G. H. Hardy

How to use this course

G. H. Hardy is remembered by the general public for two stories: his discovery of Srinivasa Ramanujan and his short book A Mathematician’s Apology. Mathematicians remember him for a body of work in analysis and number theory, most of it written with John Edensor Littlewood, that set the agenda of analytic number theory for the twentieth century. This course covers both: the life, and the theorems.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. Throughout, \(e(x)=e^{2\pi i x}\), \(p\) denotes a prime, and \(f\sim g\) means \(f/g\to1\).

Timeline

Cranleigh and Winchester, 1877–1896

Godfrey Harold Hardy was born on 7 February 1877 in Cranleigh, Surrey. His father Isaac was bursar and art master at Cranleigh School; his mother Sophia had taught at a teacher-training college in Lincoln. Neither had been to university, but both were cultured and ambitious for their children. Hardy and his younger sister Gertrude grew up in a house full of books and puzzles. Friends knew him as Harold.

He showed mathematical ability very early. He attended Cranleigh School until he was twelve and then won a scholarship to Winchester College in 1889, entering in 1890. He was a brilliant pupil, but painfully shy and self-conscious about his appearance, a trait that stayed with him all his life. He also acquired at Winchester a lifelong passion for cricket.

Trinity, 1896–1911

Hardy entered Trinity College, Cambridge, in 1896. Cambridge mathematics was dominated by the Mathematical Tripos, a competitive examination for which students trained with private coaches by drilling tricky problems. Hardy was first coached by R. R. Webb. Dissatisfied, he moved to Augustus Edward Hough Love, an applied mathematician who did him a decisive service: he told Hardy to read Camille Jordan’s Cours d’analyse. Hardy later said that this was the moment he learned what mathematics really meant. Continental analysis, with its careful definitions of limit, continuity and convergence, became his model.

He was fourth wrangler in 1898, a placing he professed to find absurd, since he despised the whole ranking system. In 1900 he was elected a Prize Fellow of Trinity, and in 1901 he shared the Smith’s Prize with James Jeans. His early papers were on the convergence of series and integrals and on orders of infinity, in the style of Paul du Bois-Reymond. He became a lecturer at Trinity in 1906.

In 1908 he published A Course of Pure Mathematics, the first English textbook to teach analysis rigorously from the real numbers upward. It transformed the teaching of mathematics in British universities and is still in print. The same year he wrote, almost in passing, a short letter to the American journal Science that founded population genetics (Lesson 10). He was elected a Fellow of the Royal Society in 1910.

Littlewood and Ramanujan, 1911–1919

In 1911 Hardy began working with John Edensor Littlewood, eight years younger and also a fellow of Trinity. The partnership lasted about thirty-five years and produced around a hundred joint papers, one of the most productive collaborations in the history of mathematics. They often wrote to each other rather than meeting, even when both were in Cambridge. They had working rules: neither was obliged to read or answer the other’s letters, and it did not matter who had contributed what. Harald Bohr joked that there were three great English mathematicians: Hardy, Littlewood and Hardy–Littlewood.

In January 1913 Hardy received a letter from Srinivasa Ramanujan, a clerk at the Madras Port Trust with almost no formal training, enclosing pages of formulas without proofs. Some were known, some were wrong, and some were unlike anything Hardy had seen. He and Littlewood concluded that the writer was a mathematician of the highest class. Hardy arranged for Ramanujan to come to Cambridge, where he arrived in April 1914. For five years they worked together, combining Ramanujan’s extraordinary insight into formulas with Hardy’s insistence on proof. Their joint papers include the asymptotic formula for partitions (Lesson 06) and the theory of the normal number of prime factors. Ramanujan was elected a Fellow of the Royal Society and of Trinity in 1918. He fell gravely ill, returned to India in 1919 and died in 1920 at the age of 32.

The famous taxi story belongs to this period. Visiting Ramanujan in hospital in Putney, Hardy remarked that his cab number, 1729, seemed dull. Ramanujan replied at once that it was the smallest number expressible as a sum of two cubes in two different ways: \(1729=1^3+12^3=9^3+10^3\).

