Average order of an arithmetic function: Difference between revisions

Revision as of 12:36, 10 August 2014

Revision as of 13:19, 10 August 2014

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It is conventional to choose an approximating function ''g'' that is [[Continuous function|continuous]] and [[Monotonic function|monotone]]. But even so an average order is of course not unique.

In cases where the limit

<math>\lim_{nN\rightarrow \infty}\text{Ave}_sum_{n \le N} f(hn)=c</math> provided this limit exists.

exists, it is said that ''f'' has a '''mean value''' ('''average value''') c.

==Examples==

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<math>\text{Ave}_{n}(h)=\frac{1}{q^{n}}\sum_{f \text{ monic},\text{ deg}(f)= n}h(f)</math>.

This is the mean value (or average value) of ''h'' on the set of monic polynomials of degree ''n''. We definesay thethat mean''g(n)'' valueis (oran '''average value)order''' of ''h'' toif be

<math>\text{Ave}_{n}(h)=O(g(n))</math>

as ''n'' tends to infinity.

In cases where the limit,

<math>\lim_{n\rightarrow\infty}\text{Ave}_{n}(h)=c</math>

exists, it is said that ''h'' has a '''mean value''' ('''average value''') ''c''.

<math>\lim_{n\rightarrow\infty}\text{Ave}_{n}(h)</math> provided this limit exists.

===Zeta function and Dirichlet series in {{math|F_q[X]}}===