Stanisław Knapowski

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

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

Thus the difference π(x) − Li(x) changes sign infinitely many times.

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]

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]

• Non-Euclidean geometry[9]
• On prime numbers in arithmetical progression[10]
• Möbius function[11]
• On a theorem of Hecke[12]
• On Linnik's theorem concerning exceptional L-zeros” (1961)
• Comparative prime number theory[8]
• On Siegel's Theorem[13]