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In this note, we use Dedekind’s eta function to prove a congruence relation between the number of representations by binary quadratic forms of discriminant $-31$ and Fourier coefficients of a weight $16$ cusp form. Our result is analogous to the classical result concerning Ramanujan’s tau function and binary quadratic forms of discriminant $-23$ . more...

A prime dividing a composite Fermat number is called a Fermat prime divisor. Such a prime p must be congruent to 1 modulo 4, and so, by the Fermat–Girard theorem, there exists integers R and S such... more...

A positive-definite diagonal quadratic form $a_{1}x_{1}^{2}+\cdots +a_{n}x_{n}^{2}\;(a_{1},\ldots ,a_{n}\in \mathbb{N})$ is said to be prime-universal if it is not universal and for every prime $p$ there are integers $x_{1},\ldots ,x_{n}$ such that $a_{1}x_{1}^{2}+\cdots +a_{n}x_{n}^{2}=p$. We determine all possible prime-universal ternary quadratic forms $ax^{2}+by^{2}+cz^{2}$ and all possible prime-universal quaternary quadratic forms $ax^{2}+by^{2}+cz^{2}+dw^{2}$. The prime-universal ternary forms are completely determined. The prime-universal quaternary forms are determined subject to the validity of two conjectures. We make no use of a result of Bhargava concerning quadratic forms representing primes which is stated but not proved in the literature. more...

We show how Liouville’s formulas for the number of representations of a positive integer by the forms x12+x22+2x32+2x42,x12+2x22+2x32+4x42,x12+x22+x32+4x42,x12+x22+4x32+4x42 , and x12+4x22+4x32+4x4... more...

Let [Formula: see text] denote a complex variable with [Formula: see text]. For a positive integer [Formula: see text] let [Formula: see text] If [Formula: see text] we define [Formula: see text] for each nonnegative integer [Formula: see text]. In this paper, we determine results of the type [Formula: see text] more...

An explicit formula is given for the representation number of each of the 75 reduced, positive-definite, integral, primitive, quaternary quadratic forms [Formula: see text], which belong to a genus with discriminant [Formula: see text] containing one and only one form class. more...

Let F m denote the set of positive-definite primitive integral quadratic forms in m variables. Let f , g ∈ F m . In this paper we introduce a new concept, namely that of g being derivable from f. This concept is based on a certain theta function identity being valid. A consequence of this concept is that if g is derivable from f then the representation number of g can be given in terms of that of f. Many examples are given, especially for diagonal ternary quadratic forms. more...

Let n be a positive integer. Let $$\delta _3(n)$$ denote the difference between the number of (positive) divisors of n congruent to 1 modulo 3 and the number of those congruent to 2 modulo 3. In 2004, Farkas proved that the arithmetic convolution sum $$\begin{aligned} D_3(n):=\sum _{j=1}^{n-1}\delta _3(j)\delta _3(n-j) \end{aligned}$$ satisfies the relation $$\begin{aligned} 3D_3(n)+\delta _3(n)={\sum _{\mathop {_{d \mid n}}\limits _{3 \not \mid d}}}d. \end{aligned}$$ In this paper, we use a result about binary quadratic forms to prove a general arithmetic convolution identity which contains Farkas’ formula and two other similar known formulas as special cases. From our identity, we deduce a number of analogous new convolution formulas. more...

Let denote the Dedekind eta function. We use a recent productto- sum formula in conjunction with conditions for the non-representability of integers by certain ternary quadratic forms to give explicitly ten eta quotientssuch that the Fourier coefficients c(n) vanish for all positive integers n in each of infinitely many non-overlapping arithmetic progressions. For example, we show that if we have c(n) = 0 for all n in each of the arithmetic progressions more...

We prove a general theorem that evaluates the Legendre symbol [Formula: see text] under certain conditions on the integers A, B, m and the prime p. The evaluation is in terms of parameters appearing in a binary quadratic form representing p. The theorem has applications to quartic residuacity. more...

