what is the sum of all prime no multiplication
Answers
Prime Numbers are infinite. So we can't multiply all of them.
- Thank you.
Answer:
Explanation:
Many infinite series that you wouldn’t ordinarily expect to be summable can in fact be assigned a finite value using zeta function regularization. For example, Euler famously computed the “sum of all positive integers”:
1+2+3+4+⋯=(−1)=−112
1+2+3+4+⋯=ζ(−1)=−112
because the Riemann zeta function ()ζ(s), whose series definition ()=11+12+13+14⋯ζ(s)=11s+12s+13s+14s+⋯ only converges for ℜ()>1ℜ(s)>1, can be analytically continued to all complex numbers ≠1s≠1.
Yeah, it sounds crazy, but my physicist friends tell me that these kinds of summations have real applications in quantum field theory! We live in a strange universe.
Unfortunately, this method still fails to compute the sum of all primes 2+3+5+7+⋯2+3+5+7⋯. Landau and Walfisz (1919) showed that the prime zeta function defined by ()=12+13+15+17+⋯P(s)=12s+13s+15s+17s+⋯ cannot be analytically continued beyond ℜ()>0ℜ(s)>0, due to the clustering of singular points along the imaginary axis arising from the nontrivial zeros of the Riemann zeta function, so we don’t get a natural value for (−1)P(−1).
This does not exclude the possibility that the sum of all primes could be assigned a finite value by some more powerful method, but I don’t know of a way to do this.
(However, you can compute the product of all primes with zeta function regularization! 2⋅3⋅5⋅7⋯=422⋅3⋅5⋅7⋯=4π2.)
