DC9 CREW AND HIGH PERFORMANCE MIL ACFT HAD AN NMAC IN ZJX CLASS E AIRSPACE.

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PASSING APPROX 16500 FT MSL; CLBING AT APPROX 2500 FPM; ATC NOTIFIED US OF VFR TFC AT 10 O'CLOCK POS; 6 MI; LEVEL 17500 FT. ACFT HDG; TYPE; AND SPD NOT STATED BY ATC. TARGET WAS NOT ON TCASII SCREEN AT THIS TIME. BEGAN A VISUAL SEARCH; BUT NO CONTACT MADE. AT APPROX 16800 FT MSL; TARGET APPEARED ON OUTER EDGE OF TCASII DISPLAY AT 10 O'CLOCK POS. TARGET WAS 'JUMPING' TO VARIOUS POS ON THE TCASII; AND THEN INSTANTLY TO THE 9 O'CLOCK POS ON THE INNER CIRCLE OF THE DISPLAY. AT APPROX 17300 FT MSL; I DECIDED TO STOP CLB; AND BEGAN A DSCNT. AS I PUSHED ACFT OVER; A TCASII WARNING 'INCREASE CLB' OCCURRED. I COMPLIED WITH THE RA; REVERSING ACFT DIRECTION AGAIN TO BEGIN CLB. AS I DID THIS; A SECOND TCASII RA OCCURRED 'CLB; CLB NOW.' I COMPLIED WITH THIS RA COMMAND. THE SUSPECT ACFT PASSED UNDERNEATH US. SEVERAL MINS LATER; I INFORMED ATC OF THE RA AND ASKED IF HE KNEW WHAT THE ACTUAL SEPARATION WAS. HIS RESPONSE WAS 'NOT MUCH FROM WHAT I CAN SEE.' HE THEN INFORMED US IT WAS A MIL ACFT (IMPLYING A HIGH RATE OF SPD). A LOGBOOK WRITE-UP WAS MADE UPON LNDG IN DTW FOR THE JUMPING CHARACTERISTIC OF THE TARGET. AS FAR AS A WAY TO PREVENT THIS OCCURRENCE IN THE FUTURE; THE ONLY PERSON THAT HAD INFO ON BOTH ACFT WAS THE CTLR; AND YET HE VECTORED ME ACROSS THE FLT PATH OF THE MIL ACFT. ATC ASSIGNED MY HDG; KNEW THE SPD AND DIRECTION OF BOTH ACFT; KNEW THAT I WAS CLBING AND YET DID NOT OFFER A HDG FOR SEPARATION 'BECAUSE I WAS ON AN IFR FLT PLAN; AND THE OTHER ACFT WAS ON A VFR FLT PLAN.' THIS IS THE PROB. ATC SHOULD STILL BE RESPONSIBLE FOR SEPARATION.