scholarly journals ONE-LEVEL DENSITY OF LOW-LYING ZEROS OF QUADRATIC HECKE L-FUNCTIONS OF IMAGINARY QUADRATIC NUMBER FIELDS

2020 ◽  
pp. 1-23
Author(s):  
PENG GAO ◽  
LIANGYI ZHAO
Keyword(s):  
Class Number ◽  
Level Density ◽  
Number Fields ◽  
Quadratic Number ◽  
Density Result ◽  
Quadratic Number Fields ◽  
Imaginary Quadratic Number

Abstract In this paper, we prove a one level density result for the low-lying zeros of quadratic Hecke L-functions of imaginary quadratic number fields of class number 1. As a corollary, we deduce, essentially, that at least $(19-\cot (1/4))/16 = 94.27\ldots \%$ of the L-functions under consideration do not vanish at 1/2.

scholarly journals Using Continued Fractions to Compute Iwasawa Lambda Invariants of Imaginary Quadratic Number Fields

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Jordan Schettler
Keyword(s):  
Continued Fractions ◽  
Class Number ◽  
Continued Fraction ◽  
Number Fields ◽  
Quadratic Number ◽  
Continued Fraction Expansion ◽  
Quadratic Number Fields ◽  
Analogous Formula ◽  
Partial Quotients ◽  
Imaginary Quadratic Number

Let l>3 be a prime such that l≡3 (mod 4) and Q(l) has class number 1. Then Hirzebruch and Zagier noticed that the class number of Q(-l) can be expressed as h(-l)=(1/3)(b1+b2+⋯+bm)-m where the bi are partial quotients in the “minus” continued fraction expansion l=[[b0;b1,b2,…,bm¯]]. For an odd prime p≠l, we prove an analogous formula using these bi which computes the sum of Iwasawa lambda invariants λp(-l)+λp(-4) of Q(-l) and Q(-1). In the case that p is inert in Q(-l), the formula pleasantly simplifies under some additional technical assumptions.

scholarly journals Evaluation of the Dedekind Eta Function

2006 ◽  
Vol 49 (1) ◽  
pp. 21-35 ◽  
Author(s):  
Robin Chapman ◽  
William Hart
Keyword(s):  
Class Number ◽  
Number Fields ◽  
Class Numbers ◽  
Modular Equations ◽  
Quadratic Number ◽  
Dedekind Eta Function ◽  
Quadratic Number Fields ◽  
Imaginary Quadratic Number

AbstractWe extend the methods of Van der Poorten and Chapman for explicitly evaluating the Dedekind eta function at quadratic irrationalities. Via evaluation of Hecke L-series we obtain new evaluations at points in imaginary quadratic number fields with class numbers 3 and 4. Further, we overcome the limitations of the earlier methods and via modular equations provide explicit evaluations where the class number is 5 or 7.

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