Riemann zeta function can be defined by the equation

$$ \zeta(s)= \lim _{x\to \infty} \left( \sum_{n\le x} \frac {1} {n^{s}}-\frac{x^{1-s}} {1-s}\right)$$ if $0<s<1$ (see for example Tom M. Apostol "Introduction to Analytic Number Theory" Springer-Verlag, 1976, p.55).

Apparently, we can present this limit as an infinite convergent sum of "elementary differences": $$\zeta(s)=-\sum_{n=0}^{\infty} \left( \frac{(n+1)^{1-s}} {1-s}-\frac{n^{1-s}} {1-s} -\frac {1} {(n+1)^{s}} \right) \, (1) $$ In fig. the area of the shaded region is equal to the modulus of the zeta function: fig.

My questions:

1.How can we show that the specified infinite convergent sum in the limit $s \to 0$ is equal to $-1/2$?

2.Can we represent $\zeta(s)$ graphically for $s \le 0$ similar way, that is, as the infinite convergent sum of some "elementary differences"?


I just found formula (1) in "The Prime Number Theorem" by J.G.O. Jameson, Cambridge University Press, 2003, p.113 (An exercise 7 after Chapter 3: "A series expression for the extended zeta function"). I also saw another formula on the wiki that the answer by Mhenni Benghorbal quotes. Is it the same expression, then how to show this?


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