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I find myself needing the asymtotics of the following summation for my work. Let $a$ be a positive real number and $p_n$ be the $n$-th prime.

$$ \sum_{k=1}^{n} [k^a - (k-1)^a]p_k $$

At $a=1$, this becomes the sum of the first $n$ primes and the asymptotics of this is well known. Moreover it is easy to prove that

$$ \sum_{k=1}^{n} [k^a - (k-1)^a]p_k = n^a p_n - \int_{2}^{p_n} \pi(x)^a dx. $$

I want an asymptotic expansion in terms of either $n$ or $p_n$ or a combination of both and get rid of the integration. (Don't ask me how many terms of the asymptotic expansion you want, do your best.)
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On a sum involving prime numbers

I find myself needing the asymtotics of the following summation for my work. Let $a$ be a positive real number and $p_n$ be the $n$-th prime.

$$ \sum_{k=1}^{n} [k^a - (k-1)^a]p_k $$

At $a=1$, this becomes the sum of the first $n$ primes and the asymptotics of this is well known. Moreover it is easy to prove that

$$ \sum_{k=1}^{n} [k^a - (k-1)^a]p_k = n^a p_n - \int_{2}^{p_n} \pi(x)^a dx. $$

I want an asymptotic expansion in terms of either $n$ or $p_n$ or a combination of both and get rid of the integration.