Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 126 / 5

Work initially with $\mathrm{Re}s>1$, where the nonholomorphic Eisenstein series and its differentiated series converge locally absolutely. For $\gamma =\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in S{L}_2(\mathbb{Z})$, the identitiesshow that each summand at $\gamma \tau$ becomes the summand indexed by $(ma+nc,mb+nd)$ at $\tau$. Reindexing the nonzero integer pairs therefore provesHere $y^s$ and the denominator powers use logarithms of positive real numbers, so there is no ambiguity for complex $s$.

For the nonnegative Laplace-Beltrami operator of the hyperbolic plane, $\Delta y^s=-y^2{\partial }_y^2{y}^s=s(1-s)y^s$. If $(m,n)=r(c,d)$ with $(c,d)$ primitive and $r>0$, choose a matrix in $S{L}_2(\mathbb{Z})$ with bottom row $(c,d)$. The corresponding summand isThe Laplace-Beltrami operator commutes with hyperbolic isometries, so this summand has eigenvalue $s(1-s)$. This also covers $m=0$, or follows there directly. Termwise differentiation now givesThe equation describes an automorphic eigenfunction; it does not assert that this noncuspidal function belongs to the square-integrable discrete spectrum.

The constant Fourier coefficient is $A_0(y,s)={\int }_0^{1}G(x+iy,s)dx$. The $m=0$ terms give $2\zeta (2s)y^s$. For fixed $m\ne 0$, unfolding the sum over $n$ givesIndeed, the substitution $u=mx+n$ introduces a factor $1/|m|$, while the translated intervals of length $|m|$ cover the real line exactly $|m|$ times, cancelling that factor. Now scale $u=|m|yt$. The beta function integral yieldswhich can be checked by inserting $(1+t^2{)}^{-s}=\Gamma (s{)}^{-1}{\int }_0^{\infty }{v}^{s-1}{e}^{-v(1+t^2)}dv$ and evaluating the inner Gaussian integral. Thus the fixed-$m$ contribution is $|m{|}^{1-2s}{y}^{1-s}\sqrt{\pi }\Gamma (s-1/2)/\Gamma (s)$. Summing over positive and negative $m$ proves the constant term of a nonholomorphic Eisenstein seriesWith the printed normalization $\xi (u)={\pi }^{-u/2}\Gamma (u/2)\zeta (u)$ of the completed Riemann zeta function, this is exactlyBoth the unfolding and the original series calculation are justified for $\mathrm{Re}s>1$. Beyond this region the identities are understood meromorphically: $G(\tau ,s)$ is the Epstein zeta function of the unit-covolume lattice $(\mathbb{Z}\tau +\mathbb{Z})/\sqrt{y}$, so question 4 supplies its continuation. The symbol $\xi$ here has the completed-zeta normalization displayed above, which has poles at zero and one; it does not include the extra polynomial factor sometimes used to define an entire xi function.