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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

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# Intuitive Number theory:

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