Lower Bounds on Class Numbers Jose Chavez Mentor: Jeffrey Stopple - - PowerPoint PPT Presentation

Lower Bounds on Class Numbers

Jose Chavez Mentor: Jeffrey Stopple UC LEADS UCSB-UCLA

Pure Mathematics and the Modern World

• Pure Mathematics has application in many areas:
• Finance
• Stochastic Processes used to understand

the system that is “The Market”

• Physics
• Biology
• Yet to be discovered
• Intellectual Pursuit

Number Theory and Our World

• Algebraic Number Theory
• RSA encryption algorithm
• Euler's Theorem, Congruence Classes,

difficulty factoring large numbers to secure information

• Analytic Number Theory
• Connections to Physics
• Backwards Heat Equation satisfied by Xi

function

ut=−Δu

Binary Quadratic Forms and Equivalence

• Quadratic Form:
• Examples of BQF's:
• Equivalence of Forms: If matrix M s.t

(change of variables)

• Example

x

2+y 2{1,0,1}F (x , y)

x

2+2y 2{1,0,2}F (x ,y)

ax

2+bxy+cy 2{a,b ,c}F (x , y)

∃ ̀ F (x ,y)=F ((x , y)M) F

'=2x 2+4xy+5y 2=F ,((x , y)[

1 1 1])=F (x+y , y)=2(x+y )2+3y

2

F

'=2x 2+4xy+5y 2 F=2x 2+3y 2

Class and Goals

• Forms that are equivalent, form an equivalence

class

• Discriminant of form:
• Class number =h(d): number of classes

associated to discriminant d

• Theorem: Suppose is a real non-principal

character, with . If has no zero in where then

• Goal-Improve the constant bounding h(d)

{a ,b ,c}=b

2−4ac

χ mod d d≥200 L(s , χ) [1−α,1]

0<α< 1 20ln(d ) .35απ

√∣d∣ <h(d)

Setting Up the Improvement

• WANT:Raise lower bound
• Change interval in hypothesis of Hecke's

Theorem

• Change lower bound on discriminant
• Introduce Polya-Vinagradov's inequality

to the body of Hecke's Theorem

• Combine techniques
• Pursue clean looking result

.35α π

√∣d∣ <h(d)

∣∑

m≤x

χD(m)∣<√Dln(D)

Results

• Take new interval
• Let d>= 10,000,000
• Employ Polya-Vinagradov Inequality
• Result:

40,000<α<1/(30ln(d))<0.002 .50α π

√∣d∣ <h(d)

Conclusions and Future Directions

• Little room for more improvement
• Class numbers are connected to fascinating

topics

• Explore the connections between class

number and Siegel zeros(zeta-zeros that contradict the Reimann Hypothesis)

Acknowledgments

• UC LEADS
• UCSB
• Jeffrey Stopple
• Those that work in UCSB summer research

programs

• Students in UCSB summer research