Hardy was a strong opponent of the First World War, which isolated him from many colleagues at Trinity. When the college deprived Bertrand Russell of his lectureship in 1916 for anti-war activity, Hardy was among his defenders; he later told the story in Bertrand Russell and Trinity (1942). Unhappy in wartime Cambridge, he accepted the Savilian Chair of Geometry at Oxford in 1919.

Rigour, divergent series, Tauberian theorems

For the general reader. What is \(1-1+1-1+\cdots\)? The series does not converge, but there are sensible ways to assign it the value \(\tfrac12\). Hardy spent much of his career on such summation methods. The deepest question is the reverse one: if a series is summable in a weak sense and its terms are small, must it actually converge? Theorems that say yes are called Tauberian, and Hardy and Littlewood found the most important of them.

Summation methods

Let \(s_n=a_0+\cdots+a_n\). The series \(\sum a_n\) is Cesàro summable (or \((C,1)\) summable) to \(s\) if \(\sigma_n=\frac{s_0+\cdots+s_n}{n+1}\to s\). It is Abel summable to \(s\) if \(\sum a_nx^n\) converges for \(|x|<1\) and tends to \(s\) as \(x\to1^-\). Convergence implies Cesàro summability to the same value, and Cesàro summability implies Abel summability (Frobenius). Neither converse holds.

Proof sketch of Hardy’s theorem

Suppose \(|a_k|\le C/k\). For \(m>n\),

Hence the average of \(s_{n+1},\dots,s_m\) differs from \(s_m\) by at most \(C(m-n)/(2n)\). Take \(n=\lfloor m/(1+\varepsilon)\rfloor\), so \((m-n)/n\approx\varepsilon\). The left side equals \(\sigma_m+\frac{n+1}{m-n}(\sigma_m-\sigma_n)\), and since \(\frac{n+1}{m-n}\approx1/\varepsilon\) is bounded while \(\sigma_m-\sigma_n\to0\), it tends to \(s\). So \(\limsup|s_m-s|\le C\varepsilon/2\) for every \(\varepsilon>0\). Littlewood’s Abelian version is much harder; Jovan Karamata gave a short proof in 1930 using Weierstrass approximation.

Tauberian theorems are the analytic engine of the prime number theorem. Hardy and Littlewood used them to give proofs of the prime number theorem, and Norbert Wiener’s general Tauberian theorem of 1932, which grew out of their work, yields it directly. Hardy’s last book, Divergent Series (1949), is the classic account of the subject.

Zeros on the critical line

For the general reader. The Riemann hypothesis says that all the non-trivial zeros of the zeta function lie on one vertical line in the complex plane. In 1914 nobody knew whether even infinitely many of them did. Hardy proved that they do. The idea is to turn the zeta function on that line into a real-valued function and show that it changes sign infinitely often.

Hardy’s function

Let \(\zeta(s)=\sum_{n\ge1}n^{-s}\) for \(\operatorname{Re}s>1\), continued meromorphically. The completed function \(\xi(s)=\tfrac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\) satisfies \(\xi(s)=\xi(1-s)\), and \(\overline{\xi(\bar s)}=\xi(s)\). Bernhard Riemann conjectured that every zero of \(\xi\) has real part \(\tfrac12\). Define

with \(\theta\) continuous and \(\theta(0)=0\). Then \(Z\) is real for real \(t\), \(|Z(t)|=|\zeta(\tfrac12+it)|\), and the real zeros of \(Z\) are exactly the zeros of \(\zeta\) on the critical line. Stirling’s formula gives \(\theta(t)=\frac t2\log\frac t{2\pi}-\frac t2-\frac\pi8+O(1/t)\).

Proof sketch

Hardy’s original proof used a Mellin-transform identity involving Jacobi’s theta function. A later and more transparent version runs as follows. Suppose \(Z(t)\) has no zeros for \(t>T_0\), so it has constant sign there. Then for large \(T\),

The right side is \(T+O(T^{1/2})\): integrate the approximate formula \(\zeta(\tfrac12+it)=\sum_{n\le t}n^{-1/2-it}+O(t^{-1/2})\) term by term; the term \(n=1\) contributes \(T\) and the others oscillate. The left side is \(o(T)\): writing \(Z(t)\) as a sum of terms \(n^{-1/2}\cos(\theta(t)-t\log n)\) with rapidly varying phase, van der Corput’s estimates show it has much smaller mean. The contradiction proves the theorem.

