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Approximations to Pi, Dedekind's Eta Function and Modular Equations

Permanent Link: http://ufdc.ufl.edu/UFE0021455/00001

Material Information

Title: Approximations to Pi, Dedekind's Eta Function and Modular Equations
Physical Description: 1 online resource (40 p.)
Language: english
Creator: Ghosh, Amitava
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: iteration, maple, pi
Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: In 1989, Bailey, Borwein and Borwein gave a method for constructing series and algorithms that converge to $\pi$ to high order using Jacobi's theta-functions. In 1997, Borwein and Garvan developed a family of functions $\alpha_p(r)$, defined in terms of Dedekind's eta-function, for constructing $p$th order algorithms that converge to $\pi$. Their method involved finding initial values $\alpha_p(r_0)$ and finding certain modular equations. Their paper included some experimental values of $\alpha_p(r_0)$. In this thesis a method for finding and proving the validity of such evaluations is given. We confirm Borwein and Garvan's evaluations as well as giving some new ones.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Amitava Ghosh.
Thesis: Thesis (M.S.)--University of Florida, 2007.
Local: Adviser: Garvan, Francis G.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021455:00001

Permanent Link: http://ufdc.ufl.edu/UFE0021455/00001

Material Information

Title: Approximations to Pi, Dedekind's Eta Function and Modular Equations
Physical Description: 1 online resource (40 p.)
Language: english
Creator: Ghosh, Amitava
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: iteration, maple, pi
Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: In 1989, Bailey, Borwein and Borwein gave a method for constructing series and algorithms that converge to $\pi$ to high order using Jacobi's theta-functions. In 1997, Borwein and Garvan developed a family of functions $\alpha_p(r)$, defined in terms of Dedekind's eta-function, for constructing $p$th order algorithms that converge to $\pi$. Their method involved finding initial values $\alpha_p(r_0)$ and finding certain modular equations. Their paper included some experimental values of $\alpha_p(r_0)$. In this thesis a method for finding and proving the validity of such evaluations is given. We confirm Borwein and Garvan's evaluations as well as giving some new ones.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Amitava Ghosh.
Thesis: Thesis (M.S.)--University of Florida, 2007.
Local: Adviser: Garvan, Francis G.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021455:00001


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APPROXIMATIONS TO xr, DEDEK(IND'S ETA FUNCTION
AND MODULAR EQUATIONS


















By

AMITAVA GHOSH


A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE

UNIVERSITY OF FLORIDA

2007


































S2007 Amitava Ghosh



































To my most loving parents, Pranab K~umar Ghosh and Mita Ghosh










TABLE OF CONTENTS


page

5

6i


LIST OF TABLES . .

ABSTRACT .....

CHAPTER

1 INTRODUCTION

1.1 Algorithms For Computing 7r
1.2 c4,(r) and p-th Order Iteration
1.3 The Goal

2 DEDEK(IND'S ETA-FITNCTION

2.1 The Function c;>
2.2 Modular Forms
2.3 Transformation Formulas ..
2.4 Finding Values of oz,(r) ...

:3 MAPLE PROGRAMS

:3.1 An Example ......
:3.2 MAPLE Steps .. ...
:3.3 MAPLE Code .. ...

4 EVALUATIONS OF EgyP(1/NV)

REFERENCES ........

BIOGRAPHICAL SKETCH ....


Construction










LIST OF TABLES

Table page

1-1 Iterations of the Borwein quartic algorithm ..... .. .. 8

1-2 Iterations of the Borwein quadratic algorithm ..... .. 11

4-1 Initial values a~,(ro) . ... . .. 37

4-2 Values of Ew,,(1/NV ) for NV = 2 ......... ... 37

4-3 Values of Ew,,(1/NV ) for NV = 3 ......... ... 37

4-4 Values of Ew,,(1/NV ) for NV = 5 ......... ... 38

4-5 Values of Ew,,(1/NV ) for NV = 6 ......... ... 38

4-6 Values of Ew,,(1/NV ) for NV = 7 ......... ... 38









Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Science

APPROXIMATIONS TO xr, DEDEK(IND'S ETA FUNCTION
AND MODULAR EQUATIONS

By

Amitava Ghosh

August 2007

C'I I!r: F.G. Garvan
Major: Mathematics

In 1989, Bailey, Borwein and Borwein gave a method for constructing series and

algorithms that converge to xr to high order using Jacobi's theta-functions. In 1997,

Borwein and Garvan developed a family of functions n,,(r), defined in terms of Dedekind s

eta-function, for constructing pth order algorithms that converge to xr. Their method

involved finding initial values oz,(ro) and finding certain modular equations. Their paper

included some experimental values of c4,(ro). In this thesis a method for finding and

proving the validity of such evaluations is given. We confirm Borwein and Garvan's

evaluations as well as giving some new ones.










CHAPTER 1
INTRODUCTION

1.1 Algorithms For Computing xr

In Bailey, Borwein and Borwein [3] an overview is given for a method for constructing

series and algorithms for the rapid computation of xr. These methods involve defining a

sequence {0~,}", recursively for which


lim as


Let p be an integer > 1. We w?-a sequence {a~,}", converges to as, to pth order or has

pth order convergence if



for some constant c > 0. For example, we give Borwein and Borwein's [2, p.700] quartic

algorithm. Define three sequences {0~,}, {8,}, {s(} by


Cto :=,


so = 22- 1

S, = (1 84 1/4

1s

4"
an, = (1 + s,)4 n-_1 i

Then as, converges quartically (4th order) to ~. We illustrate this by MAPLE. We write

and execute the following program where we create a MAPLE function called bb4.

bb4:=proc (n,D)
local a,m,i,diffl:
Digits :=D:
a:=1/3:
s:=sqrt (2) -1:
for i from 1 to n do
ss:= (1-s^4)^ (1/4) :
s:= (1-ss) / (+ss) :
a:=evalf((1+s)^4*a+4^i/3*(1-(1+s)^4)):
diff 1:=evalf (abs (1/a-Pi)) :










print (i, evalf (1/a) ,evalf (dif f1, 6));
od:
RETURN() :
end:

> read bb4:
> it2(2,50);
1, 3.14159264877417701592631401719666485312 970265 .481562e-8
2, 3.14159265358979323846264338327950288419731923 .362975e-40


Table 1-1. Iterations of the Borwein quartic algorithm



1 10-s
2 10-40
3 10-170
4 10-694
5 10-2789
6 10-11"I1



We see that the digit accuracy quadruples with each iteration. This illustrates the 4th

order convergence. Computation of gives xr correctly to 11, 170 digits. It turns out that

in the algorithm above an, = a~(16".4) where a~(r) is a certain function defined in terms of

the classical theta functions.







Then :Cqn+)

where = exp-xf) ad 04 9 4(q). n [3] orwiBrenadBie eeal


to xpes a~2T il trm o a~ ndvaious tht functos tlzn -re oua









equations for the theta functions, they were able to construct p-th order iterations that
converge to

1.2 a~,(r) and p-th Order Iteration Construction

In [5], Borwein and Garvan defined a family of functions a~,(r) for each p > 2, and
showed how to construct p-th order iterations converging to ~. The definition of a~,(r) is

given in the next chapter. And it turns out that


Qp(P> =Qp(~.Tnpp(~ 3

where m,,,(r) is defined in (2-14). For a fixed initial ro and p > 2 define the sequence

{a~n} by



Then as, converges to to p-th order. To construct the corresponding p-th order iteration,
we need 3 ingredients:

(1) Initial value a~,(ro)

(2) Initial value mp,p (ro)

(3) a modular equation giving mp,,(p2T) ill terms of 772,,(r).

