Imagine a Matematician wants to create this function:

$$F(s)=\sum_{n=0}^{\infty} \frac{1}{(n^2+1)^s}$$

Where would he start from?

More precisely, how would he know the exact values of $F(s)$ for $s> 0$? And how would he extend this result for every natural $s$? And for real $s$?

Thank you!

Imagine a mathematician wants to create this function:

$$F(s)=\sum_{n=0}^{\infty} \frac{1}{(n^2+1)^s}$$

Where would he start?

More precisely, how would he know the exact values of $F(s)$ for $s> 0$? And how would he extend this result for every natural $s$? And for real $s$?

Imagine a Matematician wants to create this function:

$$\zeta_\beta(s)=\sum_p^{\infty} \frac{1}{(p-1)^s}$$

For $p$ a prime number.

Where would he start from?

More precisely, how would he know the exact values of $\zeta_\beta(s)$ for $s> 0$? And how would he extend this result for every natural $s$? And for real $s$?

Thank you!

Imagine a Matematician wants to create this function:

$$F(s)=\sum_{n=0}^{\infty} \frac{1}{(n^2+1)^s}$$

Where would he start from?

More precisely, how would he know the exact values of $F(s)$ for $s> 0$? And how would he extend this result for every natural $s$? And for real $s$?

Thank you!

Imagine a Matematician wants to create this function:

$$\zeta_\beta(s)=\sum_n^{\infty} \frac{1}{(n^2+1)^s}$$

Where would he start from?

More precisely, how would he know the exact values of $\zeta_\beta(s)$ for $s> 0$? And how would he extend this result for every natural $s$? And for real $s$?

Thank you!

Imagine a Matematician wants to create this function:

$$\zeta_\beta(s)=\sum_p^{\infty} \frac{1}{(p-1)^s}$$

For $p$ a prime number.

Where would he start from?

More precisely, how would he know the exact values of $\zeta_\beta(s)$ for $s> 0$? And how would he extend this result for every natural $s$? And for real $s$?

Thank you!