How many zeros?

Let \(N(T)\) count zeros \(\rho=\beta+i\gamma\) with \(0<\gamma\le T\), and \(N_0(T)\) those with \(\beta=\tfrac12\). The Riemann–von Mangoldt formula gives \(N(T)=\frac T{2\pi}\log\frac T{2\pi}-\frac T{2\pi}+O(\log T)\). Hardy and Littlewood proved \(N_0(T)>KT\) for some \(K>0\) in 1921. Atle Selberg proved in 1942 that a positive proportion of zeros lie on the line, \(N_0(T)>K\,T\log T\). Norman Levinson raised the proportion above one third in 1974, and Brian Conrey above two fifths in 1989.

Partitions and the Hardy–Ramanujan formula

For the general reader. How many ways can 200 be written as a sum of positive whole numbers, ignoring order? Percy MacMahon counted them by hand: almost four trillion. Hardy and Ramanujan found a formula that gives this number to within a fraction, and in doing so they invented the method that Hardy and Littlewood would turn on the primes.

The generating function

Let \(p(n)\) be the number of partitions of \(n\), with \(p(0)=1\). Leonhard Euler observed that

Expanding each factor as \(1+q^k+q^{2k}+\cdots\) chooses how many parts equal \(k\).

The method

By Cauchy’s formula, \(p(n)=\frac1{2\pi i}\oint F(q)\,q^{-n-1}\,dq\) over a circle \(|q|=r<1\). As \(r\to1\), \(F\) is largest near \(q=1\), and also large near every root of unity \(e(a/k)\). The behaviour near each root of unity is given exactly by the modular transformation of Dedekind’s eta function \(\eta(\tau)=q^{1/24}\prod(1-q^k)\), \(q=e(\tau)\). Near \(q=1\), with \(q=e^{-t}\), this gives \(F(e^{-t})\approx\sqrt{t/2\pi}\,e^{\pi^2/6t}\). Hardy and Ramanujan split the circle into arcs around the roots of unity and evaluated each contribution: this is the first appearance of the circle method. In 1937 Hans Rademacher refined their series into a convergent one that equals \(p(n)\) exactly.

Ramanujan also discovered the congruences \(p(5n+4)\equiv0\pmod5\), \(p(7n+5)\equiv0\pmod7\) and \(p(11n+6)\equiv0\pmod{11}\). With Hardy he proved that the number of distinct prime factors \(\omega(n)\) has normal order \(\log\log n\): for every \(\varepsilon>0\), all but \(o(x)\) of the integers \(n\le x\) satisfy \(|\omega(n)-\log\log n|<(\log\log n)^{1/2+\varepsilon}\). This was the first theorem of probabilistic number theory.

The circle method and Waring’s problem

For the general reader. Every positive integer is a sum of four squares, and in 1909 David Hilbert proved that for each power \(k\) there is a number \(s\) such that every integer is a sum of \(s\) \(k\)-th powers. Hilbert’s proof gave no useful bounds. Hardy and Littlewood adapted the partition method to give an asymptotic formula for the number of representations, and with it good bounds. Their series of papers, Some problems of Partitio Numerorum, created modern additive number theory.

Set-up

Fix \(k\ge2\) and \(s\). Let \(r_{k,s}(n)\) be the number of solutions of \(x_1^k+\cdots+x_s^k=n\) in positive integers. With \(P=\lfloor n^{1/k}\rfloor\) and \(f(\alpha)=\sum_{m=1}^{P}e(\alpha m^k)\),

by orthogonality: \(\int_0^1e(h\alpha)\,d\alpha\) is 1 if \(h=0\) and 0 otherwise. The integrand is large when \(\alpha\) is close to a rational \(a/q\) with small \(q\), because the terms \(e(\alpha m^k)\) then fall into a few residue classes. The neighbourhoods of these rationals are the major arcs; the rest of the circle is the minor arcs, where one shows \(f\) is small by Weyl’s inequality.