(1) We can ah-- .1-< take ro = and so cto := a~,(l-) = ~. See (2-20). (2) Also,

mp~~r_) =p. See (2-193). (3) It turns out that there is ah~-~-,1 an algebraic relation between
mp,,(p2r) and m,,,(r).
EXAMPLE (p = 2)


m(q2) =(1-2)
1 + (4 -- m(q)) (2 + mI~q)

Here, m(q) = m2,2 T), Where q = exp(-2xfz/z/ ). We define two sequences {a~,},

{m,} by

{0~,} := a2~(22n-1)










and


{ni,z := v22,2(22n>

Then by (1-2), we have

m,z = (1-3)

and by (1-2), we have

a~n = mn-Q-la~- 1-mt1 (1-4)

Equations (1-2), (1-3) give a quadratic iteration such that





We illustrate this by 1\APLE.

quadit :=proc (n, D)
local a,m,i,diffl:
Digits :=D:
a:=1/3:
m:=2:
for i from 1 to n do
a:=evalf(m*a + 2^(i-1)/3*(1-m)):
m:=4/(1 + sqrt( (4-m)*(2+m) )):
diffl:=evalf(abs(1/a-Pi)):
1print(i~evalf(1/a),evalf(diffi,6));
od:
RETURN():
end:

> read quadit;
> quadit(5,45);
1, 3.000000000000000000000000000000000000000000 .141593
2, 3.14075448203408147040144747789402692448 8209 .838172e-3
3, 3.141592646824848796086496902723483686627542 .676494e-8
4, 3.141592653589793238286947213656120849501961 .175696e-18
5, 3.141592653589793238462643383279502884197152 .5358e-40

We see that the digit accuracy at least doubles with each iteration. This illustrates

the quadratic convergence. Computation of --gives xr to 1391 digits. A table of the first

10 iterations is given helow in Table (1-2).


1.3 The Goal

In this thesis, we concentrate on the evaluation of cn,(ro) which is the first step in

iteration construction discussed in the previous section. When N is a positive integer, we










Table 1-2. Iterations of the Borwein quadratic algorithm



1 < 10-]
2 < 10-~
3 < 10-8
4 < 10-ls
5 < 10-40
6 < 10-s
7 < 10-170
8 < 10-344
9 < 10-"9
10 < 10-1392



prove that ni,(NV) is algebraic. See Theorem 1.4.1. We also give an algorithm for finding

the nmininmal polynomial of cz>(NV) (see ('!s Ilter 2). In [5], some evaluations of cz>(NV) were

given without proof. We rigorously verify these evaluations and find new evaluations using

1\APLE. Tables of these new evaluations are given in C'! Ilpter 4.









CHAPTER 2
DEDEK(IND'S ETA-FUNCTION

2.1 The Function c0,

In this section we describe Borwein and Garvan's [5] infinite family of functions cp,,

where p is any integer greater than 1. Theoretically it is possible to find an equation

relating a~,(NV2T) With a~,(r). This relation is particularly nice when NV = p. This will give

rise to p-th order iterations with a nice form. The functions c0, are constructed from the
Dedekind eta function instead of the theta functions.

Let q := exp(2ri-r) (with S-r > 0). As usual the Dedekind eta function is defined as


7)(7) := exp(dir/12) Ir(1 -exp(2aii7))) (2-1)


= 124_n) (2-2)

Then

q(-1/ ) = ~ r).(2-3)

See [1, p.121] for a proof. Now for p > 1 (a positive integer) we define


B,3(rJ):= C():. (2-4)

where -r = izf/~ and q = exp(-2xfz/ ).) It should be noted that the functions B3

and C3 occurred naturally in the Borwein-Borwein cubic iteration [2], [4]. Define


a,(r) : (2-5)

where

Ap(r) := q p2 (2-6)
p 1C B (6

Here B = Ji~. From (2-2) w~e have


Ap(r) = 1 + O(q), (2-7)









and


4,(1/) = r,(r),(2-8)

which follows from (2-3). The definition of c0, was chosen so that it had a form analogous

to that of (1-1) and that it satisfied a transformation like (1-1) below. Using (2-3) and

(2-8) it is not hard to show that


ap(1r) (2-9)

Substituting r = gives
p+1
ap (1) = ,(2-10)
6~T
Since q 0 as r oo we see that


lim a,(r)(2-11)
r->oo T

The following theorem is given in [5, p.93].

Theorem 2.1.1. Let N, p > 1 be jfixred. We have


a (NV27) p F) mN,p (r)+ e N,,(r), (2-12)

where
egl -N I,~ ~q- 4B ( N (2 -13)



mN (2-14)
'"A,(NV2T)
Further

A, = [P(q') P(q)] (2-15)
p-1
where

P(q) := 1 -24 1 "~ = 24q (2-16)









Proof. The statement (2-12) follows easily from (2-5) and (2-6). Equation (2-15) follows

easily from the product expansion (2-2) and the definition in (2-4). O

When NV = p, the function ey, has a nice form


6p~v(r) (1- mp, (r) ,(2-17)

so that

a,(p"r) = a,(r)m~p,(r) + (1-m,(). (2-18)

The proof of (2-17) follows easily from (2-2), (2-4) and (2-15). From (2-8) and (2-14) we
have

m,,,p(1/p) = p. (2-19)

By using (2-18) and (2-19) we find that (2-9) and (2-18) give rise to two equations

involving ap,(p) and a~,(1/p). These equations may be solved easily to yield


a~p(p) = ap(1/p) = (Not a bad starting point for 1/xr). (2-20)

We know that

q(-/r)= (). (2-21)

Now let's prove

A,(1/) = r,(r),(2-22)

PROOF : (Stepl)

Let's find B,(-1/p-r) in terms of Op(-r).

Now,


B?3r, -1~ (2-23)















Step 2: Let's find C1)(-1/pr) n11 termsI of Bp(Tj.
Now,


C, _~ (2-24)












Step3: Lt's ind (--) in termls of 7, Opi(T) an1d C(i). (please n~rot that the in 'ir

mens() If we calculate (-p-1), we will see th~at:

BI 1 pC )7 2
p~ pr2 (7) (2-25)
B?3 pr 2 C/,

Now it's easy to see that as r goes to i, p is mapped to --

Step 4: Let's find q ( ) in sterns of r anld Cj(r).


B1 ( ) =t 9- )9 ()

B?3 p C/, 2.2xri T
C, T(p 1)2/
=-r(q ) +
C, 4 x

Step 5:

Let's findl q ( ) in terms of r and Bp(r).
We know that:











C(- 1) = 7 i p B(r)
p-r

1 1-, 1 p p-3 p-1
C'( )= 2p 2 [- 2 B,3(r) + 2B()
p-r 2


BI 1 p lr -2 C
B?3 pr C/,


B,1~ C,(~ 2~(p 1) 4
B?3 C/, 4 x
So,



1,I 24 C ( B
r p2 _- C r B
24 C B
r(q q
p2 -
24 C B
= r( ( ))
p2



We can show that

T T

When r = 1, we have
p+1
a,(1) =
6T
Since q 0 as r oo, we see that:


lim ap(r) =
r->oo iT

Now, we define
Ap(r)
'"" Ap(N27.









