On the Topography of Maass Waveforms for PSL(2, Z)

Computations of Eisenstein series on Fuchsian groups

“…In line with the heuristic arguments in [HR92,§ §6,7] it is natural to expect that we should have σ ≈ σ 0 as R → ∞. One also expects that σ 1 should be asymptotically negligible in the limit, thus σ ≈ πΩ/8 as R → ∞.…”

“…Table 11. Comparison between standard deviation σ 0 as predicted by [HR92] and standard deviation σ computed from our data.…”

“…Hejhal and Rackner [HR92,Hej99] and, for CM -forms on congruence subgroups of P SL(2, Z), Hejhal and Strömbergsson [HS01]. Regarding the values of the Eisenstein series on the arithmetic surface P SL(2, Z)\H, promising experimental results along with some heuristics are presented in [HR92].…”

“…Finally, we give some data in connection with a conjectural asymptotic formula for the standard deviation σ over a given rectangle F , based on [HR92]. Let y av be the average y-value over F and set…”

“…Hejhal and Rackner [HR92,Hej99] and, for CM -forms on congruence subgroups of P SL(2, Z), Hejhal and Strömbergsson [HS01]. Regarding the values of the Eisenstein series on the arithmetic surface P SL(2, Z)\H, promising experimental results along with some heuristics are presented in [HR92]. This paper is mainly concerned with the Eisenstein series on non-arithmetic surfaces without any symmetries, i.e., in a setting where, in line with the PhillipsSarnak conjecture [PS85], there are expected to be (generically) no discrete eigenfunctions.…”