Table Of ContentPREFACE
TH:E main object of this publication is to render more accessible the papers
of a great mathematician, which in their original form appeared in many
journals over a period of about 50 years. The editors have kept in view
a second object also: that of rendering the work useful to mathematicians
generally by providing introductions to groups of papers, or comments
where appropriate. These editorial additions, while not always systematic
or exhaustive, will (it is hoped) assist the reader to view Hardy’s papers in
proper perspective.
It is this second object which has led the editors to divide th.e pders
into groups (or further into subgroups) in accordance with the nature of
their subject-matter, instead of publishing them in chronological order.
The editors have been very conscious of the difficulty of making such
a classification, which is most acute in those instances in which a paper
that is primarily on one topic has subsequently proved to be of great
importance for another. There are cases in which our allocation of a paper
to one section rather than to another has been in the nature of an arbitrary
choice, but we hope that adequate cross-references are provided. It may
be a matter for regret that our policy has sometimes resulted in distribut#ing
the papers of a series (such as ‘Notes on the Integral Calculus’) among
several sections, but we believe that any arrangement which kept them
together would have been less satisfactory.
We are grateful to Professor Littlewood for his permission to include all
the Hardy-Littlewood papers, and for his approval of our policy of
treating them on the same footing as Hardy’s own papers.
THE EDITORS
ACKNOWLEDGEMENTS
THE editors are grateful to the following societies and publishers of journals who have
kindly given permission for the reproduction of Hardy’s papers. Details of the sources
are given in the List of ~qers by G. H. Ha&y, which appears at the end of each
volume.
Abbtilungen aw dem Mathematischen Seminar der Urtiversitit Hamburg: Messrs. A. Liebing
(Neudruck Journalfranz), Wiirzburg,
Acta Mathematiba: Institut M&tag&e&r, Sweden.
American Mathematical Monthly: the Mathematical Association of America.
Ann&i della Scuola Normale Euperiore de P&z: The Director of the Scuola Normale Super&e.
Annals of Muthematics: the Editors.
British Association Reports: the British Association for the Advancement of Science.
Bulletin of the American Mathematical Society: the American Mathematical Society.
Bulletin of the Calcutta Mathematical Society: the Calcutta Mathematical Soceity.
Comptes Rendus de l’Acad&mie des Scie+wes: Messrs. Gauthier-Villars, Paris,
Duke Mathematical Joumml: the Editor.
Fundamenta Mathematicae: the Editor.
Journal fiir die reine und angewatite Mathemutik: Messrs. Walter de Gruyter & Co., Berlin.
Journal of the Indian Mathematical Society: the Council of the Indian Mathematical Society.
Jourrzal of the London Mathematical Society: the Council of the London Mathematical Society.
Matemutisk Tidaakrift : Dansk Matematisk Forening, Copenhagen.
Matematiskfyaiske Meddelelser: I>& Kongelige Danske Videnskabernes Selskab, Copenhagen.
Mathematical Gazette: the Mathematical Association.
Mathematische Annalen: Springer-Verlag, Heidelberg.
Mathematische Zeitachrift : Springer-Verlag, Heidelberg.
Mind: the Editor.
Nuchrichten van der G~ellschuft der Wissenschaften zu GtittiNgen: die Akademie der Wissenschaften
zu Giittingen.
Proceedings of the Cambridge Philosophical Society: the Cambridge Philosophical Society.
Proceedings of the London Mathematical Society: the Council of the London Mathematical
Society.
Proceeding8 of the National: Academy of Sciences: the National Academy of Sciences, Washington,
D.C.
Proceeding8 of the Royal Sociely : the Council of the Royal Society.
Resdicmti de2 Circolo Matemutico di Palermo: the Editor.
Tdhoku Mathemuticul Jourruzl: the Editor.
Tranzx&ow of the American Mathematical Society: the American Mathematical Society.
Trawactiow of the Cambridge Phibsophica~ Society: the Cambridge Philosophical Society.