When NV = p and r = ,


A, ( )
"'"~ = 24(p2 1)

A() p.Az>(p)
A (p) Ap>(p)

Also, we have

a,(p2r (rp,()+~ 112~()
When r = -1, we get:



or, az(p) = a,( ) + ~(1 -p)

or, (3p)a,( ) = 3n,(p) + (p )
Also, we know that:



Putting r = p, we have :






3p~p() p+ 1 -3app)

Fr-om the two equations, we have :




Now, let's prove

P(q) := 1 24~ = 24q r
~1 q" rl
PROOF: We know that


n=l











log(1p~
n=l


log rl = log(q) +

Now, differentiatingf both sides w.r.t q,


1 0 -, .4 ,
24q 1 q" '


where il =
dq


00

(iC
n=]


1 2 4 ~ l q q
n= 1


24
r


Also let's prove :


~P(q")


P(q)l


p-l


p- 1 1-q 4"


p-1 (1 24
p-1 1


= -24p 4-1 24 .41
p-1- 41-" p-1 1- q"
n=] n =

Now's let's evaluate the left hand side (LHS) i.e. Az,. We know that


24q C
A, (r) =(
p2 _- 1


Now,


Br) = r


up)


- q")"
- qP")


So, log B, = p Cn1log(1


q") Clog(1


q'")


If we simplify the right hand side, we get :


ap~/24 -,1
4/24 ,<


-1(1 1
(~ 1











Differentiatingf both sides w.r.t. q, we get the following :


n= 1


-1 0 a
Sqn C1 qP'
n= 1


-1q 1"
n= 1


-1
"


n= 1


Now,


q~' II~~p /24 -1,c 9pn p
q1/24 -1 9


f) log(q) + p log( 1
n=1


n=1


P2
log(C,) = (


Differentiatingf both sides w.r.t. q, we get:


C, p2 _-
C, 2 4 q


n=


_ pnq"-1

1 q P


n=]


n~q-1

1 q "


So, now we have


24q C
A,(r)> = (
p2 _- C


24 C
2 (q


B
q


24 p
S- 1 24


1 qP"
n= 1


1 q" ~q
n= 1


ns
1 -
n= 1


qnn- ii 1 q~mn~
n= 1


24 p2 1
p2 1L 24 -p(p 1 ) 1
n= 1


qP~" (p + 1) 1
n= 1


n.4


- q"


1-
p-1 1 4nqP"


p- 1 1- q'n


=Right Hand Side (hence proved)

2.2 Modular Forms

Let 'F denote the complex upper half plane. SL2 ~) aCtS tranSitiVely on 'F by linear

fractional transformations











AT r = where A =b E SL2()
c-r + d'

and -r E N. Let N be a positive integer. We define


F(NV):= (: E SL2(Z) : a -da 1 (mod NV) and b -c- 0 (mod NV)


so that L(1) = SL2(Z). A subgroup of L(1) is called a congruence ;,1by.. ;,1 of level N if it

contains L(NV). We will be concerned with the following congruence subgroup


fo(N) :=e (1) : cE 0 (mod NV)


Let k be a non-negative integer and let L' c L(1) be a congruence subgroup of level N

with [L(1) : L'] < 00. Also, L(NV) c L'. A function f :'F i C is a modular form of weight

k for L' if

(i) f is holomorphic on N~,

(ii) f(A-r) = (c-r + d)k/ Tr) for all A = ,

(iii) f is holomorphic at the cusps A(oo) for Ae E (1)
i.e. there is an expansion


(cr' + d)-k f (A-r) = )1ane2xinr/N
n=o

for Im(-r) > 6.

We let M. i (L') be set of such modular forms. It turns out that this is a finite

dimensional C -vector space. We will need the valence formula


Cord( f ,(,0) = 1









if f / 0, fe E T. (F') and FT* is a proper fundamental region for F'. See Theorem 4.14

(p.98) of Rankin's book [9]. H% will need


[F() FoN) =N I(1 + -1)
plN

See Theorem 4.2.5 (p.106) of Miyake's book [8].

2.3 Transformation Formulas

We are going to assume throughout that p is prime. H% define the following




Here, f '(7) = f(7). If f (7)= F(q) where q= e2xir then

dF
f'(-r) = 2xriq and
dq


dF 1


H% note that

i 1 C' B


Let A = bcdE F(1), so that

a-rt +b
ATr =
c-r + d
and
d(A-r) 1
d-r (c-r + d)2
H% need the following

17(Ar) = (a,bh,cl d) -i~cr +d~jl (r)

See theorem 3.4 (p.52) of Tom Apostol's book on Modular Functions and Dirichlet

Series [1].















































Now,


Proposition 2.3.1. For p prime, F,(r) is a modular form of weight 2 on Fo(p).

Proof. Let A = e b (1). Then we get the following
c d


rl'(A


dA -cqr
Ir) = -ic + q()
d~r 2 --is(cr +

rl'(A-r) dA '()c/2



o'(Ar) o'() 1
= (c-r +d)2. -C (C + d)
rl(A-)qr) 2


B = a E fo (P)



B* ~ aP pp E F(1),




01 0"". 1


pB() = B*(p-r)


so that



and



Now let



so that


(2-26)


We have the following


rl'(B-r) 2 rl('
=(YT + 6) + Y(Y' + 6)
rl(Br) rl(') 2


rl'(pB-r) rl'(B*p-r)
rl(pB-r) rl(B*p-r)


= p[(y-T + 6) +(ylT + 6)]
rl(p-r) 2 p











=p(y-r + 6)2 + (y-T + 6)
fl(p-r) 2
Hence, F,(B-r) = p(y-r + 6)2 2'().

i.e. F,(B-r) = (y-T + 6)2 pp-r

So, F,(-r) is a modular form of weight 2 on Fo(p).


Proposition 2.3.2. For p prime and N r .;, integer > 2, / .

B',(r)-r B(N
Bp(r) Bp(NT~)

Then, G~,,(r) is a modular form of weight 2 on Fo(pNV)


Proof. Let A = E Fo(P)

Then we have
p-1
B,(A-r) = X(d)(c-r + d) z B,(-r),
where X(d) = ( ) (Legerndrer symbol modl p) See Corollary (2.2.12) in [6].

So,


and


Noc~r + NVp a~(NT-) + NVp
NVAr =
y-7+ 6 X((NT)+ 6


and


dA e-1 p- 1 p-3
B',Ar) = X(d)(c-r +d) z B'- ()+X(d)( 2)(c-r +d) z B,(-r)



=(c~r + 1)2 + ~c(cr + d)v
B, ( A ) B, (7-) 2

Let A =s a E fo(pNV) so that


A* =a Np E fo(P)










BJ(Ar)
B,(A-r)


2- P~ r$,
(Y' + 6)2 + ( +)
Bp(-r) 2


BJ(' r B (NT) p -
N) = ~' N(TT + 5)2
B,(NVAr) BN) 2

So, G,,w(A-r) = (y-T + 5)2G,, (-r)
Therefore, G3,,w(r) is a modular form of weight 2 on Fo(pNV).


