Pi and the AGM: A Study in Analytic Number Theory and by Jonathan M. Borwein

By Jonathan M. Borwein

Provides new learn revealing the interaction among classical research and glossy computation and complexity thought. in detail interwoven threads run notwithstanding the textual content: the arithmetic-geometric suggest (AGM) new release of Gauss, Lagrange, and Legendre and the calculation of pi[l.c. Greek letter]. those threads are carried in 3 instructions. the 1st results in nineteenth century research, particularly, the transformation thought of elliptic integrals, which necessitates a short dialogue of such themes as elliptic integrals and capabilities, theta capabilities, and modular services. the second one takes the reader into the area of analytic complexity - simply how intrinsically tough is it to calculate algebraic capabilities, undemanding features and constants, and the well-known capabilities of mathematical physics? The solutions are impressive, for the widespread equipment are usually faraway from optimum. The 3rd path leads via purposes and ancillary fabric - relatively the wealthy interconnections among the functionality concept and the quantity conception. integrated are Rogers-Ramanujan identities, algebraic sequence for pi[l.c. Greek letter], effects on sums of 2 and 4 squares, the transcendence of pi[l.c. Greek letter] and e[ital.], and a dialogue of Madelung's consistent, lattice sums, and elliptic invariants. routines.

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Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity

Offers new study revealing the interaction among classical research and sleek computation and complexity concept. in detail interwoven threads run although the textual content: the arithmetic-geometric suggest (AGM) generation of Gauss, Lagrange, and Legendre and the calculation of pi[l. c. Greek letter]. those threads are carried in 3 instructions.

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D + e)! F (xK )G(xL ) (K,L)∈sh(d,e) . p The rest of the proof is an approximation argument. Consider the discretization i/n , i = 1, . . , n of [0, 1] ; given continuous F , G on [0, 1]d and [0, 1]e , we approximate F , G by step functions as above and construct corresponding polynomials f , g . As n → ∞ , these functions are dense in Lp ([0, 1]d ) and Lp ([0, 1]e ) . 9. The constant c2 (d, e) is c2 (d, e) = d+e d 1/2 . 7. Then 2 F (xK )G(xL ) = 2 (K,L)∈sh(d,e) (K , L )∈sh(d , e ) (K , L )∈sh(d , e ) [0,1]d +e F (xK )G(xL ) F (xK )G(xL ) dx and this is equal to d+e d F 2 2 G 2 2 + (K,L)=(K ,L ) [0,1]d +e F (xK )G(xL ) F (xK )G(xL ) dx.

Choose j ∈ {0, . . , n} such that xj |ϕ(X ) = 0 . Then the vector (f0 , . . , fn ) is proportional to (ϕ∗ x0 /ϕ∗ xj , . . , ϕ∗ xn /ϕ∗ xj ) ∈ K(X)n+1 and we may assume that they are equal. ,n Z where the sums range over all prime divisors Z of X . By the valuative criterion of properness (cf. 10), the domain U of ϕ has a complement of codimension at least 2 . 7. By choosing a trivialization of (ϕ|U )∗ OPn (1) at a generic point of Z , we may view ϕ∗ (xi ) as regular functions in Z . ,n and thus ordZ (ϕ∗ xj ) deg Z.

The global height hλ is independent of the choices of F and of the section s . 7, the global height is independent of F . Its independence from the choice of s can be verified as follows. Let t be another non-zero meromorphic section of O(D) with P ∈ / supp(D(t)). 5 show that λD(s) (P, v) − λD(t) (P, v) = λs/t (P, v) for any v ∈ MF . On the other hand, the product formula shows that the global height of P relative to λs/t is 0, proving the claim. 5. As an immediate consequence the global height relative to the natural local height of a non-zero rational function is identically 0.

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