Helmut Maier (mathematician)

Helmut Maier (born October 17, 1953 in Geislingen an der Steige ) is a German mathematician who deals with analytical number theory. He is a professor at Ulm University .

Helmut Maier, Oberwolfach 2008

Maier received his doctorate from J. Ian Richards at the University of Minnesota in 1981 (Some results on prime numbers based on the application of sieve theory). From 1984 to 1995 he was a professor at the University of Georgia .

In 1985 he proved a theorem about the asymptotic distribution of prime numbers in small intervals, which showed problems of the simple probabilistic model of the distribution of prime numbers by Harald Cramér (and showed that a well-known result by Atle Selberg of 1943, that the asymptotic behavior of the density of the Prime numbers described in small intervals except for possible exceptional values, cannot be improved in this regard). He used a matrix method that was also used in other problems of analytical number theory. In 1981 Maier proved the existence of chains of any length with large distances between successive prime numbers (after Erdős had already shown this for chains of length 2).

He published with Paul Erdős and even won one of Erdős' private prizes. Both drove a taxi together in Athens (Georgia) and Maier told Erdős about a theorem he had recently proven, whereupon Erdős said that this might be one of his price problems and after checking, he paid promptly. Maier also received a prize from Erdős for a work with Gérald Tenenbaum , in which she proved a conjecture by Erdős that almost all integers have factors a, b . ${\displaystyle a<b<2a}$

Maier also improved the values ​​for the bound in Erdős' theorem on the distances between consecutive prime numbers (see prime number twin ). He could exceed the upper bound for:

${\displaystyle \Delta =\liminf _{n\rightarrow \infty }{\frac {p_{n+1}-p_{n}}{\log(p_{n})}}}$

1986 press 0.2486 .., the best value up to the breakthrough by Goldston , Yildirim and Pintz (2005).

With Carl Pomerance , he improved the results of Erdős (1935), Robert Alexander Rankin (1938) and Arnold Schönhage (1963) over a lower limit for the largest gap between consecutive prime numbers (Erdős-Rankin problem).

He also deals with exponential sums, zeta functions, and circular division polynomials.

Fonts (selection)

In addition to the works cited in the footnotes:

• with A. Hildebrand Irregularities of the distribution of primes in short intervals , J. Reine Angewandte Mathematik, Volume 397, 1989, pp. 162-193
• with Carl Pomerance : Unusually large gaps between consecutive primes , Trans. Amer. Math. Soc., Vol. 322, 1990, pp. 201-237

Web links

• Homepage in Ulm

Individual evidence

1. ↑ Helmut Maier in the Mathematics Genealogy Project (English)
2. ↑ Faculty members University of Georgia, Mathematics ( Memento of the original from March 1, 2006 in the Internet Archive ) Info: The archive link was automatically inserted and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice.  
3. ↑ Maier, Primes in short intervals The Michigan Mathematical Journal, Volume 32, 1985, pp. 221–225, Project Euclid ( Memento of the original from March 4, 2016 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. 
4. ↑ Maier Chains of large gaps between consecutive primes , Advances in Mathematics, Volume 39, 1981, pp. 257-269, online
5. ↑ David Wells Prime numbers , 2011, p. 61
6. ↑ Charles soap Erdős' hard to win prizes quietly draw bounty hunters , Science, Vol 296, May 2002, pp 39-40
7. ↑ Maier, Tenenbaum On the set of divisors of an integer , Invent. Math., Vol. 76, 1984, pp. 121-128
8. ↑ Maier Small differences between prime numbers , Michigan J. Math., Volume 35, 1988, pp. 323-344
9. ↑ You proved ${\displaystyle \Delta =0}$
10. ↑ Small gaps between consecutive primes. Recent work of Yildirim and Goldston