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The purpose of this paper is to introduce the reader to numerous types of the utmost precept, ranging from its classical formula as much as generalizations of the Omori-Yau greatest precept at infinity lately bought by way of the authors. purposes are given to a couple of geometrical difficulties within the environment of whole Riemannian manifolds, less than assumptions both at the curvature or at the quantity progress of geodesic balls.

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Providence, RI, 2002. [12] J. Jorgenson: Asymptotic behavior of Faltings’s delta function. Duke Math J. 61 (1990), 221–254. [13] J. Jorgenson, J. Kramer: Expressing Arakelov invariants using hyperbolic heat kernels. In: The Ubiquitous Heat Kernel. J. Jorgenson and L. ). AMS Contemp. Math. 398, Providence, RI, 2006, 295–309. 40 J. Jorgenson and J. Kramer [14] J. Jorgenson, J. Kramer: Non-completeness of the Arakelov-induced metric on moduli space of curves. Manuscripta Math. 119 (2006), 453–463.

9. 5. 1. With the above notations, we have 1 2g Z 1 0 Z b 0 D 23=2. v/ 0 dv : v (44) Proof. s; y/ ft b nD1 2 p 2y ! dy b y2 0 ! 2s/b y b y 0 nD1 ! 2s/b 1 1 X nD1 D 23=2. f t u which completes the proof. 2. s/ Z 1 r sinh. s=2 ir/ dr: 36 J. Jorgenson and J. Kramer Proof. v/ D f 2 1 r sinh. v/ 0 Z 1 D v s 2 0 p Z 2 D 2 1 r sinh. r 2 C1=4/t 0 1 r sinh. v= p 2/ dr ! v= 2/ v From [11], p. s/ Z 1 r sinh. s=2 t u which is the claimed formula. 3. tI z/ dx dy dt y2 r sinh. s/ dr: r 2 C 1=4 2 Proof. tI z/ dx 3=2.

1. Z/ Eisenstein series at the identity. Let 1 ; 2 be two quadratic characters unramified away from 2. 2) where • d20 D . d2 1/=2 d2 and d20 is the Kronecker symbol associated with the squarefree part of d20 . • dO1 is the part of d1 relatively prime to the squarefree part of d2 . k; l/ is even, 0 otherwise. g [BBFH07]. As such these functions have an analytic continuation to s1 ; s2 2 C and satisfy a group of 6 functional equations. 2). 1. s; wI 1; 2/ 1; D 2 be quadratic characters ramified only at 2.