Mathematical Sciences: A Conjecture Which Implies the       Riemann Hypothesis
{'Code': '03040102', 'Directorate': {'Abbreviation': 'MPS', 'LongName': 'Direct For Mathematical & Physical Scien'}, 'Division': {'Abbreviation': 'DMS', 'LongName': 'Division Of Mathematical Sciences'}}
{'SignBlockName': 'John V. Ryff', 'PO_EMAI': None, 'PO_PHON': None}
Work to be done on this project will focus on the                          solution of the celebrated Riemann hypothesis.  The                             conjecture concerns the location of the roots of a certain                      function of a complex variable defined initially when the                       real part of the variable exceeds unity.  Its name is the                       zeta-function or Riemann zeta-function, although it was known                   to Leonard Euler, appearing in 1737.  It is a function of                       fundamental importance in number theory and in the higher                       theory of special functions.  By means of analytic                              continuation the zeta function can be shown to exist and be                     differentiable for all complex values, save for unity.                          Further analysis shows that all roots of the zeta-function                      lie in a strip in which the real part of the variable lies                      between zero and one.  The object of this work will be to                       show that the roots lie along the line where the real part of                   the variable equals one-half.  It has been known since 1914                     that infinitely many roots lie along this line.  The present                    work plans to exploit new mathematical techniques in                            eventually resolving the question.  Specifically, the theory                    of Hilbert spaces whose elements are entire functions subject                   to certain analytic restrictions will be the central                            structure for the analysis.  While not all such spaces are                      particularly useful in this context, certain of them can be                     shown to have elements whose roots lie along a line.  The                       obstruction which must be cleared is that of the zeta-                          function's singularity.  If an analogous theory can be                          developed allowing for isolated singularities, the result                       will follow.  The first step in this process will be to prove                   the conjecture for a related class of functions called                          character zeta-functions.  The way appears open to achieving                    this during the course of this award.
{'FirstName': 'Louis', 'LastName': 'de Branges', 'PI_MID_INIT': None, 'PI_SUFX_NAME': None, 'PI_FULL_NAME': 'Louis de Branges', 'EmailAddress': 'branges@math.purdue.edu', 'NSF_ID': '000225690', 'StartDate': '04/01/1988', 'EndDate': None, 'RoleCode': 'Principal Investigator'}
{'Name': 'Purdue Research Foundation', 'CityName': 'WEST LAFAYETTE', 'ZipCode': '479064353', 'PhoneNumber': '3174946200', 'StreetAddress': '1281 WIN HENTSCHEL BLVD STE 2500', 'StreetAddress2': None, 'CountryName': 'United States', 'StateName': 'Indiana', 'StateCode': 'IN', 'CONGRESSDISTRICT': '04', 'CONGRESS_DISTRICT_ORG': 'IN04', 'ORG_UEI_NUM': 'C1G1LGYJF5G5', 'ORG_LGL_BUS_NAME': 'PURDUE RESEARCH FOUNDATION', 'ORG_PRNT_UEI_NUM': 'YRXVL4JYCEF5'}
{'Name': None, 'CityName': None, 'StateCode': None, 'ZipCode': None, 'StreetAddress': None, 'CountryCode': None, 'CountryName': None, 'StateName': None, 'CountryFlag': None, 'CONGRESSDISTRICT': None, 'CONGRESS_DISTRICT_PERF': '""'}
{'Code': '1261', 'Text': 'CLASSICAL ANALYSIS'}
{'url': 'https://www.nsf.gov/awardsearch/download?DownloadFileName=1988&All=true', 'xml': '8800416.xml'}