Λ(n), the von Mangoldt function, is 0 unless the argument n is a prime power , in which case it is the natural log of the prime p: p(n) – partition function p(n), the partition function, is the number of ways of representing n as a sum of positive integers, where two representations with the same summands in a different order are not counted as being different: λ(n) – Carmichael function λ(n), the Carmichael function, is the smallest positive number such that for all a coprime to n. Equivalently, it is the least common multiple of the orders of the elements of the multiplicative group of integers modulo n. For powers of odd primes and for 2 and 4, λ(n) is equal to the Euler totient function of n; for powers of 2 greater than 4 it is equal to one half of the Euler totient function of n: and for general n it is the least common multiple of λ of each of the prime power factors of n: h(n) – Class number h(n), the class number function, is the order of the ideal class group of an algebraic extension of the rationals with discriminant n. The notation is ambiguous, as there are in general many extensions with the same discriminant. See quadratic field and cyclotomic field for classical examples. rk(n) – Sum of k squares rk(n) is the number of ways n can be represented as the sum of k squares, where representations that differ only in the order of the summands or in the signs of the square roots are counted as different. D(n) – Arithmetic derivative Using the Heaviside notation for the derivative, the arithmetic derivative D(n) is a function such that if n prime, and (the product rule) Summation functions Given an arithmetic function a(n), its summation function A(x) is defined by A can be regarded as a function of a real variable. Given a positive integer m, A is constant along open intervals m < x < m + 1, and has a 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: Individual values of arithmetic functions may fluctuate wildly – as in most of the above examples.