Proposition 5.9 . [0542] Original official author HTML, exact retained edition. Historical TeX conversion verdicts remain unchanged. Cited-edition alignment and mathematical self-containment are not assessed.
Source coverage notes Source fidelity gap: the original arXiv HTML omits an author TeX footnote attached to equation e:def-omega, including its reference to Remark r:error-function. The retained HTML is preserved as published; the original author TeX remains available. TeX correspondence is not complete. Complete original source context · Original author HTML
Proposition 5.9 .
There exists some dimensional constant C n > 0 C_{n}>0 y ≤ − 1 y\leq-1
(5.116) C n − 1 ⋅ Q − 1 4 − 1 2 n ⋅ ( − y ) − 1 ⋅ e y + F ( t 0 ) Γ ( Q + 1 ) \displaystyle C_{n}^{-1}\cdot Q^{-\frac{1}{4}-\frac{1}{2n}}\cdot\frac{(-y)^{-1}\cdot e^{y+F(t_{0})}}{\Gamma(Q+1)} ≤ Ψ ♭ ( β , α , y ) ≤ C n ⋅ Q 1 4 ⋅ e y + F ( t 0 ) Γ ( Q + 1 ) , \displaystyle\leq\Tri(\beta,\alpha,y)\leq C_{n}\cdot Q^{\frac{1}{4}}\cdot\frac{e^{y+F(t_{0})}}{\Gamma(Q+1)},
(5.117) C n − 1 ⋅ Q − 1 4 ⋅ ( − y ) 1 − 2 α 4 ⋅ e y + G ( u 0 ) Γ ( Q + 1 ) \displaystyle C_{n}^{-1}\cdot Q^{-\frac{1}{4}}\cdot\frac{(-y)^{\frac{1-2\alpha}{4}}\cdot e^{y+G(u_{0})}}{\Gamma(Q+1)} ≤ Φ ♯ ( β , α , y ) ≤ C n ⋅ ( − y ) 1 − 2 α 4 ⋅ e y + G ( u 0 ) Γ ( Q + 1 ) . \displaystyle\leq\Ku(\beta,\alpha,y)\leq C_{n}\cdot\frac{(-y)^{\frac{1-2\alpha}{4}}\cdot e^{y+G(u_{0})}}{\Gamma(Q+1)}.