Proposition 2.3.3. Let p be prime and NV > 1. Then F,(-

1C (VrC(NT)


-pi) = -pr2 Fp')


Proof. By (2-26),


F, -~
= -~



pn [7 +
q l(')


rl(' +r


p 1


2 a l(p-r) 2-


29l (p
-p[p-r .


2 ll'
rl .


B,(- T)
-1) _


By (2-25),

pl-

= ~12-r2CVL(NT) p(p


- N
B (- )
1p-2OpT p"p 1
C,(r)2










C" (N-)
pNT-2 1


C" (7)
C, ( r )


O

Proposition 2.3.4. Let p be prime and NV > 1. Then F,(NTr) is a modular form of weight


2 on Fo(pNV).


Proof. Let

and


E fo(pNV), so that A*


E fo(P)


NV(a~r +P)
ypNT r+6


~(NT-)+ +Np
yp(NT-) +6


NA-r


F,(NVAr) = (yp(NT-) + 5)2F,(Vr


which implies that F,(NT-) is a modular form of weight 2 on Fo(pNV).


2.4 Finding Values of a~,(r)

The next proposition shows that computing a,(Nli) reduces to computing ei,p( ).


Proposition 2.4.1.

~(, = 1 (p + 1) N 2~

Proof. From (I-8), (I-14) we have


1 A
mN,1, ( ) -

), we have

a,(NV) = ap,()mN,p(


I, ()
,(NV)


1
) +


1 1


eg,( )v


From (I-9) and (1-12


1
Noc, ( ) +
NV


1 1
eg, ()
z/l N


pN6


1 Nplj










(P + )z~ 1 1
S- 0, (NV)+ ey,, ()
34 #
and the result follows. O


p+ 1

where
q (q) Nq" i(q")
q"p(~ (q)-q"("



(p + 1)F,(NT-)
Here using: q = e""xi r, T = NOWv,


6 N,p( 1 Gp,, (- )








=1Vj 4C~q (q) qVcp ~ (qN)

We note that

1V~

when r = J We let
C'l C'l
Gp,,s(7) N (NT-) ()

Then G3,,w(r) is a modular form of weight 2 on Fo(pNV).

Theorem 2.4.2. Let p be prime and N > 1. For suff:.~l; .. ;.01; elar.- there is a r-' '7,:~~,,:r

P(x) E C[x] of degree 2e such that

P (EN,p(1/NV)) = 0









Proof. Let p, NV be fixed. For e > 1, consider the set of monomials


(2-27)


for 0 < i, j I e. The number of such monomials is (e+ 1)2. Each monomial is a modular

form of weight 4 on Fo(pNV). The dimension of this space of modular forms is ~ -e. So for

sufficiently large the monomials in (2-27) are linearly dependent. Since the coefficients

of all q-expansions are rational, there exist aij EQ~ (not all zero) such that


as,, G,,,w()"((p + 1)F~,(NT))ti(NG,',,(rj))(-(p + 1)F,,(r))' =


and


(2-28)


where


P(x, y) = asyi y l

Therefore, x = eg,( )\ satisfies the ponlyrnomial


(2-29)


a nyxi+y = .


P(x) = P(x, x)


Fr-om the proof of the theorem, we see that we need to find a polynomial


ti, j) (xi (x y y )


Q (X ,2, 92,92)


(2-30)


with rational coefficients such that


Q Gwr) (p+ )F(NT, Gw~r, (p+ 1Fr)=


P EN,,(r), EN,,(-) = 0.









Because of the transformation formulas, we assume that aij = aji for all i, j. So, now
we consider monomials




for 0 < i, j < e and






( ,:(~((p + 1)Fz,(NT))"-'(Nz,,,n(r))"- r,(r)-(( p + 1)F,, >(T))




for 0
We can use MAPLE to find the polynomial in (2-30). We compute the q-expansion

up to qT for each monomial hk Tr) in (2-31) and (2-32).



hk(r) = bj~k q + O(q ~+) (2 33)
j=0
for 1 < k < C1). Here T> >(

Let

B (bj~"k 05jgT, 1ci
We choose I large enough so that B has a non-trivial nullspace. We use MAPLE to

find a vector :Fe Qk With :Fe N (B). This gives a linear relation between the monomials

hk Tr) and the polynomial in (2-30). If the fumetions hk Tr) are linearly independent then
some linear combination

h(r) = ,6 ki~h(r) 0. (2-35)

satisfies vio,(hz) > 21









By the valence formula,


( + 1) 1>1 '
2 5 ih < odh,( ')= 1 (2-36)



1 + (2-37)
t Np, t prime
where P' = Fo(pNV).

So,

+ 1 1 +- (2-38)
tlNp, t prime
Hence if
2NVp I1\
> 1 + (2-39)
-3 tl
t|Np, t prime
th~e fulmetion~s hk T)j will be linea~rly depen~dent. Now for a. given T with? T > '~, w

assume that we have used MAPLE to find x' E N(B) and hence a candidate polynomial

Q(X1, X2, Y1, Y2 -

Lect L(r) = Q G,;,(r( i) p+1F(TN',~) ( ),r
Then ,

L( pl-- 24L (2-40)

since we have assumed that the coefficients aij of the polynomial Q are symmetric in i and

j. It follows that

uso(L) = In (L). (2-41)

Hence to show that L = 0, we need only show that

[F : Fo(pNV)] Np ~ 1\
uso(L) > 661+ (2-42)
tlNp, t prime









CHAPTER 3
MAPLE PROGRAMS

3.1 An Example

As an example we show how to find and prove the evaluation

2752/
22,11(1/2) (3-1)
102

First we compute the first 100 decimal places of E2,11(1/2) and use the minpoly

MAPLE function to find a quadratic polynomial with small coefficients that "fits" this

evaluation. The minpoly function utilizes the Lenstra-Lenstra-Lovasz [7] lattice reduction

algorithm.

> xil :=evalf (epsilon0 (2, 11, 1/2)) ;


x11:= -3.812; :1II. 1~1 i. E: 119653372II.E.I1:'1 :C392809780953790\

266678783032151710924664419552518025154117454


> minpoly (x1, 2) ;

-75625 + 5202X2

> solve(%,2);
275 275
102 102

From this calculation we suspect the evaluation given in (3-1). The code for all MAPLE

functions used is given in section 3.3. Our MAPLE function epsilonO(N~, p, r) computes

EN,p(r) to a 100 decimal places.
First we compute q-expansions of


Gp,,(-r), F,(NTr), Gp,,(r), Fp(r)

for NV = 2, p = 11.

> dopl(2,11,100):









Here, dopl(NV, p, T) computes the q-expansions of these functions up to q Next we

find e large enough so that the matrix B in (1-31) has non-trivial nullspace.

> newdop3(5,2,11,101,60):


No polynomial relation found


Note that newdop3(e, NV, p, t, T) computes the nullspace of B module t for a large prime

t. Here, e= 5 gives a zero nullspace and hence no polynomial relation mod t. By trial and

error, we use newdop3(e, NV, p, t, T) to find the correct candidate for Now, let's do the

next step.

