``Le plus court chemin entre deux vérités dans le domaine réel passe par le domaine complexe.^{1} ''The above quote captures the depth analysis can bring when one is confronted by number theoretic questions. The oldest and most fundamental of such questions is the study of prime numbers. The first question to be answered is: Are there an infinite number of primes? This can be answered by a number of simple proofs (several other proofs are given in [7]):J. Hadamard.


> 


~ ln(x), 


=¥ Û there are an infinite number of primes in ak+q. 
c(n):= 
æ ç ç è 

ö ÷ ÷ ø 
= 

L(s,c):= 


= 


. 



ln L(s,c)= 




ln(1c(p)p^{s}) 
= 









ln  L(s,c)= 



= 

p^{s} +O(1), s® 1^{+}. (4) 

p^{1}= 

æ ç ç ç ç ç è 

+ 

+ 

ö ÷ ÷ ÷ ÷ ÷ ø 

ln L(1,c) + O(1). (5) 
0¹ L  (  1,(· / q)  )  = 


p^{1}=+¥. 

ln(p) = 

ó õ 




ln(p) = x 

 


= xln  (2p) 


 

ln(1x^{2}), (6) 
Â ln  z(s)= 


m^{1} p 

cos(tmln(p)). 
y(x) = x  x^{1/2} 
æ ç ç è 


ö ÷ ÷ ø 
+o(x^{1/2}). 
q(x) = x  x^{1/2} 
æ ç ç è 
1+ 


ö ÷ ÷ ø 
+o(x^{1/2}). 


, 

c(n) ln(p) =  x^{1/2} 
æ ç ç ç è 


ö ÷ ÷ ÷ ø 
+ o(x^{1/2}), 
y_{q, a}(x) = 

ln(p) = 

 





q_{q,a} (x) = 

ln  (p) = y_{q, a} (x)  

ln(p) + O(x^{1/3}) = y_{q, a}(x)  c_{q, a} x^{1/2} + O(x^{1/3}), 


® .00000026, 


® .9959. 
This document was translated from L^{A}T_{E}X by H^{E}V^{E}A.