Let ℕ denote the set of positive integers and ℤ the set of all integers. Let ℕ0 = ℕ ∪{0}. Let a1x2 + b1xy + c1y2 and a2z2 + b2zt + c2t2 be two positive-definite, integral, binary quadratic forms. The number of representations of n ∈ ℕ0 as a sum of these two binary quadratic forms is [Formula: see text] When (b1, b2) ≠ (0, 0) we prove under certain conditions on a1, b1, c1, a2, b2 and c2 that N(a1, b1, c1, a2, b2, c2; n) can be expressed as a finite linear combination of quantities of the type N(a, 0, b, c, 0, d; n) with a, b, c and d positive integers. Thus, when the quantities N(a, 0, b, c, 0, d; n) are known, we can determine N(a1, b1, c1, a2, b2, c2; n). This determination is carried out explicitly for a number of quaternary quadratic forms a1x2 + b1xy + c1y2 + a2z2 + b2zt + c2t2. For example, in Theorem 1.2 we show for n ∈ ℕ that [Formula: see text] where N is the largest odd integer dividing n and [Formula: see text] more...

Ramanujan's formula for the number r24(n) of representations of a positive integer n as a sum of 24 squares asserts that [Formula: see text] where [Formula: see text] and Ramanujan's function τ(n) is given by [Formula: see text] In this paper, we determine a class of formulae analogous to Ramanujan's formula for r24(n). more...

Let ℕ, \(\mathbb{N}_{0}\), ℤ, ℚ, and ℂ denote the sets of positive integers, nonnegative integers, integers, rational numbers, and complex numbers, respectively. If \(f(q)\) is a complex-valued function with $$f(q)= \sum_{n=0}^{\infty} f_{n} q^{n} \quad\bigl(q \in\mathbb{C}, \vert q \vert more...

In his famous paper, "On certain arithmetical functions", Ramanujan offers for the first time the Euler product of the Dirichlet series in which the coefficients are given by Ramanujan's tau-function. In his lost notebook, Ramanujan records further Euler products for L-series attached to modular forms, and, typically, does not record proofs for these claims. In this semi-expository article, for the Euler products appearing in his lost notebook, we provide or sketch proofs using elementary methods, binary quadratic forms, and modular forms. more...

The sum of divisors function σ(m) is defined by [Formula: see text] Let [Formula: see text] denote the upper half of the complex plane. Let η(z)[Formula: see text] be the Dedekind eta function. A class [Formula: see text] of eta quotients is given for which the Fourier series of each member of [Formula: see text] can be given explicitly. One example is [Formula: see text] where [Formula: see text] more...

The purpose of this historical remark is to observe that a slightly stronger form of a recent theorem of Zhang and Yue can be proved more easily using an elementary method given by Dirichlet in 1834. more...

Formulae, originally conjectured by Liouville, are proved for the number of representations of a positive integer n by each of the eight sextenary quadratic forms $x_1^2 +x_2^2+x_3^2 +x_4^2+x_5^2+4x_6^2$, $x_1^2 +x_2^2+x_3^2+x_4^2+4x_5^2+4x_6^2$, $x_1^2 +x_2^2+x_3^2 +4x_4^2+4x_5^2+4x_6^2$, $x_1^2+x_2^2+4x_3^2 +4x_4^2+4x_5^2+4x_6^2$, $x_1^2 +4x_2^2+4x_3^2+4x_4^2+4x_5^2+4x_6^2$, $x_1^2 +x_2^2+x_3^2 +2x_4^2+2x_5^2+4x_6^2$, $x_1^2+x_2^2+2x_3^2 +2x_4^2+4x_5^2+4x_6^2$, $x_1^2 +2x_2^2+2x_3^2+4x_4^2+4x_5^2+4x_6^2$. more...

We use the recent evaluation of certain convolution sums involving the sum of divisors function to determine the number of representations of a positive integer by certain diagonal octonary quadratic forms whose coefficients are 1, 2 or 4. more...

The convolution sum $$ \sum\limits_{\mathop {m = 1}\limits_{m \equiv a(\bmod 4)} }^{n - 1} {\sigma (m)\sigma (n - m)} $$ is evaluated for a ∈ {0, 1, 2, 3} and all n ∈ ℕ. This completes the partial evaluation given in the paper of J.G. Huard, Z.M. Ou, B.K. Spearman, K. S. Williams. more...