All Questions
Tagged with divergent-series prime-numbers
6 questions
10
votes
2
answers
2k
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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 ...
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 ...
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 ...
4
votes
0
answers
922
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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-...
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 ...
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/...