Corollary 4.11.1 . [0529] 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
Corollary 4.11.1 .
The following hold:
(1)
Let C > 0 C>0 be fixed, then for T T large, r − ≤ C r_{-}\leq C implies z ≥ − 3 4 T − 1 log T z\geq-\frac{3}{4}T^{-1}\log T .
(2)
Let c > 0 c>0 be fixed. Then for T T large if r ≤ c T − 1 log T r\leq cT^{-1}\log T for some c < 1 / 2 c<1/2 , then
log r − ≤ − 1 2 ( 1 2 − c ) log T . \log r_{-}\leq-\frac{1}{2}(\frac{1}{2}-c)\log T.
(3)
Let C ≥ 1 C\geq 1 be fixed, then for T T large, z ≥ − C z\geq-C implies log r − ≤ ( C + 1 ) T \log r_{-}\leq(C+1)T