# Stanisław Knapowski

Stanisław Knapowski (May 19, 1931 – September 28, 1967) was a Polish mathematician who worked on prime numbers and number theory. Knapowski published 53 papers despite dying at only 36 years old. [1]

Stanisław Knapowski
Knapowski expanded on the distribution of primes
BornMay 19, 1931
DiedSeptember 28, 1967 (aged 36)
Florida, USA
CitizenshipPolish
EducationPoznań University , Wrocław University and Adam Mickiewicz University
Known forPrime numbers and number theory
AwardsMazurkiewicz Prize, Rockefeller Scholarship
Scientific career
FieldsMathematician

## Life and education

Stanisław Knapowski was the son of Zifia Krysiewicz and Roch Knapowski. His father, Roch Knapowski was a lawyer in Poznań but later taught at Poznań University. The family moved to the Kielce province in south-eastern Poland after the German invasion of 1939 but returned to Poznań after the war. [1]

Stanisław completed his high school education in 1949 excelling at math and continued on at Poznań University to study mathematics. Later in 1952 he continued his studies at University of Wrocław and earned his master's degree in 1954.

Knapowski was appointment an assistant at Adam Mickiewicz University in Poznań under Władysław Orlicz and worked towards his doctorate. He studied under the direction of Pál Turán starting in Lublin in 1956. He published many of his papers with Turán and Turán wrote a short biography of his life and work in 1971 after his death.[2] Knapowski began to work in this area and finished his doctorate in 1957 “Zastosowanie metod Turaná w analitycznej teorii liczb” ("Certain applications of Turan's methods in the analytical theory of numbers").

Knapowski had the opportunity to work abroad. He spent a year in Cambridge and worked with Louis J. Mordell and listened to classes by J.W.S. Cassels and Albert Ingham. He then moved on to Belgium, France and The Netherlands.

Knapowski returned to Poznań to finish another thesis to complete a post-doctoral qualification needed to lecture at a German university.[2][1] "On new "explicit formulas" in prime number theory" in 1960.[3] In 1962 the Polish Mathematical Society awarded him their Mazurkiewicz Prize and he moved to Tulane University in New Orleans, United States. After a very short return to Poland, he left again and taught in Marburg in Germany, Gainesville, Florida and Miami, Florida.[2][4]

## Death

Knapowski was a good classical pianist. He was also an avid driver, but he died in a traffic accident where he lost control of his car, leaving the Miami airport.[2]

## Work

Knapowski expanded on the work of others in several fields of number theory, prime number theorem, modular arithmetic and non-Euclidean geometry.

### Number of times the Δ(n) prime sign changes

The relative error of ${\displaystyle {\tfrac {n}{\log n}}}$  and the logarithmic integral ${\displaystyle \operatorname {Li} (n)}$  as approximations to the prime-counting function. Both relative errors decrease to zero as ${\displaystyle n}$  grows, but the convergence to zero is much more rapid for the logarithmic integral.

Mathematicians work on primality tests to develop easier ways to find prime numbers when finding them by trial division is not practical. This has many applications in cybersecurity. There is no formula to calculate prime numbers. However, the distribution of primes can be statistically modelled. The prime number theorem, which was proven at the end of the 19th century, says that the probability of a randomly chosen number being prime is inversely proportional to its number of digits (logarithm). At the start of the 19th century, Adrien-Marie Legendre and Carl Friedrich Gauss suggested that as ${\displaystyle x}$  goes very large, the number of primes up to ${\displaystyle x}$  is asymptotic to ${\displaystyle x/\log x}$ , where ${\displaystyle \log x}$  is the natural logarithm of ${\displaystyle x}$ .

${\displaystyle \operatorname {li} (n)=\int _{0}^{n}{\frac {dt}{\log t}}}$

where the integral is evaluated at ${\displaystyle t=n}$ , also fits the distribution.