> newdop3 (6, 2, 11, 101, 60) :

This gives a polynomial relation mod t for = 6. We need to recalculate over Z. We

also need to find the minimum number of terms to prove the result.

> numterms(6,2,11);

36;



Here, numrterms(, N, p)j comnputes [r~r( which is the right hand side of (1-41).
So we must take T > 36.

> P:= newdop2a(6,2,11,50);

We successfully found P(X, Y) in this case.

> f:= factor(subs(Y=X,P));

(-75625 + 5202X2)(18X2 1925)2(X 10)6
f :
1685448

The MAPLE function newdop2a(, NV, p, T) finds the polynomial P(x, y) in (2-29) and

checks that (2-28) holds up to q If T > numterms(e, NV,p), this calculation proves

that (2-28) holds. In the calculation above, we have proved that the minimal polynomial

of E2,11 mu\1?1t~ dividel then ponlynomiarll f. A;l litt nle ;r chc ilng show that then m~~iniml









polynomial is

P(x) = 5202x2 75625

which proves the evaluation t2,11 -f

3.2 MAPLE Steps
To find and prove anl devaluation of Ey,Vp (J~ "; we,,, perfor th fllowing steps.,

STEP 1: Find numterms(e, NV,p).

STEP 2: For T > numterms(e, NV,p) use dopl(N~, p, T) to calculate the q-expansions





upto q .

STEP 3: Ch....~--- a large prime t and use newdop3(e, NV, p, t, T) for different values of

until we find B with non-trivial nullspace mod t.

STEP 4: Use newdop2a(, NV, p, T) to find and verify the polynomial P(x, y) in

(2-29).
STEP 5: Factor the polyvnomnial P(x, x) = P(x). EN,, (l ) will be,.,,, a ,,,, roo ofon f h

factors. Use the MAPLE command evalf to find the correct factor.
STEP 6j: If possible, try to fatctorize in order to identify tnw,, (() in~ terms- of radicals.





3.3 MAPLE Code
epsilonO:=proc(N,p,r)
local BB,CC,qq;
qq:=exp (-2*Pitsqrt (r) /sqrt (p)) :
BB:=qlogdiff (B(p)):
CC:=qlogdiff (C(p)):
(BB (q)-N*BB sN>C(q^N)) /(CC(q^N) -BB(q^) /(4 (^2-)
subs(q=qq,%):
end:

dopi:=proc(N,p,T)
localEPL;
globalX1,X2,Y1,Y2:










EPL:=NEWEPFUNCS(N,p,T):
X1:=EPL[1] :
X2:=EPL[2] :
Y1:=EPL[3] :
Y2:=EPL[4] :
RETURN():
end:

Digits :=100:
with(polytools):
etaq:=proc(q,i,trunk)
local k,x,zi,z,w:
zl:=(i + sqrt( iti + 24*trunk*i ) )/(6*i):
z:=1+trunc( evalf(zl) ):
x:=0:
for k from -z to z do
w:=itk*(3*k-1)/2:
if w<=trunk then
x:=x+ q^( w )*(-1)^k:
fi:
od:
RETURN(x):
end:

findpoly2:=proc() #This proc looks for a polynomial
local
x,y,q,deg1,deg2,ARGLIST,TYPELIST,CORRECTLcekdmdi2nmkjBqPOY
kkk,i,POLYg,POLYfunc,ss,numtyp,tA,0pkk,POLSTIDITCFA~ndid~n2
global A,X,Y,kk,AQ,COFMATSET:
1print('WARNING: X,Y are global.');
## This version assumes coeff matrix is symmetric
#relation between x,y of degree deg1 in x, and degree deg2 in y.
#The relation is checked to order 0(q~check).
if nargs<5 then
ERROR(' number of arguments must be 5 or 6.');
fi:
x:=args [1] :
y:=args [2] :
q:=args [3] :
degi:=args [4] :
deg2 :=args [5] :
ARGLIST:=[q,deg1,deg2];
TYPELIST:=map(whattype,ARGLIST);
POLYSET:=:
INDLIST:=[]:
COFMAT:=array(1..(degi+1),1..(deg2+1)):
COFMATSET:=:
CORRECTL:=[symbol,integer,integer] :
if TYPELIST=CORRECTL then
if nargs=6 then
check:=args[6]:
fi:
if nargs>6 then
ERROR(' findpoly can at most 6 arguments');
fi:
dimi:=(degi+1)*(degi+2)/2:
dim2:=dimi+10:
print(' dims ', dimi, dim2);
A:=array(1..dimi,1..dim2):
AQ:=array(1..(degi+1),1..(deg2+1)):










num:=0:
for k from 0 to deg1 do
for j from k to deg2 do
num:=num+1:
if i=j then
B[num]:=X~k*Y~j:
qq:=series(x~k*y~j,q,dim2+10):
else
B[num]:=X~k*Y~j+X~j*Y~k:
qq:=series(x~k*y~j+x~j*y~k,q,dim2+10):
fi:
INDLIST:=[op(INDLIST),[k+1,j+1]] :
AQ[k+1,j+1] :=degree(convert(qq,polynom),q):
for 1 from 0 to (dim2-1) do
A[num,1+1] :=coeff(qq,q,1):
od:
od:
od:
tA:=1inalg[transpose](A):
kk:=1inalg[kernel](tA):
if kk= then
printt' NO polynomial relation found. )
else
for opkk in kk do
POLY:=0;
##0pkk:=op(kk);
kkk: =opkk;
for i from 1 to num do
POLY:=POLY+B[i]*kkk[i] :
indd:=INDLIST[i] :
indl:=indd[1] :ind2:=indd[2] :
COFMAT[indi,ind2]:=kkk[i]:
od:
COFMATSET:=COFMATSET union COFMAT:
POLYg:=0;
for j from 0 to deg2 do
POLYg:=POLYg+factor(coeff(POLY,Y,deg2-j))*^dg-)
od:
if nargs=6 then
print('The polynomial is');
print(POLYg);
POLYfunc:=unapply(POLYg,X,Y):
ss:=series(POLYfunc(x,y),q,check+1):
print('Checking to order',check);
print(ss);
##RETURN(POLYg):
else
print('The polynomial is');
fi:
POLYSET:=POLYSET union POLYg:
od:
RETURN(POLYSET):
fi:
else
numtyp:=nops(TYPELIST):
ERROR(' Wrong type of argument. ARGLIST has type ',seq(TYPELIST[i],
i=1..numtyp), 'It should have type',seq(CORRECTL[i],i=1..numtyp)); fi:
end:


dop2:=proc(1,N,p,T)










local dd;
dd:=dimpoly(1,p,N):
print ("dd= ",dd);
findpoly2(L1,L2,q,1,1,T);
RETURN(%) :
end:
newnewdop2:=proc ()
local dd,1,N,p,T;
1:=args [1] :
N:=args [2] :
p:=args [3] :
LP:= args [4] :
if nargs=5 then
T:=args [5] :
fi:
dd:=dimpoly(1,p,N):
print ("dd= ",dd);
NL1:=modp (L1, LP):
NL2:=modp (L2, LP):
if nargs= 5 then
newfindpoly3(NL1,NL2,q,1,1,LP,T);
else
newfindpoly3(NL1,NL2,q,1,1,LP);
fi:
RETURN(%) :
end:


