An orthogonal test of the L-functions Ratios Conjecture, II

Abstract

ABSTRACT. 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 → ∞ through the primes or N = 1 and k → ∞. 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 both for the weighted and unweighted 1-level density (where in the weighted case we use the Petersson weights), thus showing that either formulation may be used. These two 1-level densities differ by a term of size 1/ log(k 2 N ). Finally, we show that there is another way of extending the sums arising in the Ratios Conjecture, leading to a different answer (although the answer is such a lower order term that it is hopeless to observe which is correct).

Nuclei, Primes and the Random Matrix Connection

“…70 Another problem is that the main term in the 1-level density agrees with random matrix theory, but the arithmetic of the family does not surface until we examine the lower order terms (which control the rate of convergence; see for example [103,107,133]). One promising line of research is the L-functions Ratios Conjecture [116,117], which is supported by corresponding calculations for random matrix ensembles (see [118,122,[125][126][127]131] for some recent work supporting these conjectures, especially [118] for a very accessible introduction to the method and a summary of its successes). Another approach is through hybrid product formulas [123].…”

Untitled

“…(2-3) (In the current paper, the parameter Q plays the role of jᏲ N j.) Asymptotic formulas for R Ᏺ N .˛; / have been conjectured for a variety of families Ᏺ N (see Conrey and Snaith 2007;Goes et al 2010;Huynh et al 2011;Miller 2008;2009b;Miller and Montague 2011]) and are believed to hold up to errors of size jᏲ N j 1=2C for any > 0. The evidence for the correctness of this error term is limited to test functions with small support (frequently significantly less than .…”

“…The powerful ratios conjecture of Conrey, Farmer and Zirnbauer Conrey et al 2005b] yields a formula for D 1IQ=2;Q .Á/, which is believed to hold up to an error of O .Q 1=2C /. While there have been several papers [Conrey and Snaith 2007;David et al 2013;Goes et al 2010;Huynh et al 2011;Miller 2008;2009b;Miller and Montague 2011] showing agreement between various statistics involving L-functions and the ratios conjecture's predictions, evidence for this precise exponent in the error term is limited; the reason this exponent was chosen is the "philosophy of square root cancellation". While some of the families studied have 1-level densities that agree beyond square root cancellation, it is always for small support (supp.y Á/ .…”

A unitary test of the Ratios Conjecture

“…One may weaken the Ratios Conjecture by not discarding these terms; this is done in [Mil5,MilMo], where as predicted it is found that these terms do not contribute. To provide a better test, we also do not drop these terms (see Remark 2.1 for a discussion of which terms, for this family, may be ignored).…”

“…These have been verified as accurate (up to square-root agreement as predicted!) for suitably restricted test functions for orthogonal families of cusp forms [Mil5,MilMo] and the symplectic families of Dirichlet characters [Mil3,St] and elliptic curves twisted by quadratic characters [HuyMil]. Further, these lower order terms have been used as inputs in other problems, such as modeling the first zero above the central point for certain families of elliptic curves [DHKMS].…”