▁all ▁, ▁an ▁assertion ▁sometimes ▁known ▁as ▁Le h mer ' s ▁conject ure . ▁Le h mer ▁verified ▁the ▁conject ure ▁for ▁ ▁( A post ol ▁ 1 9 9 7 , ▁p .   2 2 ). ▁The ▁following ▁table ▁summar izes ▁progress ▁on ▁finding ▁success ively ▁larger ▁values ▁of ▁ ▁for ▁which ▁this ▁condition ▁holds ▁for ▁all ▁. ▁ ▁Notes ▁ ▁References ▁ ▁Category : Mod ular ▁forms ▁Category : Mult ip lic ative ▁functions ▁Category : S rin iv asa ▁Ram an uj an <0x0A> </s> ▁D ale ▁D ues ing ▁( born ▁September ▁ 2 6 , ▁ 1 9 4 7 ) ▁is ▁an ▁American ▁bar it one . ▁As ▁an ▁opera ▁singer , ▁he ▁has ▁had ▁an ▁international ▁career ▁sp anning ▁five ▁dec ades . ▁ ▁D ues ing ▁grew ▁up ▁in ▁Mil w au kee , ▁Wisconsin . ▁He ▁studied ▁piano ▁throughout ▁child hood , ▁and ▁en rolled ▁at ▁Lawrence ▁University , ▁major ing ▁in ▁piano ▁performance . ▁He ▁switched ▁to ▁vocal ▁performance ▁while ▁in ▁college , ▁and ▁won ▁the ▁Metropolitan ▁Opera ▁Compet ition ▁in ▁his ▁final ▁year ▁of ▁study . ▁After ▁travel ing ▁to ▁Europe ▁with ▁a ▁Ful br ight ▁Sch olar ship , ▁D ues ing ▁made ▁a ▁name ▁for ▁himself ▁there . ▁ ▁D ues ing ▁has ▁performed ▁at ▁the ▁Metropolitan ▁Opera , ▁San ▁Francisco ▁Opera , ▁Ly ric ▁Opera ▁of ▁Chicago , ▁La ▁Scala ▁( as ▁Ar le c ch ino ▁in ▁Ari ad ne ▁auf ▁N ax os ,