Here \(G(k)\) is the least \(s\) such that every sufficiently large integer is a sum of \(s\) \(k\)-th powers, while \(g(k)\) requires all integers. The singular series is a product of local densities, one for each prime, counting solutions modulo prime powers; the gamma factor is the real density. In the 1930s Ivan Vinogradov greatly improved the minor-arc estimates and showed that \(G(k)=O(k\log k)\).

Goldbach and the prime-tuple conjectures

For the general reader. Is every even number larger than 2 a sum of two primes? Are there infinitely many twin primes like 11 and 13? Hardy and Littlewood could not prove either. Instead they predicted, with remarkable precision, how many such representations and pairs there should be. Their formulas have matched every computation made since, and they remain the guiding conjectures of the subject.

The conjectures of Partitio Numerorum III (1923)

Applying the circle method to \(\sum_{p\le N}(\log p)\,e(\alpha p)\), Hardy and Littlewood found that the major arcs predict the following. Let \(C_2=\prod_{p>2}\big(1-\frac1{(p-1)^2}\big)\approx0.6601618\).

In the same paper, assuming the generalized Riemann hypothesis for Dirichlet \(L\)-functions, they proved an asymptotic formula for the number of representations of a large odd number as a sum of three primes. Vinogradov removed the hypothesis in 1937, and Harald Helfgott completed the proof of the ternary Goldbach conjecture for all odd numbers above 5 in 2013. The binary Goldbach and twin prime conjectures remain open, though Yitang Zhang proved in 2013 that infinitely many pairs of primes differ by at most a bounded amount.

A second conjecture in the paper, \(\pi(x+y)\le\pi(x)+\pi(y)\), turned out to conflict with the prime \(k\)-tuples conjecture, as Douglas Hensley and Ian Richards showed in the early 1970s. Most number theorists believe the \(k\)-tuples conjecture and expect the second to fail at very large values.

Inequalities, maximal functions, Hardy spaces

For the general reader. Much of analysis is the art of estimating one quantity by another. Hardy, Littlewood and George Pólya collected and sharpened the basic estimates in their book Inequalities (1934), which made the subject a discipline of its own. Several inequalities and function spaces still bear Hardy’s name, and one of them, the maximal theorem of 1930, was explained in its original paper through a cricketer’s batting averages.

Hardy’s inequality

Proof. Substitute \(u=xt\): \(\frac{F(x)}x=\int_0^1f(xt)\,dt\). By Minkowski’s integral inequality, \(\big\|\int_0^1f(\cdot\,t)\,dt\big\|_p\le\int_0^1\|f(\cdot\,t)\|_p\,dt=\int_0^1t^{-1/p}\,dt\,\|f\|_p=\frac p{p-1}\|f\|_p\). Hardy arrived at this inequality around 1920 while looking for a simple proof of Hilbert’s inequality for double series.

The maximal theorem

For \(f\in L^1_{\mathrm{loc}}(\mathbb R^d)\) define \(Mf(x)=\sup_{r>0}\frac1{|B(x,r)|}\int_{B(x,r)}|f|\).

Hardy and Littlewood proved the one-dimensional case through a rearrangement argument; the form above, with the Vitali covering lemma, is due to Norbert Wiener (1939). Part (b) follows from (a) and the trivial bound \(\|Mf\|_\infty\le\|f\|_\infty\) by Marcinkiewicz interpolation. The weak bound (a) implies the Lebesgue differentiation theorem: averages of \(f\) over shrinking balls converge to \(f(x)\) almost everywhere. Maximal functions became a central tool of harmonic analysis.

Hardy spaces

For \(f\) holomorphic in the unit disc and \(0<p<\infty\) let \(M_p(r,f)=\big(\frac1{2\pi}\int_0^{2\pi}|f(re^{i\theta})|^pd\theta\big)^{1/p}\).

