Universität zu Köln

Berndt, Bruce C., Kim, Sun and Zaharescu, Alexandru (2018). The Circle Problem of Gauss and the Divisor Problem of Dirichlet-Still Unsolved. Am. Math. Mon., 125 (2). S. 99 - 115. WASHINGTON: MATHEMATICAL ASSOC AMER. ISSN 1930-0972

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Abstract

Let r(2)(n) denote the number of representations of the positive integer n as a sum of two squares, and let d(n) denote the number of positive divisors of n. Gauss and Dirichlet were evidently the first mathematicians to derive asymptotic formulas for Sigma(n <= x) r(2)(n) and Sigma(n <= x) d(n), respectively, as x tends to infinity. But what is the error made in such approximations? Number theorists have been attempting to answer these two questions for over one and one-half centuries, and although we think that we essentially know what these errors are, progress in proving these conjectures has been agonizingly slow. Ramanujan had a keen interest in these problems, and although, to the best of our knowledge, he did not establish any bounds for the error terms, he did give us identities that have been used to derive bounds, and two further identities that might be useful, if we can figure out how to use them. In this paper, we survey what is known about these two famous unsolved problems, with a moderate emphasis on Ramanujan's contributions.

Item Type:	Journal Article
Creators:	Creators Email ORCID ORCID Put Code Berndt, Bruce C. UNSPECIFIED UNSPECIFIED UNSPECIFIED Kim, Sun UNSPECIFIED UNSPECIFIED UNSPECIFIED Zaharescu, Alexandru UNSPECIFIED UNSPECIFIED UNSPECIFIED
URN:	urn:nbn:de:hbz:38-203226
DOI:	10.1080/00029890.2018.1401853
Journal or Publication Title:	Am. Math. Mon.
Volume:	125
Number:	2
Page Range:	S. 99 - 115
Date:	2018
Publisher:	MATHEMATICAL ASSOC AMER
Place of Publication:	WASHINGTON
ISSN:	1930-0972
Language:	English
Faculty:	Unspecified
Divisions:	Unspecified
Subjects:	no entry
Uncontrolled Keywords:	Keywords Language HIGHER-POWER MOMENTS; LATTICE POINTS; EXPONENTIAL-SUMS; SERIES; IDENTITIES; DELTA(X) Multiple languages Mathematics Multiple languages
Refereed:	Yes
URI:	http://kups.ub.uni-koeln.de/id/eprint/20322