Inter-universal Teichmuller Theory III -------------------------------------- ・Slightly modified the first display of Definition 3.8, (i) ・Corrected a misprint ("(from above)" ---> "[from above]") in Step (x) of the proof of Corollary 3.12 Inter-universal Teichmuller Theory IV ------------------------------------- ・Slightly modified (i.e., in fact, **strengthened**) the statement and proof of Proposition 1.1 ・Slightly modified (i.e., in fact, **strengthened**) the inclusions in the statement and proof of Proposition 1.2, (ii), (iii) ・Slightly modified the wording of the statement of Proposition 1.3, (i) ・Slightly modified the first two sentences of the proof of Proposition 1.3 ・Slightly modified the wording of various portions of the second paragraph of the proof of Proposition 1.3 ・Slightly modified the phrasing in the final portion of the statement of Proposition 1.4, (i) ・Slightly modified the statement and proof of Proposition 1.4, (iii), so as to take into account the **stengthened** versions of 　Proposition 1.2, (ii), (iii) ・Slightly modified the final portion of the statement of Proposition 1.8, (vii) ・At the beginning of Definition 1.9, deleted the portion concerning the notation "ord_v(-)" (since this notation is never used) and 　added an explanation of the notation "f_v" ・Corrected a misprint in the statement of Theorem 1.10 ("effective divisor" ---> "effective arithmetic divisor") ・Slightly modified the final two displays of the statement of Theorem 1.10, as well as Steps (iii) (the last two displays), (v), (viii) of the proof of Theorem 1.10, so as to take into account the **stengthened** versions of Propositions 1.2, (ii), (iii); 　1.4, (iii) ・In Step (iii) of the proof of Theorem 1.10, slightly modified the definition of "F_\ast" and added a reference to Proposition 1.8 　immediately following (R4) ・Slightly modified the the description of "\beta_e" in Step (v) of the proof of Theorem 1.10 ・Corrected misprints ("=" ---> "\le") in the second and third displays of Step (viii) of the proof of Theorem 1.10 ・Slightly modified the text surrounding and including the final display of Remark 1.10.6, (ii) ・Corrected a misprint ("3.11" ---> "3.11, (i)") in Remark 1.10.7, (ii) ・Slightly modified the wording of Remark 1.10.7, (ii), surrounding the discussion of Step (v) of the proof of Theorem 1.10 ・Slightly modified the wording (involving the term "normalized") of the discussion of notation immediately preceding the statement 　of Corollary 2.2, as well as in the statement of Corollary 2.2, (i) ・Removed unnecessary parentheses from "\delta" in the proof of Corollary 2.2 (7 instances) ・In the proof of Corollary 2.2, slightly modified the use of the word "observe" in the discussion immediately preceding (P5) and 　added an explanation of the notation "V_mod" to (P5) ・Slightly modified the discussion immediately following the phrase "apply Theorem 1.10" in the proof of Corollary 2.2 so as to 　take into account the **strengthened** version of Theorem 1.10 ・Added an explanatory phrase concerning the notation "\delta" at the beginning of Remark 2.2.1, (i) ・In Remark 2.2.1, (i), (v), slightly modified the phrasing of the first sentence that mentions the Riemann hypothesis ・Corrected a misprint ("thought of as a sort" ---> "thought of as a sort of") in the final sentence of Remark 2.2.4, (iii) ・Corrected a misprint ("and real" ---> "and the real") in the discussion immediately following (PB1) in Remark 2.3.5, (i) ・Slightly modified the wording of the text immediately preceding (PB4) in Remark 2.3.5, (ii) ("(PB2)" ---> "both (PB2) and (PB3)") ・Slightly modified the wording of the final display of Remark 2.3.5, (iii) ・Slightly modified the explanations at the beginning of Definition 3.3 and immediately following the final display of Remark 3.3.1, 　(i)