The sum of all positive integers is divergent, but the result is ( -1/12).

Crazy but it is true in Quantum Physics (String Theory) !

**2nd Proof:**

The sum of all positive integers is divergent, but the result is ( -1/12).

Crazy but it is true in Quantum Physics (String Theory) !

**2nd Proof:**

The book which changed their life:

1. GH Hardy: by Carmille Jordan’s **Cours d’Analyse:**

“I shall never forget the astonishment with which I read the remarkable work … and I learnt for the first time as I read it what mathematics really meant.”

2. Ramanujan : George Carr’s

“**A Synopsis of Elementary Results in Pure & Applied Mathematics”**

(4,400 results without proofs)

3. Riemann : Legendre’s book

4. Hardy/Littlewood:

Landau 2-volume “**Handbuch der Lehre von Der Verteilung der Primzahlen**”

(Handbook of the Theory of the Distribution of Prime Numbers)

5. Atle Selverg (Norway): Ramanujan’s “**Collected Papers**”

Note: This blogger’s mathematics ‘fire’ is rekindled by John Derbyshire’s “**Unknown Quantity”.
**

Albert Einstein owed much to Riemann’s conception, saying that if he had not been acquainted with them

“I never would have been able to develop the theory of relativity.”

Riemann intuitively found the Zeta Function ζ(s), but couldn’t prove it. Computer ‘tested’ it correct up to billion numbers.

Or equivalently (see note *)

ζ(1) = Harmonic series (Pythagorean music notes) -> diverge to infinity

(See note #)

ζ(2) = Π²/6 [Euler]

ζ(3) = not Rational number.

1. **The Riemann Hypothesis**:

*All non-trivial zeros of the zeta function have real part one-half.*

ie ζ(s)= 0 where s= ½ + bi

Trivial zeroes are s= {- even Z}:

s(-2) = 0 =s(-4) =s(-6) =s(-8)…

You might ask why Re(s)=1/2 has to do with Prime number ?

There is another **Prime Number Theorem (PNT**) conjectured by Gauss and proved by Hadamard and Poussin:

π(Ν) ~ N / log N

ε = π(Ν) – N / log N

The error ε hides in the Riemann Zeta Function’s **non-trivial zeroes,** which all lie on the Critical line = 1/2 :

All non-trivial zeroes of ζ(s) are in Complex number between ]0,1[ along real line x=1/2

2. David Hilbert:

‘*If I were to awaken after 500 yrs, my 1st question would be: Has Riemann been proven*?’

It will be proven in **future** by a young man. ‘uncorrupted’ by today’s math.

Note (*):

…[1]

… [2]

[1]-[2]:

Finally,

Or

Note #:

Let s=1

RHS: Harmonic series diverge to infinity

LHS:

Diverge to infinity => there are infinitely many primes p