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Difference between revisions of "Erdős-Fuchs theorem"

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m (Richard Pinch moved page Erdős–Fuchs theorem to Erdős-Fuchs theorem: hyphenation)
m (→‎References: isbn link)
 
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====References====
 
====References====
 
* P. Erdős, W.H.J. Fuchs, ''On a Problem of Additive Number Theory''Journal of the London Mathematical Society '''31''' (1956) 67–73 {{DOI|10.1112/jlms/s1-31.1.67}} {{ZBL|0070.04104}}
 
* P. Erdős, W.H.J. Fuchs, ''On a Problem of Additive Number Theory''Journal of the London Mathematical Society '''31''' (1956) 67–73 {{DOI|10.1112/jlms/s1-31.1.67}} {{ZBL|0070.04104}}
* D.J. Newman, "Analytic number theory", Graduate Texts in Mathematics '''177''' Springer (1998) ISBN 0-387-98308-2 {{ZBL|0887.11001}}
+
* D.J. Newman, "Analytic number theory", Graduate Texts in Mathematics '''177''' Springer (1998) {{ISBN|0-387-98308-2}} {{ZBL|0887.11001}}

Latest revision as of 16:49, 23 November 2023

2020 Mathematics Subject Classification: Primary: 11B34 [MSN][ZBL]

A statement about the number of ways that positive integers can be represented as a sum of two elements of a given set, stating that the average order of this number cannot be close to being a linear function. Given a subset $A$ of the natural numbers let $r(n)$ denote the number of ways that a natural number $n$ can be expressed as the sum of two elements of $A$ (taking order into account). We consider the average $$ R(n) = (r(1)+r(2)+\cdots+r(n) ) / n \ . $$ The theorem states that $$ R(n) = C + O\left(n^{-3/4-\epsilon}\right) $$ cannot hold unless $C=0$

References

  • P. Erdős, W.H.J. Fuchs, On a Problem of Additive Number TheoryJournal of the London Mathematical Society 31 (1956) 67–73 DOI 10.1112/jlms/s1-31.1.67 Zbl 0070.04104
  • D.J. Newman, "Analytic number theory", Graduate Texts in Mathematics 177 Springer (1998) ISBN 0-387-98308-2 Zbl 0887.11001
How to Cite This Entry:
Erdős-Fuchs theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Erd%C5%91s-Fuchs_theorem&oldid=54615