Namespaces
Variants
Actions

Difference between revisions of "Average order of an arithmetic function"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (typo)
Line 9: Line 9:
 
holds as $x$ tends to infinity.
 
holds as $x$ tends to infinity.
  
It is conventional to assume that the approximating function$g$ is [[Continuous function|continuous]] and [[Monotonic function|monotone]].
+
It is conventional to assume that the approximating function $g$ is [[Continuous function|continuous]] and [[Monotonic function|monotone]].
  
 
===Examples===
 
===Examples===

Revision as of 05:11, 10 September 2016

2020 Mathematics Subject Classification: Primary: 11A25 [MSN][ZBL]

Some simpler or better-understood function which takes the same values "on average" as an arithmetic function.

Let $f$, $g$ be functions on the natural numbers. We say that $f$ has average order $g$ if the asymptotic equality $$ \sum_{n \le x} f(n) \sim \sum_{n \le x} g(n) $$ holds as $x$ tends to infinity.

It is conventional to assume that the approximating function $g$ is continuous and monotone.

Examples

See also

References

  • G.H. Hardy; E.M. Wright (2008). An Introduction to the Theory of Numbers (6th ed.). Oxford University Press. ISBN 0-19-921986-5
  • Gérald Tenenbaum (1995). Introduction to Analytic and Probabilistic Number Theory. Cambridge studies in advanced mathematics 46. Cambridge University Press. ISBN 0-521-41261-7
How to Cite This Entry:
Average order of an arithmetic function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Average_order_of_an_arithmetic_function&oldid=39076