EDITORIAL NOTE
The work will comprise seven volumes
FOR convenience of reference, papers are numbered according to years,
e.g. 1912, 4. A complete list of HardyI’ s papers will be found at the end of
this volume (pp. 683-99) and will be reproduced at the end of each
volume. This list is based on that compiled by Titchmarsh (Journal of the
Londm i’kxthematical Society, 25 (1950), 89-101).
The date of publication of a paper, where it differs from the year men-
tioned in the reference number, is given (for the sake of its historical
interest) in the contents list of the volume containing the paper.
Where reference is made, in the corrections or comments, to the pages
of a paper, the numbers used are those of the original pagination and not
the consecutive page numbers of this volume. The joint papers with
Ramanujan are reproduced from The Cdlected Papers of S. Ramanujan
(Cambridge, 1927), and for these the ‘original pagination’ relates to that
volume and not to the first publication in a journal.
CONTENTSOFVOLUMEI
Godfrey Harold Hardy, by E, c, TITCEMARSH* I
I. DI:OPHANTXNE APPROXIMATION
Introduction 15
1912,4 (with J. E, Littlewood). Some problems of Diophantine approximation. 20
Proceedings of the 5th International: Compe8s of Mathemmticiana, Canbridge, 1912, i. 223-9.
Published 1913.
1914, 2 (with J. E. Littlewood). Some problems of Diophantine approximation. 28
I. The fractional part of nk8.
Acta MatAwmtica, 3 7, 155-91.
1914, 3 (with 5. E. Littlewood). Some problems of Diophantine approximation. 67
II. The trigonometrical series associated with the elliptic &functions.
Acta Mathematics, 37, 193-238.
1916, 3 (with J. E. Littlewood). Some problems of Diophantine approximation: 115
A remarkabIe trigonometrical series.
Proceedings of the National A.cademy of Sciences, 2, 583-6.
1916,9 (with J. E. Littlewood). Some problems of Diophantine approximation: 119
The series 2 e(h,) and the distribution of the points (h, QI).
Proceedings of the National Academy of Sciences, 3, 84-88. Published 1917.
1919, 4. A problem of Diophantine approximation. 124
JuumaL of the Indian Mathematical Society, II, 162-6.
1922,5 (with J. E, Littlewood). Some problems of Diophantine approximation: 130
A further note on the trigonometrical series associated with the elliptic theta-
functions.
Proceedkgs of the Cctw&idge Philosophical Society, 2 1, 1-5.
1922,6 (with J. E. Littlewood). Some problems of Diophantine approximation: 136
The lattice-points of a right-angled triangle.
Proceed&g& oj the Londm Mathematical Society, (2) 20, 35- 36.
1922, 9 (with J. E. Littlewood). Some problems of Diophantine approximation: 159
The lattice-points of a right-angled triangle. (Second memoir.)
Abhandlungeta auo dem Ildathsmatischen bni~~.ar der Hamburgischen Univemitlit, 1, 212-49.
Published 1921.
* Reprinted with slight changes from Obituary Noticea of Fellowaof the Royal Society, 6 (
CONTENTS
1923,3 (with J. E. Littlewood). Some problems of Diophantine approximation: 197
The analytic character of the sum of a Dirichlet’s series considered by Hecke.
Abhandlungen a.us dem Mathematischsn Seminar der Hambwgischen UniversitZit, 3, 57-68.
1923,4 (with J. E. Littlewood). Some problems of Diophantine approximation: 212
The analytic properties of certain Dirichlet’s series associated with the distri-
bution of numbers to modulus unity.
Tramactions of the Cambridge Philosophical Society, 22, 519- 33.
1925, 4 (with J. E. Littlewood). Some problems of Diophantine approximation: 227
An additional note on the trigonometrical series associated with the elliptic
theta-functions,
Acta Mathematics, 47, 189-98. Published 1926.
1930, 3 (with J. E. Littlewood). Some problems of Diophantine approximation: 237
A series of cosecants.
RuEleti~ of the Calwtta Mathematical Society, 20, 251.-66.
1946, I (with J. E. Littlewood). Notes on the theory of series (XXIV) : A curious 253
power series.
