(adsbygoogle = window.adsbygoogle || []).push({}); An apparent paradox??

W have [tex] \int_{2}^{x} dt d\pi (t) t^{2} = \sum_{p \le x}p^{2} [/tex]

also for every prime p then [tex] \sigma _{2} (p) = 1+p^{2} [/tex]

by the definition of 'divisor function' of order 2

so [tex] \sum_{p \le x}p^{2}+ \pi (x) = \int_{2}^{x} dt d\pi (t) \sigma_{2}(t) = \sum_{p \le x} \sigma _{2} (p) [/tex]

since for every prime the divisor function has only 2 numbers 1 and p then differentiating to both sides we find:

[tex] d \pi (x) x^{2} = d \pi (x) \sigma_{2} (x)+ d \pi(x) [/tex]

which is completely absurd since we could remove the derivative of the prime counting function..i believe that perhaps a derivative of second order [tex] d^{2} \pi (x) [/tex] or a factor [tex] d \pi (x) d \pi (x) [/tex] should appear.

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# An apparent paradox?

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