Eisenstein series and approximations to pi - Mathematics

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10 BRUCE C. BERNDT AND HENG HUAT CHAN then [2, p. 127, Entries 13(iii), (iv)], (4.16) and (4.17) Q(q) = z 4 2(1 + 14x2 + x 2 2), R(q) = z 6 2(1 + x2)(1 − 34x2 + x 2 2), where [2, p. 122–123, Entries 10(i), 11(iii)] (4.18) z2 := ϕ 2 (q) and x2 := 16q ψ4 (q 2 ) ϕ 4 (q) . Replacing q by −q in (4.16) and (4.17), and using (4.18), we find that (4.19) and (4.20) Q(q) = ϕ 8 (−q) − 224qϕ 4 (−q)ψ 4 (q 2 ) + 16 2 q 2 ψ 8 (q 2 ) R(q) = (ϕ 4 (−q) − 16qψ 4 (q 2 )) × (ϕ 8 (−q) + 544qϕ 4 (−q)ψ 4 (q 2 ) + 16 2 q 2 ψ 8 (q 2 )). Using the transformation formula [2, p. 43, Entry 27(ii)], in (4.19) and (4.20), we deduce that (4.21) and (4.22) ϕ(e −π/t ) = 2e −πt/4√ tψ(e −2πt ) R(e −π/√ n ) = −n 3 R(e −π √ n ) Q(e −π/√ n ) = n 2 Q(e −π √ n ). Using (4.14), (4.21), and (4.22), we may rewrite (4.13) as nP(e −π√n −π/ ) − P(e √ n ) = 2n R(e−π√ n ) Q(e−π√ − 6z n ) 2 (e −π√ n dm )Jn 1 − J1/n 2 dJ (Jn, Jn) , = 2n dm 1 − Jn − 6Jn 1 − J1/n 2 dJ (Jn, (4.23) Jn) z 2 (e −π√ n ),
where APPROXIMATIONS TO π 11 (4.24) Jk = J(e −π√ k ), k > 0. This gives the first relation between P(e −π√ n ) and P(e −π/ √ n ). Recall the definitions of Ramanujan’s function f(−q) and the Dedekind eta-function η(τ), namely, f(−q) := ∞ (1 − q k ) =: e −2πiτ/24 η(τ), q = e 2πiτ , Im τ > 0. k=1 The function f satisfies the well-known transformation formula [2, p. 43] (4.25) n 1/4 e −π√ n/24 f(e −π √ n ) = e −π/(24 √ n) f(e −π/ √ n ), n > 0. Logarithmically differentiating (4.25) with respect to n, multiplying both sides by 48n 3/2 /π, rearranging terms, and employing the definition of P (q) given in (3.1), we find that (4.26) 12 √ n π = nP(e−π√ n ) + P(e −π/ √ n ). This gives a second relation between P(e −π√ n ) and P(e −π/ √ n ). Now adding (4.23) and (4.26) and dividing by 2, we arrive at nP(e −π√n 6 ) = √ n π + n dm 1 − Jn − 3Jn 1 − J1/n 2 dJ (Jn, Jn) z 2 (e −π√n ), or, by (4.1), 1 √ Pn − Qn 6 √ = nπ 1 − J1/n 1 − Jn 1 − 3Jn n √ dm 1 − Jn 2 dJ (Jn, Jn) , where Qn is defined by (2.3), and Pn is defined by (3.2). (Be careful; Jn = Jn, where Jn is defined by (2.4).) We record the last result in the following theorem, which should be compared with Theorem 3.1. Theorem 4.1. If Pn, Qn, and Jn respectively, then are defined by (3.2), (2.3), and (4.24), (4.27) 1 √ Pn − Qn 6 √ = nπ 1 − Jntn,
Page 1 and 2: EISENSTEIN SERIES AND APPROXIMATION
Page 3 and 4: Theorem 2.1. For each positive inte
Page 5 and 6: APPROXIMATIONS TO π 5 Theorem 3.1.
Page 7 and 8: APPROXIMATIONS TO π 7 where rn is
Page 9: Using (4.8) and (4.9), we conclude
Page 13 and 14: APPROXIMATIONS TO π 13 n tn 3 0 5
Page 15 and 16: APPROXIMATIONS TO π 15 per term. T
Page 17 and 18: where APPROXIMATIONS TO π 17 λn =

Eisenstein series and approximations to pi - Mathematics

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