Proceedings of the Cambridge Philosophical Society, 42, G-90.
2. ADDITIVE NUMBER THEORY
(a) Combinatory analysis and sums of squares
Introduction 263
1916, 10. Asymptotic formulae in combinatory analysis.
Quatrikme Congrks des Mathe’maticiens Scandinaves, Stockholm, 1916, 45-53. Published 1920. 265
1917, 1 (with S. Ramanujan). Une formule asymptotique pour le nombre des 274
partitions de n,
Contptes RenduS, 164, 35-38.
1917, 4 (with S. Ramanujan). Asymptotic formulae for the distribution of 277
integers of various types.
Proceedings of the London Mathematical Society, (2) 16, 112-32.
1918, 2 (with S. Ramanujan). On the coefficients in the expansions of certain 294
modular functions.
Proceedings of th#e Royal Society, A, 95, 14655.
1918, 5 (with S. Ramanujan). Asymptotic formulae in combinatory analysis. 306
Proceedings of the London Mathematical Society, (2) 17, 7S115.
1918, 10. On the representation of a number as the sum of any number of 340
squares, and in particular of five or seven.
Proceedings of the National Academy of Sciences, 4, 189- 93,
1920, 10. On the representation of a number as the sum of any number of 345
squares, and in particular of five.
Transaction of the Americart Mathematical Society, 21, 255-84.
CONTENTS
(b) Waring’s Problem
Introduction
377
1920, 2 (with J. E. Littlewood). A new solution of Waring’s Problem.
382
Quarterly Journal of Mathematics, 48, 272-93.
1920, 5 (with J. E. Littlewood). Some problems of ‘Partitio Numerorum’: 405
I. A new solution of Waring’s Problem.
Nachrich ten van der K. Gesellschaft der Wiasens~hujten zu G&tinge%, Math.-phys. Klasse, 1920,
33-54.
1921, I (with J. E. Littlewood). Some problems of ‘Partitio Numerorum’: 427
II. Proof that every large number is the sum of at most 21 biquadrates.
Muthematische Zeitschrift, 9, 14- 27. PubIished 1920.
1922, 4 (with J. E. Littlewood). Some problems of ‘Partitio Numerorum’: 441
IV. The singular series in Waring’s Problem and the value of the number
a(k)
lkthmnutische Zeitsdrift, 12, 161-88.
1925, I (with J. E. Littlewood). Some problems of ‘Partitio Numerorum’; 469
VI. Further researches in Waring’s Problem.
Mathemutische Zeitschkft, 23, 1-37.
1928, 4 (with J. E. Littlewood). Some problems of ‘Partitio Numerorum’: 506
VIII. The number I’(k) in Waring’s Problem.
Proceed&p qf the Lortdon Mathematical Society, (2) 28, 51842.
(c) Goldbach’s Problem
Introduction 533
535
1919, 1 (with J. E. Littlewood). Note on Messrs. Shah and Wilson’s paper
entitled: ‘On an empirical formula connected with Goldbach’s Theorem’.
Proceeding8 of the Cambridge Philosophical Society, 19, 245-54.
1922, 1. Goldbach’s Theorem. 545
Matematisk Tidmkrijt B, 1922, 1-16.
561
1922, 3 (with J. E. Littlewood). Some problems of ‘Partitio Numerorum’:
III. On the expression of a number as a sum of primes.
Acta Mathematics, 44, I-70.
631
1922, 8 (with J. E. Littlewood). Summation of a certain multiple series.
Proceedings of the London Mathematical Society, (2) 20, xxx. Published 1921.
632
1924, 6 (with J. E. Littlewood). Some problems of ‘Partitio Numerorum’ :
V. A further contribution to the study of Goldbach’s Problem.
Proceedings of the Lo&on, Mathematical Society, (2) 22, 46-56. Published 1923.
CONTENTS
1920, Il. Some famous problems of the Theory of Numbers and in particular 647
Waring’s Problem.
Arrangement of the Volumes.
681
Complete list of Hardy’s mathematical papers, 683
