This incursion into the realm of elementary and probabilistic number theory of continued fractions, via modular forms, allows us to study the alternating sum of coefficients of a continued fraction, thus solving the longstanding open problem of their limit law.



f  (  T^{j}(x_{0})  )  =  ó õ 

f(y) dl(y). 
æ ç ç è 

ö ÷ ÷ ø 
= (1)^{(c1)(d1)/4} 
æ ç ç è 

ö ÷ ÷ ø 
. 
æ ç ç è 

ö ÷ ÷ ø 
=(1) 

, 
s(d,c)= 

((hd/c))((h/c)), 
h(z) = e 


(1  e 

) 
ln h 
æ ç ç è 

ö ÷ ÷ ø 
= 

(1) 
a b 
c d 
ln h(z) = 

 


, 
s(c, d) = 

+ 

+ 

 s(d, c). 
s(d,c)= 

æ ç ç è 
3+ 

 

(1)^{i}a_{i} 
ö ÷ ÷ ø 
. 
a b 
c d 
1') f(gz)=c(g) 
æ ç ç è 

ö ÷ ÷ ø 

f(z)
(for gÎ G), and 2')
ó õó õ 

  f(x+iy)    ^{2} 

<¥. 
á f,gñ
=ó õó õ 

f(z)g(z)y^{r} 

. 
S(m,n,c)=å  e 

, 
S(m,n,c,c,G)=åc(g)  e 

, 
a c 
c d 
1 q 
0 1 
1 q 
0 1 

S(m, n, c, c, G) = 

t_{j}(m, n, c, G) 

+ O(N 

), (2) 
t_{j}(m, n, c, G) = 

e 


e 

=S  (  1,1,c,c_{r},SL(2,Z)  )  , 


  { n<N:f(n)<x }    =F(x). 



e^{itf(n)}=g(t), 
ó õ 


dy = e 

, 
  { 0<d<c<N:gcd(d,c)=1 }    = 

+O(Nln N) 

e 

~ e 


. 

e 

= 

e 

+ O 
æ è 
N^{2}(ln  N) 

ö ø 
. 