dimpoly:=proc (1,p,N)
local S,x,t;
S:=numtheory [f actor set] (p*N) :
x:=1:
for t in S do
x:=x* (1+1/t) :
od:
RETURN(xtp*N*1/3);
end:


cosetdim:=proc (N)
local S,x,t;
S:=numtheory [f actorset] (N) :
x:=1:
for t in S do
x:=x* (1+1/t) :
od:
RETURN(x*N);
end:


numterms:=proc(1,N,p)
l*cosetdim(N*p) /6;
end:


newdop2:=proc ()
local dd,1,N,T;
global p:
1:=args [1] :
N:=args [2] :










p:=args [3] :
if nargs=4 then
T:=args [4] :
f i:
dd:=dimpoly(1,p,N):
print ("dd= ",dd);
if nargs= 4 then
newf indpoly2( [X1,X2], [Y1,Y2],q,1,1,T);
else
newf indpoly2( [X1,X2], [Y1,Y2],q,1,1);
f i:
RETURN(%) :
end:

newdop2a:=proc ()
local dd,1,N,T;
global p:
1:=args [1] :
N:=args [2] :
p:=args [3] :
if nargs=4 then
T:=args [4] :
f i:
dd:=dimpoly(1,p,N):
print ("dd= ",dd);
if nargs= 4 then
newf indpoly2a([X1,X2], [Y1,Y2],q,1,1,T);
else
newf indpoly2a([X1,X2], [Y1,Y2],q,1,1);
f i:
RETURN(%) :
end:





CHAPTER 4
EVALUATIONS OF Ew,p(1/NV)

In this chapter we tabulate the results for evaluations of Ew,,(1/NV) using the methods
of Cl'I I'lter 2.

Table 4-1. Initial values a~,(ro)


p ro ap,(ro)
2 3 (32 224)/12
2 4 (32/ 2)/7
2 5 (321 42/)/12
2 6 (132 12)/33
2 6 (132 12)/33
3 2 (22 24)/9
3 4 (40~ 4)/9
3 5 (1021 1424)/45
3 6 (282/ 924)/75
3 7 (2 /2- 22/ 1)/9


ap(ro)
(32 2)/6
(102 9)/24
(621 52/)/30
(21 /1- 2520)/105
(212 20)/39
(902 17 /)/348
(12 /2- 142/ + 7)/63
(2002/ 336)/525
(452/ 1424)/81


J49/27+ (1/3) /j


(14/9)@


Table 4-2. Values of Ew,,(1/NV) for NV

3 J
5 ~


e numterms :oPN
2 4 12
2 4 12


4 24
6i 36i
6 42
8 72
14 210


11 275 2
102



29 539 2
111


where :i= $5617+636 8


Table 4-3. Values of E ,,(1/NV) for NV


p E3,p 1 3
3
5
7 -140\+ 2
11 -675 +200 T
386
13 741+546


Il umterms
2 4


8 64
8 75


[r : ro(pN)1
12
24
32















Table 4-4. Values
p e5p(/)
2 2
3 4
5 t 2


of tN,p(1/N)
numterms
6
16
10


Table 4-5. Values
p E6,p (1/6)
2 4 7s
3 4+ 6


of tN,p(1/N)
numterms
16
36


Table 4-6. Values of Ew,,(1/NV) for NV = 7
p 87,p(1/7) -e numterms [F : Fo(pV)1


2 "`
8
3 -140+J2
18


4 16i
4 22


for NV = 5


18
24
30


for NV = 6i
[F : Fo (pV)1
24
36;









REFERENCES


[1] Tom M. Apostol, M~odular Functions and Diricklet Series in Number The .-;
Springer-15 11 I_ 2nd edition, 1990.

[2] J.M. Borwein and P.B. Borwein, A cubic counterpart of Jacobi's identity and the
AGM, Trans. Amer. M~ath. Soc., 323 1991, 691-701.

[3] J.M. Borwein, P.B. Borwein and D.H. Bailey, R Il.. Ilriii .Il modular equations,
and approximations to pi or how to compute one billion digits of pi, Amer. M~ath.
M~onib~le; 96 (1989), 201-219.

[4] J.M. Borwein, P.B. Borwein and F.G. Garvan, Some cubic modular identities of
R .... ...ni ii Trans. Amer. M~ath. Soc., 343 1994, 35-47.

[5] J.M. Borwein and F.G. Garvan, Approximations to xr via the Dedekind eta function.
Organic M~athematics (Burna~l.;; BC, 1995), C'jS Conf. Proc., 20, Amer. Math.
Soc., Providence, RI, 1997, 89-115.

[6] F.G. Garvan, Some congruences for partitions that are p-cores, Proc. London M~ath.
Soc. (S) 66, no. 3, 1993, 449-478.

[7] A.K(.Lenstra, H.W.Lenstra and L.Lovasz, Factoring polynomials with rational
coefficients. M~ath. Ann. 261, no.4, 1982, 515-534.

[8] Toshitsune Miyake, M~odular forms, Translated from the 1976 Japanese original by
Yoshitaka Maeda. Reprint of the first 1989 English edition. Springer Monographs in
Mathematics. Springer-15 11 I_ Berlin, 2006.

[9] Robert A. Rankin, M~odular Forms and Functions, Cambridge University Press,
Cambridge-New York-Melbourne, 1977.









BIOGRAPHICAL SKETCH

Amitava Ghosh was born on December 10, 1979, in Durgapur (West Bengal), India.

He received his high school education at St.Xavier's School, Durgapur. He then went to

DAV Model School for eleventh and twelfth grade. He participated in the International

Mathematics Olympiad training camp at Mumbai in 1995 and 1996. He has independent

proofs of the Bertrand's conjecture regarding primes and the Gauss's reciprocity laws.

He also has worked on the construction of Abelian Galois groups of any arbitrary order.

In 2002, he graduated from Regional Engineering College (University of Burdwan)

with a bachelor's degree in metallurgical engineering. After completing his diploma, he

went to the Institute of Mathematical Sciences-Chennai, India as a visiting student in

mathematics. In the fall of 2004, he entered the University of Florida-Gainesville to

undertake a Master of Science degree in mathematics by research.

Currently, he finished his master's degree in mathematics from the University of

Florida-Gainesville via the thesis option.





