Acta Mathematica

On the representation of numbers in the form ax2+by2+cz2+dt2

H. D. Kloosterman

Note

An account of the principal results of this paper has been published in the ‘Verslagen van de Koninklijke Akademie van Wetenschappen’, Amsterdam, 31 Oct. '25.

Article information

Source
Acta Math., Volume 49, Number 3-4 (1926), 407-464.

Dates
First available in Project Euclid: 14 February 2017

https://projecteuclid.org/euclid.acta/1487102066

Digital Object Identifier
doi:10.1007/BF02564120

Rights
1926 © Almqvist & Wiksells Boktryckeri-A.-B.

Citation

Kloosterman, H. D. On the representation of numbers in the form ax 2 + by 2 + cz 2 + dt 2. Acta Math. 49 (1926), no. 3-4, 407--464. doi:10.1007/BF02564120. https://projecteuclid.org/euclid.acta/1487102066

Literatur

• For the litterature on this subject I refer to the article of Bohr-Cramér (Die neuere Entwicklung der analytischen Zahlentheorie) in the ‘Enzyklopaedie der Mathematischen Wissenschaften’.
• ‘Over het splitsen van geheele positieve getallen in een som van kwadraten’, Groningen, 1924.
• L. E. Dickson, ‘History of the theory of numbers’, Vol. III (1923), Ch. X.
• In my paper ‘On the representation of numbers in the form ax2+by2+cz2+dt2, Proc. London Math. Soc., 25 (1926), 143–173, I have proved some of Liouville's formulae and some new formulae by means of methods due to Hardy and Mordell.
• Proc. Camb. Phil. Soc., 19 (1917), 11–21.
• A new solution of Waring's problem, Quarterly J. of pure and applied math., vol. 48 (1919), p. 272–293.
• Two representations n=ax ${}_{1}^{2}$ +by ${}_{1}^{2}$ +cz ${}_{1}^{2}$ +dt ${}_{1}^{2}$ and n=ax ${}_{2}^{2}$ +by ${}_{2}^{2}$ +cz ${}_{2}^{2}$ +dt ${}_{2}^{2}$ will be considered as the same if and only if x1=x2, y1=y2, z1=z2, t1=t2.
• ‘On certain trigonometrical sums and their applications in the theory of numbers’, Trans. Camb. Phil. Soc. 22 (1918), 259–276. The formula (1. 56) has already been given by J. C. Kluyver, ‘Eenige formules aangaande de getallen kleiner dan n en ondeelbaar met n’, Versl. Kon. Akad. v. Wetensch., Amsterdam, 1906.
• Of course the p1 occurring here and the p1 of the lemma's 2, 2*, 3* have quite a different meaning.
• It can be proved as follows, that v1, V1 exist. Consider the system of numbers v1A ${}_{1}^{2}$ +V1ϖ ${}_{1}^{2ξ1}$ , if v1 runs through all numbers, less than and prime to ϖ ${}_{1}^{ξ1}$ and V1 through all numbers, less than and prime to A1. Then these numbers are all incongruent mod q and they are prime to q. Further the system consists of ϕ(ϖ ${}_{1}^{ξ1}$ )ϕ(A1)=ϕ(q) numbers. Therefore one of them must be≡v (modq).
• We denote by (M) the number which is ≡M (modq) and for which 0≤(M)< q.
• See footnote1 on p. 421. Of course the p1 occurring here and the p1 of the lemma's 2, 2*, 3* have quite a different meaning.
• If λj≡o (modϖ), then S would be 0.
• Of course it is also possible, that S(nj) tends to zero, but not as quickly as 1/nj, if nj→∞. But the disenssion of the singular series shows, that in this case, we can always find another sequence, for which the condition 3° holds.
• S. Ramanujan, On the expression of a number in the form ax2+by2+cz2+dt2, Proc. Camb. Phil. Soc. 19 (1917), footnote on p. 14.
• Comptes Rendus, Paris, 170 (1920), 354.
• Meditationes algebraicae, Cambridge, ed. 3, 1782, 349.