely many complex quadratic fields whose 2-class groups are cyclic of order $2^k$. The proof combines the circle method with an algebraic criterion for a complex quadratic ideal class to be a square.

Source: http://arxiv.org/abs/1211.2605v1

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Sep 18, 2013
09/13

by
Steven J. Miller; Brooke Orosz; Daniel Scheinerman

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We explicitly construct infinite families of MSTD (more sums than differences) sets. There are enough of these sets to prove that there exists a constant C such that at least C / r^4 of the 2^r subsets of {1,...,r} are MSTD sets; thus our family is significantly denser than previous constructions (whose densities are at most f(r)/2^{r/2} for some polynomial f(r)). We conclude by generalizing our method to compare linear forms epsilon_1 A + ... + epsilon_n A with epsilon_i in {-1,1}.

Source: http://arxiv.org/abs/0809.4621v2

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Sep 21, 2013
09/13

by
Duc Khiem Huynh; Steven J. Miller; Ralph Morrison

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We compare the L-Function Ratios Conjecture's prediction with number theory for the family of quadratic twists of a fixed elliptic curve with prime conductor, and show agreement in the 1-level density up to an error term of size X^{-(1-sigma)/2} for test functions supported in (-sigma, sigma); this gives us a power-savings for \sigma

Source: http://arxiv.org/abs/1011.3298v3

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94

Jul 20, 2013
07/13

by
Scott Arms; Steven J. Miller; Alvaro Lozano-Robledo

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We give several new constructions for moderate rank elliptic curves over $\mathbb{Q}(T)$. In particular we construct infinitely many rational elliptic surfaces (not in Weierstrass form) of rank 6 over $\mathbb{Q}$ using polynomials of degree two in $T$. While our method generates linearly independent points, we are able to show the rank is exactly 6 \emph{without} having to verify the points are independent. The method generalizes; however, the higher rank surfaces are not rational, and we need...

Source: http://arxiv.org/abs/math/0406579v1

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6.0

Jun 29, 2018
06/18

by
Steven J. Miller; Dawn Nelson; Zhao Pan; Huanzhong Xu

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A positive linear recurrence sequence is of the form $H_{n+1} = c_1 H_n + \cdots + c_L H_{n+1-L}$ with each $c_i \ge 0$ and $c_1 c_L > 0$, with appropriately chosen initial conditions. There is a notion of a legal decomposition (roughly, given a sum of terms in the sequence we cannot use the recurrence relation to reduce it) such that every positive integer has a unique legal decomposition using terms in the sequence; this generalizes the Zeckendorf decomposition, which states any positive...

Topics: Number Theory, Probability, Mathematics

Source: http://arxiv.org/abs/1607.04692

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82

Sep 21, 2013
09/13

by
Murat Kologlu; Gene Kopp; Steven J. Miller; Yinghui Wang

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

Source: http://arxiv.org/abs/1008.3204v1

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16

Jun 28, 2018
06/18

by
Megumi Asada; Sarah Manski; Steven J. Miller; Hong Suh

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A More Sums Than Differences (MSTD) set is a set $A$ for which $|A+A|>|A-A|$. Martin and O'Bryant proved that the proportion of MSTD sets in $\{0,1,\dots,n\}$ is bounded below by a positive number as $n$ goes to infinity. Iyer, Lazarev, Miller and Zhang introduced the notion of a generalized MSTD set, a set $A$ for which $|sA-dA|>|\sigma A-\delta A|$ for a prescribed $s+d=\sigma+\delta$. We offer efficient constructions of $k$-generational MSTD sets, sets $A$ where $A, A+A, \dots, kA$ are...

Topics: Number Theory, Mathematics

Source: http://arxiv.org/abs/1509.01657

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70

Jul 20, 2013
07/13

by
Geoffrey Iyer; Oleg Lazarev; Steven J. Miller; Liyang Zhang

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We review the basic theory of More Sums Than Differences (MSTD) sets, specifically their existence, simple constructions of infinite families, the proof that a positive percentage of sets under the uniform binomial model are MSTD but not if the probability that each element is chosen tends to zero, and 'explicit' constructions of large families of MSTD sets. We conclude with some new constructions and results of generalized MSTD sets, including among other items results on a positive percentage...

Source: http://arxiv.org/abs/1107.2719v1