PAGE 1

1

PAGE 2

2

PAGE 3

3

PAGE 4

page LISTOFTABLES ..................................... 5 ABSTRACT ........................................ 6 CHAPTER 1INTRODUCTION .................................. 7 1.1AlgorithmsForComputing 7 1.2p(r)andp-thOrderIterationConstruction ................. 9 1.3TheGoal .................................... 10 2DEDEKIND'SETA-FUNCTION .......................... 12 2.1TheFunctionp 12 2.2ModularForms ................................. 19 2.3TransformationFormulas ............................ 21 2.4FindingValuesofp(r) ............................. 25 3MAPLEPROGRAMS ................................ 30 3.1AnExample ................................... 30 3.2MAPLESteps .................................. 32 3.3MAPLECode .................................. 32 4EVALUATIONSOF~N;P(1=N) ........................... 37 REFERENCES ....................................... 39 BIOGRAPHICALSKETCH ................................ 40 4

PAGE 5

Table page 1-1IterationsoftheBorweinquarticalgorithm ..................... 8 1-2IterationsoftheBorweinquadraticalgorithm ................... 11 4-1Initialvaluesp(r0) .................................. 37 4-2Valuesof~N;p(1=N)forN=2 ............................ 37 4-3Valuesof~N;p(1=N)forN=3 ............................ 37 4-4Valuesof~N;p(1=N)forN=5 ............................ 38 4-5Valuesof~N;p(1=N)forN=6 ............................ 38 4-6Valuesof~N;p(1=N)forN=7 ............................ 38 5

PAGE 6

6

PAGE 7

3 ]anoverviewisgivenforamethodforconstructingseriesandalgorithmsfortherapidcomputationof.Thesemethodsinvolvedeningasequencefng1n=1recursivelyforwhichlimn!1n=1 2 ,p.700]quarticalgorithm.Denethreesequencesfng,fsng,fsngby0:=1 3s0=p

PAGE 8

IterationsoftheBorweinquarticalgorithm 2)23(q):=1Xn=qn24(q):=1Xn=(1)n:qn2Then dq4(q).In[ 3 ],Borwein,BorweinandBaileywereabletoexpress(p2r)intermsof(r)andvariousthetafunctions.Utilizingp-ordermodular 8

PAGE 9

5 ],BorweinandGarvandenedafamilyoffunctionsp(r)foreachp2,andshowedhowtoconstructp-thorderiterationsconvergingto1 3.See(2-20).(2)Also,mp;p(1 1+p

PAGE 10

1+p 10

PAGE 11

IterationsoftheBorweinquadraticalgorithm 5 ],someevaluationsofp(N)weregivenwithoutproof.WerigorouslyverifytheseevaluationsandndnewevaluationsusingMAPLE.TablesofthesenewevaluationsaregiveninChapter4. 11

PAGE 12

5 ]innitefamilyoffunctionsp,wherepisanyintegergreaterthan1.Theoreticallyitispossibletondanequationrelatingp(N2r)withp(r).ThisrelationisparticularlynicewhenN=p.Thiswillgiverisetop-thorderiterationswithaniceform.ThefunctionspareconstructedfromtheDedekindetafunctioninsteadofthethetafunctions.Letq:=exp(2i)(with=>0).AsusualtheDedekindetafunctionisdenedas():=exp(i=12)1Yn=1(1exp(2in)) (2{1)=q1=241Yn=1(1qn) (2{2)Then i():(2{3)See[ 1 ,p.121]foraproof.Nowforp>1(apositiveinteger)wedene 2 ],[ 4 ].Dene B C_B B):(2{6)Here_B=dB dq.From( 2{2 )wehave 12

PAGE 13

2{3 ).Thedenitionofpwaschosensothatithadaformanalogoustothatof( 1{1 )andthatitsatisedatransformationlike( 1{1 )below.Using( 2{3 )and( 2{8 )itisnothardtoshowthat 3p 6p limr!1p(r)=1 5 ,p.93]. 3p BNqN_B B(qN) C(qN)qN_B B(qN));(2{13)and :(2{16) 13

PAGE 14

2{12 )followseasilyfrom( 2{5 )and( 2{6 ).Equation( 2{15 )followseasilyfromtheproductexpansion( 2{2 )andthedenitionin( 2{4 ). WhenN=p,thefunctionN;phasaniceform 2{17 )followseasilyfrom( 2{2 ),( 2{4 )and( 2{15 ).From( 2{8 )and( 2{14 )wehave 2{18 )and( 2{19 )wendthat( 2{9 )and( 2{18 )giverisetotwoequationsinvolvingp(p)andp(1=p).Theseequationsmaybesolvedeasilytoyield 3(Notabadstartingpointfor1=):(2{20)Weknowthat i():(2{21)Nowlet'sprove (2{23)=p(1 i() 14

PAGE 15

i)pp(p) i()=(p i)p iCp()Step2:Let'sndCp(1=p)intermsofBp().Now,Cp1 (2{24)=p(1 i)pp() i)pp() i(p)=(p i)p iBp()Step3:Let'sndB0p d).IfwecalculateBp p)q_Cp 2:2i:ip 15

PAGE 16

2i1p 2Bp()C0(1 2[p1 2p3 2Bp()+p1 2B0p()]B0p C(1 B(1 Cq_B B)=r(24 C_B B))So,Ap(1 3p 6p 16

PAGE 17

3(1p)or,(3p)p(1 3p 3p(p) 3=p(1 PROOF:Weknowthat=q1=24:1Yn=1(1qn) 17

PAGE 18

24log(q)+1Xn=1log(1qn)Now,dierentiatingbothsidesw.r.tq,_ =1 24q+1Xn=1nqn1 dqor,q_ =1 24q1Xn=1nqn1 =1241Xn=1nqn p1(1241Xn=1nqpn p11Xn=1nqpn p21(_C C_B B)Now,Bp()=p() 18

PAGE 19

24)log(q)+p1Xn=1log(1qpn)1Xn=1log(1qn)Dierentiatingbothsidesw.r.t.q,weget:_Cp 24q+p1Xn=1pn:qpn1 p21(_C C_B B)=24 Cq_B B)=24 24p21Xn=1n:qpn 24p(p+1)1Xn=1n:qpn p11Xn=1n:qpn 19

PAGE 20

c+d;whereA=0B@abcd1CA2SL2(Z);and2H.LetNbeapositiveinteger.Wedene(N):=8><>:0B@abcd1CA2SL2(Z):ad1(modN)andbc0(modN)9>=>;;sothat(1)=SL2(Z).Asubgroupof(1)iscalledacongruencesubgroupoflevelNifitcontains(N).Wewillbeconcernedwiththefollowingcongruencesubgroup0(N):=8><>:0B@abcd1CA2(1):c0(modN)9>=>;Letkbeanon-negativeintegerandlet0(1)beacongruencesubgroupoflevelNwith[(1):0]<1.Also,(N)0.Afunctionf:H!Cisamodularformofweightkfor0if (i) (ii) (iii) 12 20

PAGE 21

9 ].Wewillneed[(1):0(N)]=NYpjN(1+1 8 ]. df().Iff()=F(q)whereq=e2ithenf0()=2iqdF dqandqdF dq=1 2if0()WenotethatFp()=1 c+dandd(A) (c+d)2Weneedthefollowing(A)=(a;b;c;d)p 1 ]. 21

PAGE 22

Proof. d=p 2p d=0() 2c(c+d)(2{26)NowletB=0B@1CA20(p)sothatB=0B@p=p1CA2(1);0B@p0011CAB=B0B@p0011CA;pB()=B(p)Wehavethefollowing0(B) 2(+)Now,p0(pB) 2: p(+)] 22

PAGE 23

2Bp();where(d)=(d p)(Legendresymbolmodp).SeeCorollary(2.2.12)in[ 6 ].So,B0p(A)dA d=(d)(c+d)p1 2B0p()+(d)(p1 2)(c+d)p3 2Bp()andB0p(A) 2c(c+d)vLetA=0B@1CA20(pN)sothatNA=N+N +=(N)+N N(N)+andA=0B@N=N1CA20(p) 23

