Past exam of the analytic number theory and elliptic curves course of the University of Oxford / 2010 / A / 1

For $N=\lfloor x\rfloor$, interchange a finite sum and an integral:This proves the Abel's summation formula, including the term at $n=1$.

Apply it with $f(n)=1$, $F(t)=1/t$. Writing $\{t\}=t-\lfloor t\rfloor$ givesThe last two terms are $O(x^{-1})$, proving the first expansion. For the refinement let $B_1(t)=\{t\}-1/2$. It has mean zero on each unit interval, so its primitive $P(t)={\int }_0^t{B}_1(u)du$ is bounded and periodic. Integration by parts givesThus for integer $x$, where $\{x\}=0$,For the zeta representation, integrating each defining integral by parts yieldsSumming from $1$ to $N-1$ and substituting into the definition of $\zeta$ givesSince $0\le x-n\le 1$,This is uniform on compact subsets of $\mathrm{Re}s>0$, so the series is holomorphic there and limits at $s=1$ are legitimate after subtracting the pole. Expanding $N^{1-s}=1-(s-1)\log N+O((s-1{)}^2)$ givesThe final tail is $O(N^{-1})$, and the harmonic expansion sends the remaining difference to $\gamma$. Therefore