Past exam of the analytic number theory and elliptic curves course of the University of Oxford / 2009 / A / 1 / Solution

absolute convergence depends only on $\sigma =\mathrm{Re}s$, since $|a_n{n}^{-s}|=|a_n|n^{-\sigma }$. Let $A$ be the set of real $\sigma$ for which $\sum |a_n|n^{-\sigma }$ converges. Comparison with a convergent member shows that $A$ is upward closed. The hypotheses give a member of $A$ and a real number outside $A$, so its infimum ${\sigma }_0$ is finite. If $\sigma >{\sigma }_0$, some $\tau \in A$ lies below $\sigma$, giving convergence by comparison; if $\sigma <{\sigma }_0$, convergence would contradict the definition of the infimum. Thus ${\sigma }_0=\inf A$ is the abscissa of absolute convergence of a Dirichlet series. This argument leaves convergence on the boundary open.

Partition $1,\dots ,n$ according to $d=n/\gcd (k,n)$. Writing $k=(n/d)j$ identifies the part indexed by $d\mid n$ with the integers $1\le j\le d$ coprime to $d$. There are $\varphi (d)$ of them. Counting the entire partition proves the totient divisor-sum identity

We use the multiplication theorem for absolutely convergent Dirichlet series: their product is the Dirichlet series of the Dirichlet convolution of their coefficients. absolute convergence justifies rearrangement of the double sum. Here $0<\varphi (n)\le n$, so its series converges absolutely for $\sigma >2$, andThe Euler product implies $\zeta (s)\ne 0$ for $\sigma >1$. Division gives

We finally use the simple pole of the Riemann zeta function at one, of residue one, and $0<\zeta (2)<\infty$. The quotient therefore tends to $+\infty$ as real $s\downarrow 2$. If $\sum \varphi (n)n^{-2}$ converged, its nonnegative terms would dominate the series at every real $s>2$, giving a uniform finite bound and a contradiction. Thus the series diverges at two; comparison then excludes absolute convergence at every $\sigma <2$. Combined with the convergence already proved, the abscissa is exactly two. This is the totient Dirichlet series and its absolute convergence boundary.