Guido Hoheisel Explained

Guido Karl Heinrich Hoheisel (14 July 1894 – 11 October 1968) was a German mathematician and professor of mathematics at the University of Cologne.

Academic life

He did his PhD in 1920 from the University of Berlin under the supervision of Erhard Schmidt.^[1] During World War II Hoheisel was required to teach classes simultaneously at three universities, in Cologne, Bonn, and Münster.^[2] His doctoral students include Arnold Schönhage.

Hoheisel contributed to the journal Deutsche Mathematik.

Selected results

Hoheisel is known for a result on gaps between prime numbers:^[3] He proved that if π(x) denotes the prime-counting function, 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 then

p_n+1 − p_n < p_n^θ,

for all sufficiently large n. He showed that one may take

θ = 32999/33000 = 1 - 0.000(03),

with (03) denoting periodic repetition.

Selected works

• Gewöhnliche Differentialgleichungen 1926;^[4] 2nd edition 1930;^[5] 7th edition 1965
• Partielle Differentialgleichungen 1928; 3rd edition 1953
• Aufgabensammlung zu den gewöhnlichen und partiellen Differentialgleichungen 1933^[6]
• Integralgleichungen 1936;^[7] revised and expanded 2nd edition 1963
• Existenz von Eigenwerten und Vollständigkeitskriterium 1943
• Integral equations translated by A. Mary Tropper [1968, c1967]

Notes and References

1. .
2. .
3. G. Hoheisel, Primzahlprobleme in der Analysis, Berliner Sitzungsberichte, pages 580-588, (1930)
4. Cohen, A.. Review: Gewöhnliche Differentialgleichungen by G. Hoheisel. Bull. Amer. Math. Soc.. 1929. 35. 1. 136–137. 10.1090/s0002-9904-1929-04716-5. free.
5. Longley, W. R.. Review: Gewöhnliche Differentialgleichungen by G. Hoheisel. Bull. Amer. Math. Soc.. 1932. 38. 7. 478–479. 10.1090/s0002-9904-1932-05447-7. free.
6. Longley, W. R.. Review: Aufgabensammlung zu den gewöhnlichen und partiellen Differentialgleichungen by G. Hoheisel. Bull. Amer. Math. Soc.. 1933. 39. 9. 652–653. 10.1090/s0002-9904-1933-05695-1. free.
7. Longley, W. R.. Review: Integralgleichungen by G. Hoheisel. Bull. Amer. Math. Soc.. 1937. 43. 1. 14–15. 10.1090/s0002-9904-1937-06480-9. free.