Harold M. Stark

Here are two principal threads of my research. This is hardly all inclusive, but represents what I think I am best known for. I have tried to make this slightly understandable to non-mathematicians (at the cost of not mentioning some of my favorite results! My work on the Brauer-Siegel theorem is probably the biggest omission.)-Harold Stark. My research in number theory covers analytic number theory, algebraic number theory, transcendence theory and modular forms. Perhaps my principal interest is to apply analytic methods to obtain algebraic results. An algebraic number field is a generalization of the rational numbers. A quadratic field occurs when every number in the field satisfies a quadratic equation. If the roots of all these equations are real, the field is a real quadratic field; otherwise the field is a complex quadratic field. Inside a number field is an analogue of the ordinary integers: the algebraic integers of the field. Ordinary integers have the property that every integer is uniquely expressible as plus or minus one times a product of primes. Gauss conjectured that there are precisely nine complex quadratic fields whose integers have unique factorization. I gave in 1967 the first complete accepted proof of this conjecture. This result was obtained from a study of the value at s=1 of analytic objects, the zeta and L-functions attached to complex quadratic fields. Over a period of two decades, this led to the "Stark Conjectures", which relate the values at s=1 over a general field k to algebraic numbers in a corresponding field extension of k. Some fascinating instances of this conjecture allow us to numerically generate class fields from the values of L-functions. More than one computer number theory package now generates certain class fields in this way.