Guido Karl Heinrich Hoheisel (14 July 1894 – 11 October 1968) was a

German German(s) may refer to: * Germany (of or related to) **Germania (historical use) * Germans, citizens of Germany, people of German ancestry, or native speakers of the German language ** For citizens of Germany, see also German nationality law **Ger ...

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

and professor of mathematics at the

University of Cologne The University of Cologne (german: Universität zu Köln) is a university in Cologne, Germany. It was established in the year 1388 and is one of the most prestigious and research intensive universities in Germany. It was the sixth university to ...

Academic life

He did his PhD in 1920 from the

University of Berlin Humboldt-Universität zu Berlin (german: Humboldt-Universität zu Berlin, abbreviated HU Berlin) is a German public research university in the central borough of Mitte in Berlin. It was established by Frederick William III on the initiative o ...

under the supervision of

Erhard Schmidt Erhard Schmidt (13 January 1876 – 6 December 1959) was a Baltic German mathematician whose work significantly influenced the direction of mathematics in the twentieth century. Schmidt was born in Tartu (german: link=no, Dorpat), in the Govern ...

. During

World War II World War II or the Second World War, often abbreviated as WWII or WW2, was a world war that lasted from 1939 to 1945. It involved the vast majority of the world's countries—including all of the great powers—forming two opposin ...

Hoheisel was required to teach classes simultaneously at three universities, in

Cologne Cologne ( ; german: Köln ; ksh, Kölle ) is the largest city of the German western States of Germany, state of North Rhine-Westphalia (NRW) and the List of cities in Germany by population, fourth-most populous city of Germany with 1.1 m ...

Bonn The federal city of Bonn ( lat, Bonna) is a city on the banks of the Rhine in the German state of North Rhine-Westphalia, with a population of over 300,000. About south-southeast of Cologne, Bonn is in the southernmost part of the Rhine-Ruhr r ...

, and

Münster Münster (; nds, Mönster) is an independent city (''Kreisfreie Stadt'') in North Rhine-Westphalia, Germany. It is in the northern part of the state and is considered to be the cultural centre of the Westphalia region. It is also a state distr ...

. His doctoral students include

Arnold Schönhage Arnold Schönhage (born 1 December 1934 in Lockhausen, now Bad Salzuflen) is a German mathematician and computer scientist. Schönhage was professor at the Rheinische Friedrich-Wilhelms-Universität, Bonn, and also in Tübingen and Konstanz. ...

. Hoheisel contributed to the journal

Deutsche Mathematik ''Deutsche Mathematik'' (German Mathematics) was a mathematics journal founded in 1936 by Ludwig Bieberbach and Theodor Vahlen. Vahlen was publisher on behalf of the German Research Foundation (DFG), and Bieberbach was chief editor. Other editors w ...

Selected results

Hoheisel is known for a result on

gaps between prime numbers A prime gap is the difference between two successive prime numbers. The ''n''-th prime gap, denoted ''g'n'' or ''g''(''p'n'') is the difference between the (''n'' + 1)-th and the ''n''-th prime numbers, i.e. :g_n = p_ - p_n.\ W ...

: He proved that if π(x) denotes the

prime-counting function In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number ''x''. It is denoted by (''x'') (unrelated to the number ). History Of great interest in number theory is t ...

, then there exists a constant θ < 1 such that :π(''x'' + ''x''^θ) − π(''x'') ~ ''x''^θ/log(''x''), as ''x'' tends to infinity, implying that if ''p''_''n'' denotes the ''n''-th

prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

then :''p''_''n''+1 − ''p''_''n'' < ''p''_''n''^''θ'', for all

sufficiently large In the mathematical areas of number theory and analysis, an infinite sequence or a function is said to eventually have a certain property, if it doesn't have the said property across all its ordered instances, but will after some instances have pa ...

''n''. He showed that one may take :''θ = 32999/33000 = 1 - 0.000(03),'' with (03) denoting periodic repetition.

Selected works

*''Gewöhnliche Differentialgleichungen'' 1926; 2nd edition 1930; 7th edition 1965 *''Partielle Differentialgleichungen'' 1928; 3rd edition 1953 *''Aufgabensammlung zu den gewöhnlichen und partiellen Differentialgleichungen'' 1933 *''Integralgleichungen'' 1936; revised and expanded 2nd edition 1963 *''Existenz von Eigenwerten und Vollständigkeitskriterium'' 1943 *''Integral equations'' translated by A. Mary Tropper 968, c1967

References

{{DEFAULTSORT:Hoheisel, Guido 20th-century German mathematicians 1968 deaths 1894 births