scholarly journals On the Distribution of Periods of Holomorphic Cusp Forms and Zeroes of Period Polynomials

2020 ◽  
Author(s):  
Asbjørn Christian Nordentoft
Keyword(s):  
Asymptotic Behavior ◽  
Method Of Moments ◽  
Cusp Form ◽  
Joint Distribution ◽  
Random Variables ◽  
Kloosterman Sums ◽  
Limiting Distribution ◽  
Independent Random Variables ◽  
Cusp Forms ◽  
Holomorphic Cusp Forms

Abstract In this paper, we determine the limiting distribution of the image of the Eichler–Shimura map or equivalently the limiting joint distribution of the coefficients of the period polynomials associated to a fixed cusp form. The limiting distribution is shown to be the distribution of a certain transformation of two independent random variables both of which are equidistributed on the circle $\mathbb{R}/\mathbb{Z}$, where the transformation is connected to the additive twist of the cuspidal $L$-function. Furthermore, we determine the asymptotic behavior of the zeroes of the period polynomials of a fixed cusp form. We use the method of moments and the main ingredients in the proofs are additive twists of $L$-functions and bounds for both individual and sums of Kloosterman sums.

scholarly journals Limit theorems for triangular arrays under a relaxed asymptotic negligibility condition

1987 ◽  
Vol 42 (1) ◽  
pp. 117-128 ◽  
Author(s):  
Miloslav Jirina
Keyword(s):  
Asymptotic Behavior ◽  
Classical Theory ◽  
Limit Theorems ◽  
Random Variables ◽  
Triangular Array ◽  
Independent Random Variables ◽  
Triangular Arrays

AbstractLet {Xnk} be a triangular array of independent random variables satisfying the so-called tail-negligibility condition, i.e. such that Prob{|Xnk| > a} → 0 as both k, n → ∞. It is also assumed that for each fixed k, Xnk converges in distribution as n → ∞. Theorems on the asymptotic behavior of the row sums of the array, analogous to those of the classical theory under the uniform negligibility condition, are presented.

Representations and limit theorems for extreme value distributions

1978 ◽  
Vol 15 (03) ◽  
pp. 639-644 ◽  
Author(s):  
Peter Hall
Keyword(s):  
Stochastic Process ◽  
Order Statistics ◽  
Joint Distribution ◽  
Limit Theorems ◽  
Random Variables ◽  
Canonical Representation ◽  
Extreme Value ◽  
Limiting Distribution ◽  
Extreme Value Distributions ◽  
Independent Identically Distributed

LetXn1≦Xn2≦ ··· ≦Xnndenote the order statistics from a sample ofnindependent, identically distributed random variables, and suppose that the variablesXnn, Xn,n–1, ···, when suitably normalized, have a non-trivial limiting joint distributionξ1,ξ2, ···, asn → ∞. It is well known that the limiting distribution must be one of just three types. We provide a canonical representation of the stochastic process {ξn,n≧ 1} in terms of exponential variables, and use this representation to obtain limit theorems forξnasn →∞.

scholarly journals Large sums of Hecke eigenvalues of holomorphic cusp forms

2019 ◽  
Vol 31 (2) ◽  
pp. 403-417
Author(s):  
Youness Lamzouri
Keyword(s):  
Cusp Form ◽  
Riemann Hypothesis ◽  
Modular Group ◽  
Fourier Coefficients ◽  
Automorphic Forms ◽  
Character Sums ◽  
Cusp Forms ◽  
Hecke Eigenvalues ◽  
Sign Change ◽  
Holomorphic Cusp Forms

