Helmut Maier in 2008
17 October 1953 |
|Institutions||University of Ulm
University of Michigan
Institute for Advanced Study, Princeton
|Alma mater||University of Ulm
University of Minnesota (Ph.D.)
|Doctoral advisor||J. Ian Richards|
|Known for||Maier's matrix method
Helmut Maier (born October 17, 1953) is a German mathematician and professor at the University of Ulm, Germany. He is known for his contributions in analytic number theory and mathematical analysis and particularly for the so-called Maier's matrix method as well as Maier's theorem for primes in short intervals. He has also done important work in exponential sums and trigonometric sums over special sets of integers and the Riemann zeta function. His research has been groundbreaking and deeply influential.
Helmut Maier graduated with a Diploma in Mathematics from the University of Ulm in 1976, under the supervision of Hans-Egon Richert. He received his Ph.D. from the University of Minnesota in 1981, under the supervision of J. Ian Richards.
Research and academic positions
Maier's Ph.D. thesis was an extension of his paper H. Maier, Chains of large gaps between consecutive primes, Advances in Mathematics, 39 (1981), 257–269. In this paper Maier applied for the first time what is now known as Maier's matrix method. This method later on led him and other mathematicians to the discovery of unexpected irregularities in the distribution of prime numbers. There have been various other applications of Maier's Matrix Method, such as on irreducible polynomials and on strings of consecutive primes in the same residue class.
After postdoctoral positions at the University of Michigan and the Institute for Advanced Study, Princeton, Maier obtained a permanent position at the University of Georgia. While in Georgia he proved that the usual formulation of the Cramér model for the distribution of prime numbers is wrong. This was a completely unexpected result. Jointly with Carl Pomerance he studied the values of Euler's φ(n)-function and large gaps between primes. During the same period Maier investigated as well the size of the coefficients of cyclotomic polynomials and later collaborated with Sergei Konyagin and E. Wirsing on this topic. He also collaborated with Hugh Lowell Montgomery on the size of the sum of the Möbius function under the assumption of the Riemann Hypothesis. Maier and Gérald Tenenbaum in joint work investigated the sequence of divisors of integers, solving the famous propinquity problem of Paul Erdős. Since 1993 Maier is a Professor at the University of Ulm, Germany.
Helmut Maier and Michael Th. Rassias have worked on problems related to the Estermann zeta function and the Nyman-Beurling criterion on the Riemann Hypothesis involving the distribution of certain cotangent sums. They have also studied large gaps between consecutive prime numbers containing perfect powers of prime numbers building upon work of Kevin Ford, Ben Joseph Green, Sergei Konyagin, James Maynard and Terence Tao.
Other collaborators of Helmut Maier include Paul Erdős, C. Feiler, John Friedlander, Andrew Granville, D. Haase, A. J. Hildebrand, Michel Laurent Lapidus, J. W. Neuberger, A. Sankaranarayanan, A. Sárközy, Wolfgang P. Schleich, Cameron Leigh Stewart.
- Lagarias, Jeffrey (2013). "Euler's constant: Euler's work and modern developments". Bulletin of the American Mathematical Society. 50 (4): 572.
- Pomerance, C.; Rassias, M. Th. (2015). Analytic Number Theory. Springer, New York. pp. v–vi.
- Granville, Andrew (1994). "Unexpected irregularities in the distribution of prime numbers". Proc. Intern. Congress Math., Zürich: 388–399.
- Monks, K.; Peluse, S.; Ye, L. (2013). "Strings of special primes in arithmetic progressions, (English summary)". Arch. Math. 101 (3): 219–234. doi:10.1007/s00013-013-0544-x.
- Shiu, D. K. L. (2000). "Strings of congruent primes". J. London Math. Soc. 61 (2): 359–373. doi:10.1112/s0024610799007863.
- Ford, K.; Green, B. J.; Konyagin, S.; Maynard, J.; Tao, T. (2015). "Long gaps between primes". arXiv: [math.NT].
- Maier, H.; Rassias, M. Th. (2016). "Large gaps between consecutive prime numbers containing perfect k-th powers of prime numbers". Journal of Functional Analysis.