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?

 

[Swammerdami]

Speaking of transcendental numbers, exactly one week ago someone found a breathtaking -- almost unbelievable -- connection between the two most famous transcendental numbers. I didn't post immediately because I hoped there'd be some explanation. But AFAICT the entire math community has been stupefied into silence.

π⁴ + π⁵ = e⁶​

You can verify this amazing result by plugging 2.71828^6 - 3.14159^4 - 3.14159^5 into Googol. Six figs in, and the result is zero to six figs.
2.71828^6 - 3.14159^4 - 3.14159^5

 

[lpetrich]

(Me on the Riemann zeta function being defined as an infinite sum...)

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?

Poles of Riemann Zeta Function - ProofWiki
Exactly one, at 1, and it's a simple one:

zeta(s) = 1/(s-1) + (Euler gamma constant) + O(s)

One does not use the series directly for Re(s) <= 1, but instead some alternative expression that gives the series for Re(s) > 1.

 

[lpetrich]

To illustrate this convergence, I will use a much simpler example: the geometric series.

Finite:
G(a,n) = 1 + a + a² + a³ + ... + aⁿ
= sum over k from 0 to n of a^k

Infinite:
G(a) = 1 + a + a² + a³ + ...
= sum over k from 0 to infinity of a^k

The finite one can easily be summed. Multiply by 1 - a:
1 - a + a - a² + a² - a³ + ... - aⁿ⁺¹
= 1 - aⁿ⁺¹

Thus, for a != 1, G(a,n) = (1 - aⁿ⁺¹) / (1 - a)
Also, G(1,n) = n + 1

Turning to G(a) its series form is G(a,n) in the limit of n -> infinity, that is, the limit of n being arbitrarily large.

In that limit, aⁿ⁺¹ tends to 0 if |a| < 1, to infinity if |a| > 1, to 1 if a = 1, and oscillates with absolute value 1 if |a| = 1 and a != 1.

That gives us G(a) = 1/(1-a) for |a| < 1, G(a) = infinity for |a| > 1 or a = 1, and G(a) = undefined for |a| = 1 and a != 1.

But note that the first expression is finite for all a != 1. That means that we can use an extended definition of G(a):

G(a) = 1/(1-a)

one that is equal to the series definition for |a| < 1. Doing so is an example of analytic continuation, the process used to define values of zeta(s) for Re(s) <= 1, where the function's series definition does not converge.