- #1
eljose
- 492
- 0
"Intuitive" Number theory:
Now i would like to play a game called "conjecture"..we have that for asymptotic behaviour:
[tex] \pi(x)=li(x) [/tex] where here "li" means the Logarithmic integral..
my conjecture is that for the sum:
[tex] \sum_{p}^{x}p^{n}=li(x^{n+1} [/tex]
i have checked it for n=-1,0,1 and it seems to work, the "justification" is that for example for sum over integers:
[tex] \int_{0}^{n}x^{k}=1+2^{k}+3{k}+... k\rightarrow{\infty} [/tex]
for the primes case there is an extra weight function [tex]\pi(x)-\pi(x-1) [/tex] so our sum would be equal to the integral:
[tex]\int_{0}^{n}(\pi(x)-\pi(x-1))x^{k} [/tex]
but using PNT [tex]\pi(x)-\pi(x-1)=1/ln(x) [/tex] so all this becomes:
[tex]\int_{0}^{n}x^{k}/ln(x)\rightarrow{Li(n^{k+1} [/tex]
where the last expression comes from using tables to compute the integral, of course for any analyitc function on R we have:
[tex] \sum_{p}f(p)=\sum_{n=0}^{\infty}a_{n}li(x^{n+1} [/tex]
for x------->oo
Now i would like to play a game called "conjecture"..we have that for asymptotic behaviour:
[tex] \pi(x)=li(x) [/tex] where here "li" means the Logarithmic integral..
my conjecture is that for the sum:
[tex] \sum_{p}^{x}p^{n}=li(x^{n+1} [/tex]
i have checked it for n=-1,0,1 and it seems to work, the "justification" is that for example for sum over integers:
[tex] \int_{0}^{n}x^{k}=1+2^{k}+3{k}+... k\rightarrow{\infty} [/tex]
for the primes case there is an extra weight function [tex]\pi(x)-\pi(x-1) [/tex] so our sum would be equal to the integral:
[tex]\int_{0}^{n}(\pi(x)-\pi(x-1))x^{k} [/tex]
but using PNT [tex]\pi(x)-\pi(x-1)=1/ln(x) [/tex] so all this becomes:
[tex]\int_{0}^{n}x^{k}/ln(x)\rightarrow{Li(n^{k+1} [/tex]
where the last expression comes from using tables to compute the integral, of course for any analyitc function on R we have:
[tex] \sum_{p}f(p)=\sum_{n=0}^{\infty}a_{n}li(x^{n+1} [/tex]
for x------->oo
Last edited: