By Goldfeld D., Broughan G.A.

This e-book presents a wholly self-contained creation to the idea of L-functions in a mode available to graduate scholars with a uncomplicated wisdom of classical research, advanced variable thought, and algebra. additionally in the quantity are many new effects now not but present in the literature. The exposition presents whole particular proofs of ends up in an easy-to-read structure utilizing many examples and with out the necessity to be aware of and keep in mind many complicated definitions. the most issues of the booklet are first labored out for GL(2,R) and GL(3,R), after which for the final case of GL(n,R). In an appendix to the publication, a collection of Mathematica services is gifted, designed to permit the reader to discover the speculation from a computational perspective.

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0)) and en = (0, 0, . . , 0, 1). We now integrate F(z) over a fundamental domain for Ŵn \hn . It follows that F(z) d ∗ z = f (0) · Vol(Ŵn \hn ) + 2 Ŵn \hn ℓ>0 f (ℓen · z) d ∗ z. 11) Pn \hn The factor 2 occurs because −In (In = n × n identity matrix) acts trivially on hn . The computation of the integral above requires some preparations. We may express z ∈ hn in the form ⎛ ⎞ y1 y2 · · · yn−1 t ⎜ y1 y2 · · · yn−2 t ⎜ .. ⎜ z = x ·⎜ ⎜ ⎝ . 10). It follows that ⎞ ⎛ ⎛ 1 x1,2 x1,3 1 x1,n ⎜ ⎟ ⎜ 1 x2,3 1 x2,n ⎟ ⎜ ⎜ ⎟ ⎜ ⎜ .

3) . √ Remarks The bound 23 is implicit in the work of Hermite, and a proof can be found in (Korkine and Zolotareff, 1873). 2 is a well known theorem due to Siegel (1939). For the proof, we follow the exposition of Borel and Harish-Chandra (1962). 4) ∗ where t,u denotes the subset of matrices t,u · Z n which have determinant 1 ∗ ∗ and ◦ denotes the action of S L(n, Z) on 0,∞ . Note that every element in a,b Discrete group actions 16 is of the form ⎛ 1 x1,2 x1,3 ⎜ 1 x2,3 ⎜ ⎜ .. ⎜ . ⎜ ⎝ ⎞ ⎛ ⎞ dy1 y2 · · · yn−1 ⎟ ⎜ ⎟ dy1 y2 · · · yn−2 ⎟ ⎟ ⎜ ⎟ ⎟ ⎜ .

5) now becomes F(z) d xd y 2ζ (2) f ((0, 0)). 6) and solve for the volume, we obtain 2ζ (2) . f ((0, 0)) − fˆ((0, 0)) · vol(Ŵ\h2 ) = f ((0, 0)) − fˆ((0, 0)) · π Since f is arbitrary, we can choose f so that f ((0, 0)) − fˆ((0, 0)) = 0. It follows that Vol(Ŵ\h2 ) = π 2ζ (2) = . 1 using induction on n. 1 requires two preliminary lemmas which we straightaway state and prove. 6 Volume of S L(n, Z)\S L(n, R)/S O(n, R) the group of all matrices of the form ⎛ 1 ⎜ .. ⎜ . ⎜ ⎝ 31 ⎞ u1 .. ⎟ . ⎟ ⎟ 1 u n−1 ⎠ 1 with u i ∈ R (respectively, u i ∈ Z), for i = 1, 2, .