Editing Arithmetic function (section)

== Summation functions ==
Given an arithmetic function ''a''(''n''), its '''summation function''' ''A''(''x'') is defined by
<math display="block"> A(x) := \sum_{n \le x} a(n) .</math>
''A'' can be regarded as a function of a real variable. Given a positive integer ''m'', ''A'' is constant along [[open interval]]s ''m'' < ''x'' < ''m'' + 1, and has a [[Classification of discontinuities|jump discontinuity]] at each integer for which ''a''(''m'') ≠ 0.

Since such functions are often represented by series and integrals, to achieve pointwise convergence it is usual to define the value at the discontinuities as the average of the values to the left and right:
<math display="block"> A_0(m) := \frac 1 2 \left(\sum_{n < m} a(n) +\sum_{n \le m} a(n)\right) = A(m) - \frac 1 2 a(m) .</math>

Individual values of arithmetic functions may fluctuate wildly – as in most of the above examples. Summation functions "smooth out" these fluctuations. In some cases it may be possible to find [[Asymptotic analysis|asymptotic behaviour]] for the summation function for large ''x''.

A classical example of this phenomenon<ref>Hardy & Wright, §§ 18.1–18.2</ref> is given by the [[divisor summatory function]], the summation function of ''d''(''n''), the number of divisors of ''n'':
<math display="block">\liminf_{n\to\infty} d(n) = 2</math>
<math display="block">\limsup_{n\to\infty}\frac{\log d(n) \log\log n}{\log n} = \log 2</math>
<math display="block">\lim_{n\to\infty}\frac{d(1) + d(2)+ \cdots +d(n)}{\log(1) + \log(2)+ \cdots +\log(n)} = 1.</math>

An '''[[average order of an arithmetic function]]''' is some simpler or better-understood function which has the same summation function asymptotically, and hence takes the same values "on average".  We say that  ''g'' is an ''average order'' of ''f''   if
<math display="block"> \sum_{n \le x} f(n) \sim \sum_{n \le x} g(n) </math>

as ''x'' tends to infinity.  The example above shows that ''d''(''n'') has the average order log(''n'').<ref>{{cite book | title=Introduction to Analytic and Probabilistic Number Theory | author=Gérald Tenenbaum | series=Cambridge studies in advanced mathematics | volume=46 | publisher=[[Cambridge University Press]] | pages=36–55 | year=1995 | isbn=0-521-41261-7 }}</ref>