**Stanislaw Knapowski**'s mother was Zifia Krysiewicz and his father was Roch Knapowski who, at the time of Stanislaw's birth, was a lawyer in Poznan but was later professor at Poznan University. Stanislaw was brought up in Poznan until the Second World War began. Hitler declared war on Poland on 31 August 1939 and the German attack began on 1 September. The German armies rapidly overcame the Polish resistance with Warsaw surrendering on 27 September and all Polish resistance came to an end on 6 October. The Knapowski family were now living in an occupied country.

The occupying German forces expelled thousands of the Polish people from the west of the country and the Knapowskis were forced to settle in the Kielce province in south-eastern Poland. Four German extermination camps were located in Kielce and young Stanislaw grew up there in very difficult circumstances. After the war ended in 1945 efforts were made to allow displaced people to return to their homes, and the Knapowskis returned to Poznan where Stanislaw attended the local gymnasium, excelling in mathematics. He completed his school education in 1949 and at that time entered Poznan University to continue his study of mathematics. In 1952 Knapowski went to Wroclaw University to continue his studies. He was appointed as an assistant at Wroclaw while still an undergraduate, being awarded his master's degree in 1954.

Knapowski returned to the Adam Mickiewicz University in Poznan where he was appointed as an assistant. He also worked under Wladyslaw Orlicz towards his doctorate but was greatly influenced by Turán. This came about in September 1956 when Turán gave a series of lectures on a new analytic method in Lublin. Turán writes [4]:-

[

Knapowski]came to Lublin and in several conversations I realised his quick and deep understanding. We spoke much on possible applications of the method in analytic number theory.

Indeed this is precisely what Knapowski began to work on and he obtained his doctorate in 1957 from the Adam Mickiewicz University in Poznan for his thesis *Zastosowanie metod Turaná w analitycznej teorii liczb* Ⓣ . After this Knapowski went abroad. His first visit was to England where he spent just under a year working in Cambridge. He worked closely with Mordell and attended courses by Cassels and Ingham. He continued his continental tour making shorter visits to other British universities and then universities in Belgium, France and Holland.

Back in the Adam Mickiewicz University in Poznan Knapowski submitted his habilitation thesis *On new "explicit formulae" in prime number theory* which was published in *Acta Arithmetica* in 1960. Turán was one of Knapowski's examiner's for his habilitation. Two years later the Polish Mathematical Society awarded him their Mazurkiewicz Prize. During the academic year 1962-63 he taught at Tulane University in New Orleans, USA, returning to Poland during 1963. However, just over a year later, he left Poland and began teaching at a number of universities in Germany and the United States. In particular he taught at Marburg in Germany and Gainesville and Miami in the United States.

Knapowski died when only 36 years old [4]:-

He apparently lost control of his car when returning from the airport in Miami...

Despite his short mathematical career, Knapowski published 53 paper, mostly in algebra and number theory. Eighteen of these papers were written jointly with Turán. In [4] Turán writes of their collaboration:-

Most of the time our mathematical connection was by correspondence. Though this was mainly mathematical, his style and even handwriting indicated from the very beginning his highly cultured personality. This impression was confirmed by personal contact. With the assistance of the Polish Academy of Sciences he made longer visits to Hungary; we met on my short visits to Poland ... But also we met each other at some Western universities; particularly fruitful years were the summers of1963and1964at Ann Arbor and Columbus, Ohio. The long talks during evening strolls, whose main theme was mathematics and especially the further course of our joint work, was intermingled with discussions on music, literature and life, discussion which carefully concealed the serious behind jokes and usually ended, before returning home, at a student-association building with piano where he played Chopin and Liszt attracting a large audience. Car driving was one of his main hobbies; we made large excursions by car and according to my experience he was a safe driver(apart from a single occasion). It certainly did not occur to me that this will be fatal for him.

Two areas of number theory which received particular attention from Knapowski were the distribution of primes in different residue classes modulo *k*, and the sign changes in the remainder term in the prime number formula. Before indicating his contribution to the second of these areas we give some background.

On experimental evidence, after extensive calculation, Legendre in 1798 and Gauss in 1793 (according to a letter he wrote 50 years later) suggested that for large *n* the density of primes behaves like the function 1/log(*n*). Gauss's estimate, the logarithmic integral

li(

n) = ∫dt/ log(t)

where the integral is evaluated at *t* = *n*, also fits the distribution. Let π(*n*) be the number of primes less than *x* and let

Δ(

n) = π(n) - li(n).

Riemann conjectured in 1859 that Δ(*n*) was always negative, a result disproved by Littlewood in 1914. However, only in 1955 did Skewes give an upper bound for the smallest value of *n* for which Δ(*n*) became positive. However, no lower bound was known at that time. It was Knapowski who went much further and investigated *V*(*N*) the number of times Δ(*n*) changes sign in the interval Δ(*n*). His first paper on this topic *On sign changes of the difference *π(*x*) - *li*(*x*) appeared in *Acta Arithmetica* in 1962.

Among Knapowski's other number theory papers we mention: *On prime numbers in arithmetical progression* (1958), *On the Möbius function* (1958), *Contributions to the theory of the distribution of prime numbers in arithmetical progressions* (1961, 1962), *On Linnik's theorem concerning exceptional L-zeros* (1961), and *Further developments in the comparative prime number theory* (8 papers). He also wrote on other mathematical topics such as *On some criteria for indecomposability of polynomials* (1955) and *A theorem from finite group theory* (1956).

**Article by:** *J J O'Connor* and *E F Robertson*

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