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

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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]].
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It is conventional to assume that the approximating function $g$ is [[Continuous function|continuous]] and [[Monotone function|monotone]].
  
 
===Examples===
 
===Examples===
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* The average order of $\phi(n)$, the [[Euler totient function]] of $n$, is $ \frac{6}{\pi^2} n$;
 
* The average order of $\phi(n)$, the [[Euler totient function]] of $n$, is $ \frac{6}{\pi^2} n$;
 
* The average order of $r(n)$, the number of ways of expressing $n$ as a [[sum of two squares]], is $\pi$;
 
* The average order of $r(n)$, the number of ways of expressing $n$ as a [[sum of two squares]], is $\pi$;
* The [[Prime number theorem|Prime Number Theorem]] is equivalent to the statement that the [[von Mangoldt function]] $\Lambda(n)$ has average order 1.
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* The [[Prime number theorem|Prime Number Theorem]] is equivalent to the statement that the [[Mangoldt function|von Mangoldt function]] $\Lambda(n)$ has average order 1.
  
 
===See also===
 
===See also===
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===References===
 
===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  
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* 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
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* Gérald Tenenbaum (1995). Introduction to Analytic and Probabilistic Number Theory. Cambridge studies in advanced mathematics '''46'''. Cambridge University Press. {{ISBN|0-521-41261-7}}

Latest revision as of 19:41, 17 November 2023

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