Arithmetic Consequences of Jacobi’s Two–squares Theorem
نویسندگان
چکیده
There is a well-known formula due to Jacobi for the number r2(n) of representations of the number n as the sum of two squares. This formula implies that the numbers r2(n) satisfy elegant arithmetic relations. Conversely, these arithmetic properties essentially imply Jacobi’s formula. So it is of interest to give direct proofs of these arithmetic relations, and this we do.
منابع مشابه
The Parents of Jacobi's Four Squares Theorem Are Unique
Jacobi’s four squares theorem asserts that the number of representations of a positive integer n as a sum of four squares is 8 times the sum of the positive divisors of n, which are not multiples of 4. A formula expressing an infinite product as an infinite sum is called a product-to-sum identity. The product-to-sum identities in a single complex variable q from which Jacobi’s four squares form...
متن کاملInfinite Families of Exact Sums of Squares Formulas, Jacobi Elliptic Functions, Continued Fractions, and Schur Functions
In this paper we derive many infinite families of explicit exact formulas involving either squares or triangular numbers, two of which generalize Jacobi’s 4 and 8 squares identities to 4n2 or 4n(n + 1) squares, respectively, without using cusp forms. In fact, we similarly generalize to infinite families all of Jacobi’s explicitly stated degree 2, 4, 6, 8 Lambert series expansions of classical t...
متن کاملCLOSURE OF THE CONE OF SUMS OF 2d-POWERS IN CERTAIN WEIGHTED `1-SEMINORM TOPOLOGIES
In [3] Berg, Christensen and Ressel prove that the closure of the cone of sums of squares ∑R[X]2 in the polynomial ring R[X] := R[X1, . . . , Xn] in the topology induced by the `1-norm is equal to Pos([−1, 1]n), the cone consisting of all polynomials which are non-negative on the hypercube [−1, 1]n. The result is deduced as a corollary of a general result, also established in [3], which is vali...
متن کاملPartial Fractions and Four Classical Theorems of Number Theory
Michael D. Hirschhorn For some years Jacobi’s two and four squares theorems (stated below) have been sources of considerable fascination to me, so much so that I have continually sought the simplest, most direct proofs of them. Indeed, I have in the past presented two proofs of each [1,2,3,4]. It is my intention to give now what I regard as even simpler proofs of these two theorems as well as r...
متن کاملLattice Points on Circles, Squares in Arithmetic Progressions and Sumsets of Squares
Let σ(k) denote the maximum of the number of squares in a+b, . . . , a+kb as we vary over positive integers a and b. Erdős conjectured that σ(k) = o(k) which Szemerédi [30] elegantly proved as follows: If there are more than δk squares amongst the integers a+b, . . . , a+kb (where k is sufficiently large) then there exists four indices 1 ≤ i1 < i2 < i3 < i4 ≤ k in arithmetic progression such th...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 1998