▁death s ▁Category : Count s ▁of ▁R ennes ▁Category : D uk es ▁of ▁Brit t any ▁Category : 1 0 th - century ▁Bre ton ▁people ▁Category : 1 1 th - century ▁Bre ton ▁people <0x0A> </s> ▁In ▁number ▁theory , ▁T ate ' s ▁th esis ▁is ▁the ▁ 1 9 5 0 ▁Ph D ▁th esis ▁of ▁ ▁completed ▁under ▁the ▁super vision ▁of ▁Emil ▁Art in ▁at ▁Pr inc eton ▁University . ▁In ▁it , ▁T ate ▁used ▁a ▁translation ▁invariant ▁integration ▁on ▁the ▁locally ▁compact ▁group ▁of ▁id eles ▁to ▁lift ▁the ▁z eta ▁function ▁tw isted ▁by ▁a ▁He cke ▁character , ▁i . e . ▁a ▁ ▁He cke ▁L - function , ▁of ▁a ▁number ▁field ▁to ▁a ▁z eta ▁integral ▁and ▁study ▁its ▁properties . ▁Using ▁harm onic ▁analysis , ▁more ▁precisely ▁the ▁Po isson ▁summ ation ▁formula , ▁he ▁proved ▁the ▁functional ▁equation ▁and ▁mer omorphic ▁continu ation ▁of ▁the ▁z eta ▁integral ▁and ▁the ▁He cke ▁L - function . ▁He ▁also ▁located ▁the ▁pol es ▁of ▁the ▁tw isted ▁z eta ▁function . ▁His ▁work ▁can ▁be ▁viewed ▁as ▁an ▁elegant ▁and ▁powerful ▁reform ulation ▁of ▁a ▁work ▁of ▁Er ich ▁He cke ▁on ▁the ▁proof ▁of ▁the ▁functional ▁equation ▁of ▁the ▁He cke ▁L - function . ▁Er ich ▁He cke ▁ ▁used ▁a ▁generalized ▁the ta ▁series ▁associated ▁to ▁an ▁algebraic ▁number ▁field ▁and ▁a ▁lattice ▁in ▁its ▁ring ▁of ▁integers . ▁ ▁I was awa –