Integrating prime counting function 1

For any $n > 2$ we have $\int_1^\infty \frac{\pi(x)}{x^n}\,dx \; =\; \int_1^\infty \sum_{p \le x} \frac{1}{x^n}\,dx \; = \; \sum_p \int_p^\infty \frac{1}{x^n}\,dx$ using Fubini's Theorem to reverse the order of summation and integration. Thus $\int_1^\infty \frac{\pi(x)}{x^n}\,dx \; = \; \sum_p \frac{1}{(n-1)p^{n-1}} \; =\; \tfrac{1}{n-1}P(n-1) \;.$ In our case ( $n=8$ ), we obtain the answer $7+7 = \boxed{14}$ .

Our goal is to turn this into a sum so we can use the prime zeta function. Using that intuition, we can integrate by parts leaving us with 1/7 the integral of the derivative of the function over x^7. Since we are integrating our derivative, hitting infinity only at the infinitesimal point at the prime itself, we can sum 1/7(sum of the seventh powers of the reciprocals of the primes). Hence 7+7=14.