Godfrey Harold Hardy, who died on December 1, 75 years ago, was one of the most important English mathematicians of the last century. Between the two world wars he lived in the universe of the universities of Cambridge and Oxford, shared with people like Bertrand Russell or John Maynard Keynes, who cultivated his friendship. His posts with John Edensor Littlewood are universally regarded as the archetype of a fruitful mathematical collaboration. The quality of his work led Harald Bohr to state: “There are three great English mathematicians, Hardy, Littlewood and Hardy-Littlewood.”
His other celebrated collaborator was the self-taught mathematical genius Srinivasa Ramanujan, whom he discovered and helped promote. His fascinating scientific and vital adventures were described by Hardy in the Biography which he wrote after Ramanujan’s untimely death. In it he stated: “My relationship with Ramanujan is the only romantic incident of my life.”
With his two collaborators he developed the so-called circle method, an instrument of number theory that has allowed to obtain results of great importance. Among them, a formula to express the number of partitions of a positive integer as a sum of others; significant advances in problems on the representations of integers as sums of powers; and Goldbach’s ternary problem, which states that every integer greater than five can be expressed as the sum of three primes.
Along with number theory, harmonic analysis is also the core of his research. He was an actor and witness to the paradigm shift that Lebesgue’s measurement theory and the irruption of functional analysis brought about, when the emphasis shifted from the study of the properties of the so-called special functions to that of the classes or spaces of functions. That turn was driven by the mathematical foundation of the emerging quantum mechanics, as well as by the resolution of the equations in partial derivatives of other more classical theories of physics.
Hardy lived a lifetime among theorems, mainly at Trinity College, Cambridge University. His immense work, the complete collection of all his articles, was compiled into eight volumes, of about eight hundred pages. In addition, he was the author of several monographs, among which stands out Introduction to Number Theorywritten in collaboration with Edward Wright and still in use as a textbook for college courses.
In ‘Apology for a Mathematician’, written when Hardy was 62 years old, he explains that age has stripped him of the necessary energies for the deep thought that mathematical research needs
His work deserves a special mention Apology of a mathematician, written when he was already 62 years old, in which he describes the nature of mathematical research and its relationship with artistic creation. The work oozes the bitterness of someone who, according to himself, age has stripped of the energy necessary for the deep thought that mathematical research needs.
In all his work, the precision and beauty of his prose stands out, as well as the forcefulness of his opinions and aphorisms: “There is no place in the world for ugly mathematics”; “It is never worth a valuable person to waste time expressing a majority opinion. By definition, there are already too many people to do that”; “A science is called applied when it contributes to promoting the difference in wealth or directly threatens human life. Number theory does not satisfy any of these hypotheses”, are some examples.
His comments on applied science now seem profoundly wrong, but I think they must be qualified taking into account the context, his pacifist ideals and the tough times he lived through. With an eccentric character and somewhat distant treatment, he liked silk shirts, he hated having to talk on the phone or see his image reflected, so he covered all the mirrors in his room.
Although an avowed atheist, he claimed that the gods were his personal enemies, so when he sailed to visit Harald Bohr in Copenhagen, he fantasized about sending cards to friends claiming to have solved the Riemann Hypothesis. In this way, he said, he would avert the danger of shipwreck, since the gods would not allow Hardy to go down in history with the glory of having achieved the answer to such an important and elusive enigma.
The Riemann hypothesis, considered by many as the Holy Grail of mathematics, linked to the precise knowledge of the succession of primes among all integers, postulates that the non-trivial zeros of the so-called zeta function are complex numbers, located on a line vertical, abscissa 1/2. Surely, it was the problem that Hardy would have most wanted to solve and to which he dedicated time and effort. He was able to prove the existence of infinitely many such zeros on that line, which opened the door for others to underestimate the proportion of zeros that are located there. But the Riemann hypothesis is still waiting for your answer.
Antonio Cordoba Beard He is a professor emeritus at the Autonomous University of Madrid, a member of the ICMAT and an honorary academic at the Academy of Sciences of the Region of Murcia.
Coffee and Theorems is a section dedicated to mathematics and the environment in which they are created, coordinated by the Institute of Mathematical Sciences (ICMAT), in which researchers and members of the center describe the latest advances in this discipline, share meeting points between the mathematics and other social and cultural expressions and remember those who marked their development and knew how to transform coffee into theorems. The name evokes the definition of the Hungarian mathematician Alfred Rényi: “A mathematician is a machine that transforms coffee into theorems.”
Edition and coordination: Timon Agate (ICMAT).
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