The prime counting function ${\displaystyle \pi (n)}$  is defined as the number of primes not greater than ${\displaystyle n}$ .[5]

And ${\displaystyle \Delta (n)=\pi (n)-\operatorname {li} (n)}$

Bernhard Riemann stated that ${\displaystyle \Delta (n)}$  was always negative but J.E. Littlewood later disproved this. In 1914 J.E. Littlewood proved that there are arbitrarily large values of x for which

${\displaystyle \pi (x)>\operatorname {Li} (x)+{\frac {1}{3}}{\frac {\sqrt {x}}{\log x}}\log \log \log x,}$

and that there are also arbitrarily large values of x for which

${\displaystyle \pi (x)<\operatorname {Li} (x)-{\frac {1}{3}}{\frac {\sqrt {x}}{\log x}}\log \log \log x.}$

Stanley Skewes then added an upper bound on the smallest natural number ${\displaystyle x}$ :

${\displaystyle \pi (x)>\operatorname {li} (x),}$

Knapowski followed this up and published a paper on the number of times ${\displaystyle \Delta (n)}$  changes sign in the interval ${\displaystyle \Delta (n)}$ .[6]

### Modular arithmetic

Knapowski worked in other areas of number theory. One area was on the distribution of prime numbers in different residue classes modulo ${\displaystyle k}$ .

Modular arithmetic modifies usual arithmetic by only using the numbers ${\displaystyle \{0,1,2,\dots ,n-1\}}$ , for a natural number ${\displaystyle n}$  called the modulus. Any other natural number can be mapped into this system by replacing it by its remainder after division by ${\displaystyle n}$ .[7]

The distribution of the primes looks random, without a pattern. Take a list of consecutive prime numbers and divide them by another prime (like 7) and keep only the remainder (this is called reducing them modulo 7). The result is a sequence of integers from 1 to 6. Knapowski worked to determine the parameters of this modular distribution[8]

## References

1. ^ a b c "Stanislaw Knapowski biography". www-history.mcs.st-andrews.ac.uk. Archived from the original on 2018-07-08. Retrieved 2019-01-04.
2. ^ a b c d Turán, Paul (1971). "Stanisław Knapowski (19 V 193 – 28 IX 1967)". Colloquium Mathematicum. 23 (2): 309–321. doi:10.4064/cm-23-2-309-321. ISSN 0010-1354.
3. ^ Knapowski, Stanisław (1960). "On new "explicit formulas" in prime number theory II" (PDF). Acta Arithmetica. 6: 23–35. doi:10.4064/aa-6-1-23-35. Archived (PDF) from the original on 2018-07-23. Retrieved 2019-01-09.
4. ^ Browkin, J. (2017). "Stanisław Knapowski". Wiadomości Matematyczne. 14 (1). doi:10.14708/wm.v14i1.1966. ISSN 2543-991X.
5. ^ Crandall & Pomerance 2005, p. 6 Archived 2019-03-23 at the Wayback Machine.
6. ^ Knapowski, Stanisław (1962). "On sign-changes of the difference π(x)-li(x)". Acta Arithmetica. 7 (2): 107–119. doi:10.4064/aa-7-2-107-119. ISSN 0065-1036.
7. ^ Kraft & Washington (2014), Proposition 5.3 Archived 2019-03-23 at the Wayback Machine, p. 96.
8. ^ a b Knapowski, Stanisław (1955). "On the greatest prime factors of certain products". Annales Polonici Mathematici. 2 (1): 56–63. doi:10.4064/ap-2-1-56-63. ISSN 0066-2216.
9. ^ Coxeter, H. S. M.; Kulczycki, S.; Knapowski, S. (1962). "Non-Euclidean Geometry". The American Mathematical Monthly. 69 (9): 937. doi:10.2307/2311278. ISSN 0002-9890. JSTOR 2311278.
10. ^ Knapowski, Stanisław (1958). "On prime numbers in an arithmetical progression". Acta Arithmetica. 4 (1): 57–70. doi:10.4064/aa-4-1-57-70. ISSN 0065-1036.
11. ^ Knapowski, Stanisław (1958). "On the Möbius function". Acta Arithmetica. 4 (3): 209–216. doi:10.4064/aa-4-3-209-216. ISSN 0065-1036.
12. ^ Knapowski, S. (1969). "On a theorem of Hecke". Journal of Number Theory. 1 (2): 235–251. Bibcode:1969JNT.....1..235K. doi:10.1016/0022-314X(69)90043-2. ISSN 0022-314X.
13. ^ Knapowski, Stanisław (1968). "On Siegel's Theorem". Acta Arithmetica. 14 (4): 417–424. doi:10.4064/aa-14-4-417-424. ISSN 0065-1036.