function as That is, if n is odd, is the sum of the kth powers of the divisors of n, that is, and if n is even it is the sum of the kth powers of the even divisors of n minus the sum of the kth powers of the odd divisors of n. Adopt the convention that Ramanujan's if x is not an integer. Divisor sum convolutions Here "convolution" does not mean "Dirichlet convolution" but instead refers to the formula for the coefficients of the product of two power series: The sequence is called the convolution or the Cauchy product of the sequences an and bn. These formulas may be proved analytically (see Eisenstein series) or by elementary methods. where τ(n) is Ramanujan's function. Since σk(n) (for natural number k) and τ(n) are integers, the above formulas can be used to prove congruences for the functions. See Ramanujan tau function for some examples. Extend the domain of the partition function by setting This recurrence can be used to compute p(n). Class number related Peter Gustav Lejeune Dirichlet discovered formulas that relate the class number h of quadratic number fields to the Jacobi symbol. An integer D is called a fundamental discriminant if it is the discriminant of a quadratic number field. This is equivalent to D ≠ 1 and either a) D is squarefree and D ≡ 1 (mod 4) or b) D ≡ 0 (mod 4), D/4 is squarefree, and D/4 ≡ 2 or 3 (mod 4). Extend the Jacobi symbol to accept even numbers in the "denominator" by defining the Kronecker symbol: Then if D < −4 is a fundamental discriminant There is also a formula relating r3 and h. Again, let D be a fundamental discriminant, D < −4. Then Prime-count related Let be the nth harmonic number. Then is true for every natural number n if and only if the Riemann hypothesis is true. The Riemann hypothesis is also equivalent to the statement that, for all n > 5040, (where γ is the Euler–Mascheroni constant). This is Robin's theorem. Menon's identity In 1965 P Kesava Menon proved This has been generalized by a number of mathematicians. For example, B. Sury N. Rao where a1, a2, ..., as are integers, gcd(a1, a2, ..., as, n) = 1. László Fejes Tóth where m1 and m2 are odd, m = lcm(m1, m2). In fact, if f is any