AbstractLet f be a Hecke cusp form of weight k for the full modular group, and let {\{\lambda_{f}(n)\}_{n\geq 1}} be the sequence of its normalized Fourier coefficients. Motivated by the problem of the first sign change of {\lambda_{f}(n)}, we investigate the range of x (in terms of k) for which there are cancellations in the sum {S_{f}(x)=\sum_{n\leq x}\lambda_{f}(n)}. We first show that {S_{f}(x)=o(x\log x)} implies that {\lambda_{f}(n)<0} for some {n\leq x}. We also prove that {S_{f}(x)=o(x\log x)} in the range {\log x/\log\log k\to\infty} assuming the Riemann hypothesis for {L(s,f)}, and furthermore that this range is best possible unconditionally. More precisely, we establish the existence of many Hecke cusp forms f of large weight k, for which {S_{f}(x)\gg_{A}x\log x}, when {x=(\log k)^{A}}. Our results are {\mathrm{GL}_{2}} analogues of work of Granville and Soundararajan for character sums, and could also be generalized to other families of automorphic forms.

scholarly journals Joint simulation of backward and forward recurrence times in a superposition of independent renewal processes

1991 ◽  
Vol 28 (04) ◽  
pp. 930-933
Author(s):  
C. Y. Teresa Lam
Keyword(s):  
Joint Distribution ◽  
Random Variables ◽  
Independent Random Variables ◽  
Renewal Processes ◽  
Life Distribution ◽  
Recurrence Times ◽  
Joint Simulation ◽  
Total Life

It is shown that, in a superposition of finitely many independent renewal processes, an observation from the limiting (when t →∞) joint distribution of backward and forward recurrence times at t can be simulated by simulating an observation of the pair (UW, (1 – U)W), where U and Ware independent random variables with U ~ uniform(0, 1) and W distributed according to the limiting total life distribution of the superposition process.

Joint simulation of backward and forward recurrence times in a renewal process

1989 ◽  
Vol 26 (02) ◽  
pp. 404-407 ◽  
Author(s):  
B. B. Winter
Keyword(s):  
Renewal Process ◽  
Joint Distribution ◽  
Random Variables ◽  
Independent Random Variables ◽  
Recurrence Times ◽  
Joint Simulation

It is shown that, in a renewal process with inter-arrival distributionF,an observation from the asymptotic (whent→∞) joint distribution of backward and forward recurrence times attcan be simulated by simulating an observation of the pair (UW, (1 –U)W), whereUandWare independent random variables withU~ uniform(0, 1) andWdistributed according to the length-biased version ofF.

On the probability that a random point is the jth nearest neighbour to its own kth nearest neighbour

1986 ◽  
Vol 23 (01) ◽  
pp. 221-226 ◽  
Author(s):  
Norbert Henze
Keyword(s):  
Poisson Process ◽  
Random Variables ◽  
Arbitrary Point ◽  
Limiting Distribution ◽  
Poisson Processes ◽  
Random Point ◽  
Independent Random Variables ◽  
Nearest Neighbour ◽  
Homogeneous Poisson Process

In a homogeneous Poisson process in R d , consider an arbitrary point X and let Y be its kth nearest neighbour. Denote by Rk the rank of X in the proximity order defined by Y, i.e., Rk = j if X is the jth nearest neighbour to Y. A representation for Rk in terms of a sum of independent random variables is obtained, and the limiting distribution of Rk, as k →∞, is shown to be normal. This result generalizes to mixtures of Poisson processes.

scholarly journals Central values of additive twists of cuspidal L-functions

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Asbjørn Christian Nordentoft
Keyword(s):  
Asymptotic Formula ◽  
Cusp Form ◽  
Dirichlet Character ◽  
Cusp Forms ◽  
Modular Symbols ◽  
Central Values ◽  
Arithmetic Distribution ◽  
Holomorphic Cusp Forms ◽  
Normally Distributed

Abstract Additive twists are important invariants associated to holomorphic cusp forms; they encode the Eichler–Shimura isomorphism and contain information about automorphic L-functions. In this paper we prove that central values of additive twists of the L-function associated to a holomorphic cusp form f of even weight k are asymptotically normally distributed. This generalizes (to k ≥ 4 {k\geq 4} ) a recent breakthrough of Petridis and Risager concerning the arithmetic distribution of modular symbols. Furthermore, we give as an application an asymptotic formula for the averages of certain “wide” families of automorphic L-functions consisting of central values of the form L ⁢ ( f ⊗ χ , 1 / 2 ) {L(f\otimes\chi,1/2)} with χ a Dirichlet character.

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