{
  "schema": "open-mathematics-proof-routes/v1",
  "course": "LG-MF",
  "lesson": "LG-MF-19",
  "license": "CC0-1.0",
  "source": "src/the-rankin-selberg-method.md",
  "reader": "LG-MF-19.html",
  "results": [
    {
      "id": "LG-MF-19-continuous-diagonal-mellin",
      "statement": "Let k>=2 be even, f=sum_(n>=1)a_n q^n and g=sum_(n>=1)b_n q^n belong to S_k(SL_2(Z)), using unnormalized coefficients. Their Fourier series and products converge absolutely uniformly on every compact positive-height rectangle. For each sigma>1, Fourier orthogonality and the nonnegative continuous-series interchange give integral_0^infinity integral_0^1 |f(x+iy)|^2 y^(k+sigma-2) dx dy=Gamma(k+sigma-1)(4*pi)^(-(k+sigma-1))*sum_(n>=1)|a_n|^2 n^(-(k+sigma-1))<infinity. Hence D_(f,g)(u)=sum_(n>=1)a_n conjugate(b_n)n^(-u) converges absolutely locally uniformly for Re(u)>k. For u=t+k-1, Re(t)>1, the locally uniform diagonal Fourier series has finite summed absolute Mellin integrals and may be integrated termwise.",
      "proof": "LG-MF-19.html#1-unfolding-with-absolute-convergence",
      "result": "Lemma 1.1, (1.4), and the last part of Theorem 1.2"
    },
    {
      "id": "LG-MF-19-continuous-eisenstein-unfolding",
      "statement": "Let k>=2 be even, f,g be full-level cusp forms with unnormalized Fourier coefficients, H=f*conjugate(g)*y^k, Gamma=SL_2(Z), Gamma_infinity={plus or minus T^r}, and E(z,t)=sum_(Gamma_infinity\\Gamma)Im(gamma z)^t. For Re(t)=sigma>1, the positive coset series E(z,sigma) converges locally uniformly, the absolute strip integrand |H|y^(sigma-2) is integrable, and the continuous-series and projected-tile identities give integral_F |H|E(z,sigma)dmu=integral_0^infinity integral_0^1 |H(x+iy)|y^(sigma-2)dxdy<infinity. Consequently the complex continuous coset series unfolds and I_(f,g)(t)=integral_F H E(z,t)dmu=Gamma(t+k-1)(4*pi)^(-(t+k-1))*D_(f,g)(t+k-1), without an additional factor two. This uses the retained real plane integral change-of-variables theorem.",
      "proof": "LG-MF-19.html#1-unfolding-with-absolute-convergence",
      "result": "Theorem 1.2, (1.5)-(1.8), and Solution 1",
      "retained_prerequisites": [
        "The real plane change-of-variables theorem for the M\u00f6bius substitutions; the Jacobian computation alone does not supply that integral theorem."
      ]
    },
    {
      "id": "LG-MF-19-continuous-parameter-integration",
      "statement": "For the completed Eisenstein theta formula (2.4a), the entire remainder is jointly continuous in (t,z), holomorphic in t for each z, and for every compact t-set has, after multiplication by H and density y^(-2), the integrable cusp majorant C_A*y^(A+3/2+k-2)*exp(-4*pi*y), with a bounded compact-remainder majorant. LG-MF-01 Lemma 0.3 therefore makes its integral entire. The remaining rational terms give J_(f,g)(t)=integral_F H E*(z,t)dmu meromorphic on C with only possible simple poles at 0 and 1, residues -<f,g>/2 and +<f,g>/2, and reflection J(t)=J(1-t).",
      "proof": "LG-MF-19.html#2-the-completed-function-and-its-residue",
      "result": "Lemma 2.1, using the stated formulas from LG-MF-18"
    },
    {
      "id": "LG-MF-19-continuous-gamma-route",
      "statement": "The Gamma integral is holomorphic on Re(z)>0 and the Gauss integral limit is locally uniform there by fixed continuous integral majorants and compact uniform logarithm estimates. The earlier explicit finite-product proof then gives the meromorphic continuation and entire reciprocal used in Theorem2.2, without a general measurable convergence theorem.",
      "proof": "LG-MF-19.html#continuous-integral-route-for-the-gamma-facts",
      "result": "Continuous integral route for the Gamma facts, with NT-ZETA-03 Theorems1.1-1.2 and LG-MF-01 Lemma0.3",
      "retained_prerequisites": [
        "Real completeness and elementary continuous integration/real calculus",
        "The earlier complex locally uniform-limit and identity theorems"
      ]
    }
  ],
  "earlier_proofs": [
    {
      "lesson": "LG-MF-01",
      "result": "Lemma 0.3, nonnegative and absolutely integrable continuous series; rectangular and holomorphic-parameter integrals",
      "reader": "LG-MF-01.html#analytic-facts-for-subsequent-lessons"
    },
    {
      "lesson": "LG-MF-07",
      "result": "Lemma 4.0, local finiteness, compact boundary-nullity and width-one projected strip integration",
      "reader": "LG-MF-07.html#continuous-integrals-and-projected-modular-tiles"
    },
    {
      "lesson": "LG-MF-07",
      "result": "Theorem 2.1, invariant Petersson and Fourier coefficient bounds",
      "reader": "LG-MF-07.html#2-a-coefficient-estimate-from-compactness"
    },
    {
      "lesson": "LG-MF-18",
      "result": "Fourier and completed theta formulas, Theorems 2.2 and 3.2 and Corollary 3.3",
      "reader": "LG-MF-18.html"
    },
    {
      "lesson": "NT-ZETA-03",
      "result": "Theorems 1.1-1.2, Gamma integral and entire reciprocal",
      "reader": "../NT-ZETA/NT-ZETA-03.html#from-an-integral-to-an-entire-reciprocal"
    }
  ],
  "retained_prerequisites": [
    "Real completeness, elementary continuous Riemann integration and compactness.",
    "The real plane change-of-variables theorem for the M\u00f6bius substitutions; the Jacobian computation alone does not supply that integral theorem.",
    "General measurable fundamental-domain integration for the arbitrary-domain assertion in LG-MF-07 Theorem 1.1."
  ],
  "scope": "The continuous Eisenstein coset unfolding and diagonal Fourier-Mellin interchanges, and the specified Gaussian parameter integral. These records do not prove general measurable integration, the M\u00f6bius integral substitution theorem, Wiener-Ikehara, the Ramanujan-Petersson purity theorem, the historical quantitative Rankin-Selberg remainder, or higher-rank local/global integral theorems. The lesson retains those exact required obligations and useful alternatives."
}