Are there really infinite transcendental numbers?

[SLD]

 Riemann zeta function - Riemann Zeta Function -- from Wolfram MathWorld

is defined by a sum formula that is only valid for Re(s) > 1. That function has a pole, an infinite value, at s = 1, but one can go around that pole in the complex plane: analytic continuation.

Riemann Zeta Function Zeros -- from Wolfram MathWorld
The trivial ones are for s = -2, -4, -6, -8, -10, ...

The nontrivial ones have form s = sr + i*si, where 0 < sr < 1. The Riemann hypothesis states that all such zeros have sr = 1/2.  Riemann hypothesis - Riemann Hypothesis -- from Wolfram MathWorld It is related to the prime-number theorem, a hypothesis about the asymptotic distribution of prime numbers.  Prime number theorem - Prime Number Theorem -- from Wolfram MathWorld

The Riemann hypothesis has yet to be proved, despite the efforts of mathematicians for a century and a half, though it has been verified for the first 10^(13) nontrivial zeros.

So basically even though real part of the number is less than 1, the summation to infinity still converges, and in some cases along the real line 1/2 it actually goes to zero.

Does it go to infinity, I.e., not converge, for some numbers along the 1/2 “strip”? Or at other spots off the x axis?

Those would be poles right?

I wonder if you can creat a conformal map of the function?

 

[Swammerdami]

Note that

ζ(-1) = 1+2+3+4+... = -1/12​

That is: The zeta function at -1 is a divergent series that looks like ∞ but which, using analytic continuation can be treated as summing to the (unlikely looking!) value -1/12. This peculiar-looking summation was discovered by Leonhard Euler and later rediscovered by Srinivasa Ramanujan.

 

[SLD]

 Riemann zeta function - Riemann Zeta Function -- from Wolfram MathWorld

is defined by a sum formula that is only valid for Re(s) > 1. That function has a pole, an infinite value, at s = 1, but one can go around that pole in the complex plane: analytic continuation.

Riemann Zeta Function Zeros -- from Wolfram MathWorld
The trivial ones are for s = -2, -4, -6, -8, -10, ...

The nontrivial ones have form s = sr + i*si, where 0 < sr < 1. The Riemann hypothesis states that all such zeros have sr = 1/2.  Riemann hypothesis - Riemann Hypothesis -- from Wolfram MathWorld It is related to the prime-number theorem, a hypothesis about the asymptotic distribution of prime numbers.  Prime number theorem - Prime Number Theorem -- from Wolfram MathWorld

The Riemann hypothesis has yet to be proved, despite the efforts of mathematicians for a century and a half, though it has been verified for the first 10^(13) nontrivial zeros.

I just wonder how Riemann actually calculated these numbers without a calculator.

And are we sure that the series actually converges to zero at these points? Or is just that we know it seems to?