The space \(H^p\) of holomorphic \(f\) with \(\sup_{r<1}M_p(r,f)<\infty\) is the Hardy space, a name given by Frigyes Riesz in 1923. Hardy spaces are basic objects in complex analysis, operator theory and signal processing.

A letter on genetics: Hardy–Weinberg

For the general reader. Hardy liked to say that his mathematics had no practical use. One of his most cited results contradicts him. In 1908 his cricketing friend, the geneticist Reginald Punnett, asked him why a dominant trait does not spread until it takes over a population. Hardy answered in a short letter showing that, without selection, the proportions of genotypes stay fixed from one generation to the next.

The model

Consider one gene with two alleles \(A\) and \(a\), in a large population with random mating, no selection, mutation or migration. Let the genotypes \(AA:Aa:aa\) occur in proportions \(u:2v:w\), with \(u+2v+w=1\). The allele frequencies are \(P=u+v\) for \(A\) and \(Q=v+w\) for \(a\). Random mating is equivalent to drawing two alleles independently from the gene pool.

Hardy published the result in Science as “Mendelian proportions in a mixed population”. The German physician Wilhelm Weinberg found it independently the same year. In the English-speaking world it was called Hardy’s law until Curt Stern pointed out Weinberg’s priority in 1943.

Oxford and Cambridge, 1919–1939

Hardy later described his years at Oxford as the happiest of his life. As Savilian Professor of Geometry and Fellow of New College from 1919, he lectured on analysis and number theory, ran a lively research class, and spent much of the summer at cricket. The Partitio Numerorum papers with Littlewood, who stayed at Cambridge, appeared from 1920 to 1928. He received the Royal Medal of the Royal Society in 1920. He took his politics seriously: he was president of the Association of Scientific Workers, a union of scientists, from 1924 to 1926, and he served two terms as president of the London Mathematical Society (1926–28 and 1939–41). He spent the academic year 1928–29 at Princeton on an exchange.

In 1931 he returned to Cambridge as Sadleirian Professor of Pure Mathematics and again as a fellow of Trinity. He built there the strongest school of analysis in Britain. His doctoral students included Mary Cartwright, Edward Wright, Edward Linfoot, Cyril Offord, Harry Pitt, Richard Rado, Robert Rankin, Donald Spencer, Tirukkannapuram Vijayaraghavan and the statistician I. J. Good. With Wright he wrote An Introduction to the Theory of Numbers (1938), still among the most widely read number theory books, and with Littlewood and Pólya Inequalities (1934).

He received the De Morgan Medal of the London Mathematical Society in 1929 and the Chauvenet Prize of the Mathematical Association of America in 1932 for an expository article. His standards for writing became proverbial: clear, economical and elegant, with the reader’s effort always in mind.

The Apology, last years, legacy

In 1939 Hardy suffered a coronary thrombosis. It ended his cricket and tennis and, by his own account, his creative mathematics. In 1940 he published A Mathematician’s Apology, a short and melancholy essay defending pure mathematics as a creative art. In it he argued that real mathematics must be serious and beautiful, that its value does not depend on applications, and that mathematics is a young man’s game. The book has been read by generations of students; its claim that number theory would never have warlike uses was overturned within a few decades by cryptography. The same year he published Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, and the Royal Society awarded him the Sylvester Medal.

He retired from the Sadleirian chair in 1942. His sister Gertrude, who had never married either, looked after him. His health and spirits declined, and in early summer 1947 he attempted suicide; he survived. That year the Royal Society awarded him its highest honour, the Copley Medal. He died in Cambridge on 1 December 1947, aged 70, while his sister was reading to him from a history of Cambridge University cricket. Divergent Series appeared in 1949, and his collected papers were published in seven volumes between 1966 and 1979.

Legacy

• Analytic number theory: the circle method, the Hardy–Littlewood conjectures, and the study of zeros of \(\zeta(s)\) on the critical line remain central research programmes.
• Analysis: Tauberian theorems, the maximal function, Hardy’s inequality and Hardy spaces are standard tools.
• Mathematical culture: he made rigorous analysis the basis of British mathematical education, rescued Ramanujan’s genius for the world, and set a standard of mathematical prose.
• Outside mathematics: the Hardy–Weinberg principle is the starting point of population genetics.
• Honours: the London Mathematical Society awards the Hardy Lectureship in his name, and his phrase-making in the Apology is quoted wherever mathematicians explain what they do.

