Summation functions "smooth out" these fluctuations. In some cases it may be possible to find asymptotic behaviour for the summation function for large x. A classical example of this phenomenon is given by the divisor summatory function, the summation function of d(n), the number of divisors of n: 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 as x tends to infinity. The example above shows that d(n) has the average order log(n). Dirichlet convolution Given an arithmetic function a(n), let Fa(s), for complex s, be the function defined by the corresponding Dirichlet series (where it converges): Fa(s) is called a generating function of a(n). The simplest such series, corresponding to the constant function a(n) = 1 for all n, is ς(s) the Riemann zeta function. The generating function of the Möbius function is the inverse of the zeta function: Consider two arithmetic functions a and b and their respective generating functions Fa(s) and Fb(s). The product Fa(s)Fb(s) can be computed as follows: It is a straightforward exercise to show that if c(n) is defined by then This function c is called the Dirichlet convolution of a and b, and is denoted by . A particularly important case is convolution with the constant function a(n) = 1 for all n, corresponding to multiplying the generating function by the zeta function: Multiplying by the inverse of the zeta function gives the Möbius inversion formula: If f is multiplicative, then so is g. If f is completely multiplicative, then g is multiplicative, but may or may not be completely multiplicative. Relations among the functions There are a great many formulas connecting arithmetical functions with each other and with the functions of analysis, especially powers, roots, and the exponential and log functions. The page divisor sum identities contains many more generalized and related examples of identities involving arithmetic functions. Here are a few examples: Dirichlet convolutions where λ is the Liouville function. Möbius inversion Möbius inversion Möbius inversion Möbius inversion Möbius inversion where λ is the Liouville function. Möbius inversion Sums of squares For all (Lagrange's four-square theorem). where the Kronecker symbol has the values There is a formula for r3 in the section on class numbers below. where . where Define the