On a certain representation of the Riemann zeta function

Question

Let $\zeta$ denote the Riemann zeta function. In this answer: https://mathoverflow.net/a/314066/133634, @Paul Garret considers the representation

$$\frac{\zeta(s)}{s} = \int_1^\infty (\sum_{1 \le n \le x} 1) x^{-s-1}dx$$ for $\Re(s)>1$, and then claims (without proof) that this representation yields

$$\frac{\zeta(s)}{s}+\frac{\zeta(s)}{1-s}= \int_0^\infty ((\sum_{1 \le n \le x} 1-x)-x(\sum_{1 \le n \le x} \frac1n-\log x-\gamma)) x^{-s-1}dx$$ for $\Re(s)\in(0,1)$, where $\gamma$ is the Euler constant.

But i can't see how the second representation follows from the first. Can anyone please explain ?

Answer 1

Let $0<\Re(s)<1$ unless otherwise indicated. Then $$\int_0^\infty (\sum_{1 \le n \le x} 1-x) x^{-s-1}\,dx=-\int_0^1 x^{-s}+\int_1^\infty (\sum_{1 \le n \le x} 1-x) x^{-s-1}\,dx.$$ On the right hand side, the first integral equals $1/(s-1)$. The second integral equals $\zeta(s)/s-1/(s-1)$ as can be seen by proving it first for $\Re(s)>1$ and then continuing it analytically to $\Re(s)>0$. Hence $$\int_0^\infty (\sum_{1 \le n \le x} 1-x) x^{-s-1}\,dx=\frac{\zeta(s)}{s},\qquad 0<\Re(s)<1.\tag{1}$$ Now consider $$S(x):=\sum_{1 \le n \le x} \frac1n-\log x-\gamma,\qquad x>0.$$ Note that $S(x)\ll\log x$ when $x<1$, and $S(x)\ll 1/x$ when $x\geq 1$. Then, the remaining part of the integral is, by integration by parts, $$\int_0^\infty S(x)x^{-s}\,dx=\frac{1}{s-1}\int_0^\infty x^{1-s}\,dS(x).$$ Now we argue for the integral on the right hand side similarly as before: $$\int_0^\infty x^{1-s}\,dS(x)=\int_0^{1-} x^{1-s}\,dS(x)+\int_{1-}^\infty x^{1-s}\,dS(x).$$ On the right hand side, the first integral equals $1/(s-1)$. The second integral equals $\zeta(s)-1/(s-1)$ as can be seen by proving it first for $\Re(s)>1$ and then continuing it analytically to $\Re(s)>0$. Hence $$\int_0^\infty x^{1-s}\,dS(x)=\zeta(s),\qquad 0<\Re(s)<1,$$ and therefore $$\int_0^\infty S(x)x^{-s}\,dx=\frac{\zeta(s)}{s-1},\qquad 0<\Re(s)<1.\tag{2}$$ Subtracting $(2)$ from $(1)$, the claimed identity follows.