Connections

People linked to G. H. Hardy across this collection. A name without a link has no page here yet. See G. H. Hardy in the network of all 337 lives →

Teachers and advisors

• Augustus Edward Hough Love
• E. T. Whittaker

Students

• Cyril Offord
• Donald Spencer
• E. M. Wright
• Edward Linfoot
• Harry Pitt
• I. J. Good
• Mary Cartwright
• Richard Rado
• Robert Rankin
• Tirukkannapuram Vijayaraghavan

Influenced by

• Bernhard Riemann
• David Hilbert
• Karl Weierstrass

Influenced

• Atle Selberg
• G. I. Taylor
• Norbert Wiener 1913-1932
• Paul Erdős

Colleagues and collaborators

• Gösta Mittag-Leffler
• James Jeans 1901
• Olga Taussky-Todd
• George Pólya
• John Edensor Littlewood 1911-1947
• Srinivasa Ramanujan 1913-1920
• Wilhelm Weinberg

Friends

• Bertrand Russell

Also mentioned in this life

• Niels Henrik Abel
• Augustin-Louis Cauchy
• Augustus De Morgan
• Richard Dedekind
• Peter Gustav Lejeune Dirichlet
• Leonhard Euler
• Georg Frobenius
• Carl Gustav Jacob Jacobi
• Camille Jordan
• Henri Lebesgue
• Hermann Minkowski
• Frigyes Riesz
• Hermann Weyl

Fields

• analytic number theory 4 lives

On CoursesHub

• Ramanujan & Hardy · History of Mathematics

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Godfrey Harold Hardy, MacTutor History of Mathematics.
2. G. H. Hardy, Wikipedia.
3. E. C. Titchmarsh, Godfrey Harold Hardy, 1877–1947, Obituary Notices of Fellows of the Royal Society 6 (1949).
4. G. H. Hardy, A Mathematician’s Apology, Cambridge University Press, 1940; reissued with a foreword by C. P. Snow, 1967.
5. R. Kanigel, The Man Who Knew Infinity: A Life of the Genius Ramanujan, Scribner, 1991.
6. J. E. Littlewood, A Mathematician’s Miscellany, Methuen, 1953.

Mathematics

1. G. H. Hardy, A Course of Pure Mathematics, Cambridge University Press, 1908.
2. G. H. Hardy, Sur les zéros de la fonction \(\zeta(s)\) de Riemann, C. R. Acad. Sci. Paris 158 (1914).
3. G. H. Hardy, The mean value of the modulus of an analytic function, Proc. London Math. Soc. (2) 14 (1915), 269–277.
4. G. H. Hardy and S. Ramanujan, Asymptotic formulae in combinatory analysis, Proc. London Math. Soc. (2) 17 (1918).
5. G. H. Hardy and J. E. Littlewood, Some problems of “Partitio numerorum” I: A new solution of Waring’s problem, Nachr. Ges. Wiss. Göttingen (1920).
6. G. H. Hardy and J. E. Littlewood, Some problems of “Partitio numerorum” III: On the expression of a number as a sum of primes, Acta Math. 44 (1923).
7. G. H. Hardy and J. E. Littlewood, A maximal theorem with function-theoretic applications, Acta Math. 54 (1930), 81–116.
8. G. H. Hardy, Mendelian proportions in a mixed population, Science 28 (1908).
9. G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, Cambridge University Press, 1934.
10. G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Oxford, 1938.
11. G. H. Hardy, Divergent Series, Oxford, 1949.
12. R. C. Vaughan, Hardy’s legacy to number theory, J. Austral. Math. Soc. (Ser. A) 65 (1998), 238–266.
13. R. C. Vaughan, The Hardy–Littlewood Method, 2nd ed., Cambridge, 1997.
14. E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, 2nd ed. revised by D. R. Heath-Brown, Oxford, 1986.