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On the representation of a number as a sum of squares

Published online by Cambridge University Press:  18 May 2009

Karl-Bernhard Gundlach
Karl-Bernhard Gundlach
Affiliation:
Fachbereich Mathematik, der Universität Marburg, Lahnberge, 3550 Marburg/Lahn, Germany
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It is well known that the number Ak(m) of representations of a positive integer m as the sum of k squares of integers can be expressed in the form

where Pk(m) is a divisor function, and Rk(m) is a remainder term of smaller order. (1) is a consequence of the fact that

is a modular form for a certain congruence subgroup of the modular group, and

with

where Ek(z) is an Eisenstein series and is a cusp form (as was first pointed out by Mordell [9]). The result (1) remains true if m is taken to be a totally positive integer from a totally real number field K and Ak(m) is the number of representations of m as the sum of k squares of integers from K (at least for 2|k, k>2, and for those cases with 2+k which have been investigated). then are replaced by modular forms for a subgroup of the Hilbert modular group with Fourier expansions of the form (10) (see section 2).

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 19 , Issue 2 , July 1978 , pp. 173 - 197
Copyright
Copyright © Glasgow Mathematical Journal Trust 1978

References

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