PAGE 24

2(+)NB0p(NA) 2(+)So,Gp;N(A)=(+)2Gp;N()Therefore,Gp;N()isamodularformofweight2on0(pN). 2{26 ),0(1 2{25 ),Gp;N(1 2N]N[p2C0p() 24

PAGE 25

Proof. p(N)+So,Fp(NA)=(p(N)+)2Fp(N)whichimpliesthatFp(N)isamodularformofweight2on0(pN). 2p 25

PAGE 26

3p B(q)NqN_B B(qN) C(qN)qN_B B(qN)=Gp;N() (p+1)Fp(N)Hereusingq=e2i,=ip (p+1)Fp(1 26

PAGE 27

2{27 )arelinearlydependent.Sincethecoecientsofallq-expansionsarerational,thereexistaij2Q(notallzero)suchthatX0i;j`ai;jGp;N()i((p+1)Fp(N))`i(N~Gp;N())j((p+1)Fp())`j=0and 27

PAGE 28

2{30 ).Wecomputetheq-expansionuptoqTforeachmonomialhk()in( 2{31 )and( 2{32 ). 2.Here,T>`(`+1) 2.Let 2:(2{34)WechoosellargeenoughsothatBhasanon-trivialnullspace.WeuseMAPLEtondavector~x2Qkwith~x2N(B).Thisgivesalinearrelationbetweenthemonomialshk()andthepolynomialin( 2{30 ).Ifthefunctionshk()arelinearlyindependentthensomelinearcombination 2. 28

PAGE 29

2vi1(h)X2Ford(h;;0)=4l[(1):0] 12(2{36) =`Np 2,weassumethatwehaveusedMAPLEtond~x2N(B)andhenceacandidatepolynomialQ(x1;x2;y1;y2).LetL()=QGp;N();(p+1)Fp(N);N~Gp;N();(p+1)Fp().Then, 6=`Np 29

PAGE 30

~2;11(1=2)=275p 102:(3{1)Firstwecomputetherst100decimalplacesof~2;11(1=2)andusetheminpolyMAPLEfunctiontondaquadraticpolynomialwithsmallcoecientsthat\ts"thisevaluation.TheminpolyfunctionutilizestheLenstra-Lenstra-Lovasz[ 7 ]latticereductionalgorithm.>x11:=evalf(epsilon0(2,11,1/2));x11:=3:8128306828686386119653372466437938392809780953790n26667878303215171092466441955251802515411177465014>minpoly(x11,2);75625+5202X2>solve(%,2);275 102p 102p 3{1 ).ThecodeforallMAPLEfunctionsusedisgiveninsection3.3.OurMAPLEfunctionepsilon0(N;p;r)computes~N;p(r)toa100decimalplaces.Firstwecomputeq-expansionsofGp;N();Fp(N);~Gp;N();Fp()forN=2,p=11.>dop1(2,11,100):

PAGE 31

6whichistherighthandsideof(1-41).SowemusttakeT>36.>P:=newdop2a(6,2,11,50);WesuccessfullyfoundP(X;Y)inthiscase.>f:=factor(subs(Y=X,P));f:=(75625+5202X2)(18X21925)2(X10)6 2{29 )andchecksthat( 2{28 )holdsuptoqT.IfT>numterms(`;N;p),thiscalculationprovesthat( 2{28 )holds.Inthecalculationabove,wehaveprovedthattheminimalpolynomialof~2;111 2mustdividethepolynomialf.Alittlecheckingshowsthattheminimal 31

PAGE 32

2=275p 102. 2{29 ).STEP5:FactorthepolynomialP(x;x)=P(x).~N;p1

PAGE 37

Table4-1. Initialvaluesp(r0) 23(3p 32(2p Table4-2. Valuesof~N;p(1=N)forN=2 3p 624125p 324127p 24242411275p 1026363613p 38724529539p 1111421090 Table4-3. Valuesof~N;p(1=N)forN=3 31 2241255p 741624714p 66323211675p 3868644813p 387556 37

PAGE 38

Valuesof~N;p(1=N)forN=5 2p 6261837p 1541624510 321030 Table4-5. Valuesof~N;p(1=N)forN=6 27+8p 4441624344+9p 5063636 Table4-6. Valuesof~N;p(1=N)forN=7 2p 841624314p 1842232 38

PAGE 39

[1] TomM.Apostol,ModularFunctionsandDirichletSeriesinNumberTheory,Springer-Verlag,2ndedition,1990. [2] J.M.BorweinandP.B.Borwein,AcubiccounterpartofJacobi'sidentityandtheAGM,Trans.Amer.Math.Soc.,3231991,691{701. [3] J.M.Borwein,P.B.BorweinandD.H.Bailey,Ramanujan,modularequations,andapproximationstopiorhowtocomputeonebilliondigitsofpi,Amer.Math.Monthly,96(1989),201{219. [4] J.M.Borwein,P.B.BorweinandF.G.Garvan,SomecubicmodularidentitiesofRamanujan,Trans.Amer.Math.Soc.,3431994,35{47. [5] J.M.BorweinandF.G.Garvan,ApproximationstoviatheDedekindetafunction.OrganicMathematics(Burnaby,BC,1995),CMSConf.Proc.,20,Amer.Math.Soc.,Providence,RI,1997,89{115. [6] F.G.Garvan,Somecongruencesforpartitionsthatarep-cores,Proc.LondonMath.Soc.(3)66,no.3,1993,449{478. [7] A.K.Lenstra,H.W.LenstraandL.Lovasz,Factoringpolynomialswithrationalcoecients.Math.Ann.261,no.4,1982,515{534. [8] ToshitsuneMiyake,Modularforms,Translatedfromthe1976JapaneseoriginalbyYoshitakaMaeda.Reprintoftherst1989Englishedition.SpringerMonographsinMathematics.Springer-Verlag,Berlin,2006. [9] RobertA.Rankin,ModularFormsandFunctions,CambridgeUniversityPress,Cambridge-NewYork-Melbourne,1977. 39

PAGE 40

AmitavaGhoshwasbornonDecember10,1979,inDurgapur(WestBengal),India.HereceivedhishighschooleducationatSt.Xavier'sSchool,Durgapur.HethenwenttoDAVModelSchoolforeleventhandtwelfthgrade.HeparticipatedintheInternationalMathematicsOlympiadtrainingcampatMumbaiin1995and1996.HehasindependentproofsoftheBertrand'sconjectureregardingprimesandtheGauss'sreciprocitylaws.HealsohasworkedontheconstructionofAbelianGaloisgroupsofanyarbitraryorder.In2002,hegraduatedfromRegionalEngineeringCollege(UniversityofBurdwan)withabachelor'sdegreeinmetallurgicalengineering.Aftercompletinghisdiploma,hewenttotheInstituteofMathematicalSciences{Chennai,Indiaasavisitingstudentinmathematics.Inthefallof2004,heenteredtheUniversityofFlorida{GainesvilletoundertakeaMasterofSciencedegreeinmathematicsbyresearch.Currently,henishedhismaster'sdegreeinmathematicsfromtheUniversityofFlorida{Gainesvilleviathethesisoption. 40