In 1919 Cramér was appointed assistant professor at the University of Stockholm. He began to produce a series of papers on analytic number theory, and he addressed the Scandinavian Congress of Mathematicians in 1922 on Contributions to the analytic theory of numbers detailing his work on the topic up to that time. One interesting paper by Cramér over this period which we should note is one he published in 1920 discussing prime number solutions x, y to the equation ax + by = c, where a, b, c are fixed integers. Note that if a = b = 1 then the question of whether this equation has a solution for all c is Goldbach's conjecture, while if a = 1, b = -1, c = 2, then the question about prime solutions to x = y + 2 is the twin prime conjecture. Cramér's work in prime numbers is put into the context of the history of prime number theory from Eratosthenes to the mid 1990s in .
It was not only through his work on number theory that Cramér was led towards probability theory. He also had a second job, namely as an actuary with the Svenska Life Assurance Company. This led him to study probability and statistics which then became the main area of his research. In 1927 he published an elementary text in Swedish Probability theory and some of its applications. In 1929 he was appointed to a newly created chair in Stockholm, becoming the first Swedish professor of Actuarial Mathematics and Mathematical Statistics.
Cramér became interested in the rigorous mathematical formulation of probability in work of the French and Russian mathematicians such as Paul Lévy, Sergei Bernstein, and Aleksandr Khinchin in the early 1930s, but in particular the axiomatic approach of Kolmogorov. The results of his studies were written up in his Cambridge publication Random variables and probability distributions which appeared in 1937. This was to lead to later work on stationary stochastic processes. By the mid 1930s Cramér's attention had turned to look at the approach of the English and American statisticians such as Fisher, Neyman and Egon Pearson (Karl Pearson's son). These he described as admirable but :-
... not quite satisfactory from the point of view of mathematical rigour.Masani in  describes the beginnings of Cramér's work on stochastic processes as follows:-
The first phase, beginning at the start of World War II, is devoted to extending the 1934 results of Khinchin on univariate stationary stochastic processes to multivariate stationary stochastic processes, and to studying the connections between Khinchin's work and the earlier cognate work on generalised harmonic analysis by Norbert Wiener [in] 1930.During World War II Cramér was to some extent cut off from the rest of the academic world. However he gave shelter to W Feller who was forced out of Germany by Hitler's anti-Jewish policies in 1934. By the end of World War II Cramér had written his masterpiece Mathematical Methods of Statistics. The book was first published in 1945, and republished as recently as 1999. The book combines the two approaches to statistics described above and the latest reprinting is described as follows:-
In this classic of statistical mathematical theory, Harald Cramér joins the two major lines of development in the field: while British and American statisticians were developing the science of statistical inference, French and Russian probabilists transformed the classical calculus of probability into a rigorous and pure mathematical theory. The result of Cramér's work is a masterly exposition of the mathematical methods of modern statistics that set the standard that others have since sought to follow.In 1950 Cramér became the President of Stockholm University. Despite holding this post until he retired in 1961, Cramér still found time to undertake research despite the large administrative burden placed on him. The second phase of Cramér's work on stochastic processes :-
... began around 1950 and lasted until the early 1980s... [It] is devoted to the analysis of non-stationary processes, specifically to determining the extent to which the representations available for stationary process survive for non-stationary ones.Cramér's Collected Works were published in 1994. Paul Embrechts, in his review of the two volumes, writes:-
One finds treated such fields as number theory, function theory, mathematical statistics, probability and stochastic processes, demography, insurance risk theory, functional analysis and the history of mathematics. Such highlights as the probabilistic method in the study of asymptotic properties of prime numbers, the spectral analysis of stationary processes, the mathematical foundation of inference and the fundamental work on risk theory all add up to a brilliant career as a scientist.Another reviewer writes:-
This book is a classic, not least for its combination of lucidity and rigour. ... It belongs on the shelf of anyone interested in statistical methods.We should give two specific results which we have not mentioned previously which will be remembered as major contributions, namely his work on the central limit theorem and his beautiful theorem that if the sum of two independent random variables is normal then all are normal.
There have been many tributes to Cramér. Edward Phragmen wrote:-
Harald Cramér belonged to a generation of mathematicians for which it was self-evident that mathematics constitutes one of the highest forms of human thought, perhaps even the highest. For these mathematicians numbers were a necessary form of human thought, and the science of numbers was a central humanistic discipline with a cultural value of its own, completely independent of its role as auxiliary science in technical or other areas. This does not however mean that they underestimated the importance of 'using theoretical knowledge to obtain practical know-how'.Blom in  sums up Cramér's contribution with simple but effective words:-
He was a great scientist and a good man.
Article by: J J O'Connor and E F Robertson
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