8
votes
2answers
191 views

What is the sum of all complex integers?

In line with $$\zeta(-1)=-1/12$$ Could we, by considering $$f(s)=\sum_{a,b\in\mathbb Z,\;(a,b)\neq(0,0)}\frac{1}{(a+bi)^{s}}$$ Evaluate the sum of all complex integers?
4
votes
2answers
178 views

Sum of divergent series

I saw a lot of article in Math SE like Why does 1+2+3+⋯=−1/12? and S=1+10+100+100+10000+…=−1/9? How is that and lot of others. Also I saw this one of Ramanujan summation but I do not get the ...
-3
votes
2answers
116 views

If the set of natural numbers is closed under addition, how can we have the result that the sum of all the natural numbers to infinity is -1/12 [duplicate]

As seen here and on this wikipedia page the sum of all the natural numbers to infinity is -1/12. $\sum_{n=1}^\infty n = \frac{-1}{12}$ but the set of natural numbers is closed under addition and ...
3
votes
1answer
285 views

An example of divergent series with the Lerch function

I am often working with divergent series all around being this the bread and butter for a theoretical physicist. Thanks to the excellent work of Hardy these have lost their mystical Aurea and so, they ...
11
votes
1answer
296 views

Regularizing divergent series and Bernoulli numbers

Here is the actual problem I need a proof for: (I made an error writing down the equation initially) $$B_{k+1} = \frac{(-1)^k}{2^{k+2}-2} \sum_{q=0}^{k} \binom{k+1}{q} 2^q B_q$$ Below is my ...