- Suppose \(\{ a _ { n } \} _ { n = 1 } ^ { \infty }\) is a sequence of complex numbers such that \(a _ { n } = a _ { m }\) if \(n \equiv m\) mod \(q\) for some positive integer \(q .\) Define the Dirichlet L-series associated to \(\left\{ a _ { n } \right\}\) by
\[ L (s) = \sum_ {n = 1} ^ {\infty} \frac {a _ {n}}{n ^ {s}} \quad \mathrm{for} \operatorname{Re} (s) > 1. \]
Also, with \(a _ { 0 } = a _ { q }\) , let
\[ Q (x) = \sum_ {m = 0} ^ {q - 1} a _ {q - m} e ^ {m x}. \]
Show, as in Exercises 15 and 16 of the previous chapter, that
\[ L (s) = \frac {1}{\Gamma (s)} \int_ {0} ^ {\infty} \frac {Q (x) x ^ {s - 1}}{e ^ {q x} - 1} d x, \quad \mathrm{for} \operatorname{Re} (s) > 1. \]
Prove as a result that \(L ( s )\) is continuable into the complex plane, with the only possible singularity a pole at \(s = 1\) . In fact, \(L ( s )\) is regular at \(s = 1\) if and only if \(\textstyle \sum _ { m = 0 } ^ { q - 1 } a _ { m } = 0\) . Note the connection with the Dirichlet \(L ( s , \chi )\) series, taken up in Book I, Chapter 8, and that as a consequence, \(L ( s , \chi )\) is regular at \(s = 1\) if and only if \(\chi\) is a non-trivial character.
Solution
Because \(a_n\) has period \(q\), \[ \sum_{n=1}^\infty a_ne^{-nx} =\sum_{k=0}^\infty\sum_{r=1}^q a_re^{-(kq+r)x} =\frac{\sum_{r=1}^q a_re^{-rx}}{1-e^{-qx}} =\frac{Q(x)}{e^{qx}-1}. \tag{1} \] For \(\Re s>1\), absolute convergence permits termwise integration, and \[ \int_0^\infty e^{-nx}x^{s-1}\,dx=\Gamma(s)n^{-s}. \] Using (1), \[ \Gamma(s)L(s) =\int_0^\infty\frac{Q(x)x^{s-1}}{e^{qx}-1}\,dx, \] which proves the asserted formula.
For continuation, split the integral at \(1\). The integral over \([1,\infty)\) is entire in \(s\), because its integrand decays exponentially in \(x\) uniformly on compact subsets of the \(s\)-plane.
Near \(x=0\), the function \[ H(x)=\frac{xQ(x)}{e^{qx}-1} \] is holomorphic, with \[ H(0)=\frac{Q(0)}q=\frac1q\sum_{r=1}^q a_r. \] Write its Taylor expansion \(H(x)=\sum_{m\ge0}c_mx^m\). For each \(N\), \[ \int_0^1\frac{Q(x)x^{s-1}}{e^{qx}-1}\,dx =\sum_{m=0}^N\frac{c_m}{s+m-1}+R_N(s), \] where \(R_N\) is holomorphic for \(\Re s>-N\). Multiplication by \(1/\Gamma(s)\) cancels every possible pole at \(s=0,-1,-2,\ldots\). Hence \(L(s)\) continues meromorphically to the whole plane with only a possible pole at \(s=1\).
At \(s=1\), its residue is \[ \frac{c_0}{\Gamma(1)}=\frac1q\sum_{r=1}^q a_r. \] Therefore \(L\) is regular at \(s=1\) exactly when \[ \sum_{r=1}^q a_r=0. \] For a nontrivial Dirichlet character modulo \(q\), this sum is zero, whereas for the principal character it is nonzero; this gives the stated application to Dirichlet \(L\)-series.