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10 votes
2 answers
2k views

Abel summation of the alternating series of primes?

Consider the ordinary generating function of the sequence of primes ($2+3x+5x^2+7x^3 + ...$); by the ratio test and the prime number theorem, its radius of convergence is $1$. Thus, we might well ask ...
Sridhar Ramesh's user avatar
8 votes
1 answer
855 views

Is it possible to sum the divergent series with prime coefficients?

It is known that the series $$ P := \sum_{n=1}^{\infty} p_{n} \qquad \text{where } p_{n} \text{ is the n'th prime} $$ cannot be summed by means of (prime) zeta function regularization. (The result was ...
Max Lonysa Muller's user avatar
7 votes
2 answers
976 views

Regularizing the sum of all primes

In the spirit of a similar question for the harmonic series, is there a way to regularize the (divergent) sum of all primes? $$ \sum_{p \text{ prime}} p $$ Neither of these questions obtained a ...
user76284's user avatar
  • 2,203
4 votes
0 answers
922 views

Guessing of $n$th prime from "super- regularized" product of primes

( I've been thinking about asking this for a long time . Though this is not rigorous; It can be thought of as heuristic or extraction of information from different viewpoint.) We know "super-...
TPC's user avatar
  • 790
2 votes
0 answers
238 views

Possible regularisation for sum of function of primes

Consider the following sum of function of primes: $$-\sum_{p}\ln\left( 1 - \frac{1}{(ep)^{1/2}} \right){\ln(p)}$$ Here $p$ runs through all primes and $e$ is Euler's constant. We can see that the sum ...
Zaza's user avatar
  • 149
2 votes
0 answers
199 views

What is the regularized sum of the following series (sum of all primes but spaced with zeros in place of non-primes)?

The sum over primes: $$\sum_{k=0}^\infty \{\text{k if k is prime, 0 otherwise}\}$$ I know that there is no known method to ascribe a reasonable value to the sum of the primes https://www.quora.com/...
Anixx's user avatar
  • 10.1k