Emil Grosswald

Arithmetic progressions that consist only of primes

Abstract Let Nm(x) be the number of arithmetic progressions that consist of m terms, all primes and not larger than x, and set F m (x) = C m x 2 log m x (Cm explicitly given). It is shown that Hardy and Littlewoods prime k-tuple conjecture implies that Nm(x) = Fm(x){1 + Σj=1N ajlog−j x + O((log x)−N−1)}, (here the bracket represents an asymptotic series with explicitly computable coefficients). This formula holds rather trivially for m = 1 and m = 2. It is proved here for m = 3, by the Vinogradov version of the Hardy-Ramanujan-Littlewood circle method.

Remarks concerning the values of the riemann zeta function at integral, odd arguments

It is known that for a = 2n + 1, ζ(a) = πar(a) − Ga(i), where Ga(τ) has a Fourier expansion Ga(τ) = Σn = 1∞ cn(a)e2πinτ and r(a) is rational. For a slightly more general function Ga(τ, χ) = Σn = 1∞ cn(a, χ) e2πinτk [which, for k = 1 reduces to Ga(τ)] it is shown that if χ(n) is a nonprincipal, primitive congruence character modulo k, then Ga(i, χ) = πaτ(χ)r(a, χ). Here χ is the conjugate character, τ(χ) is a Gaussian sum, and r(a, χ) is algebraic. If χ(n) is also real, then k−12π−aGa(i, χ) is actually rational.

Dedekind-Rademacher Sums

(1971). Dedekind-Rademacher Sums. The American Mathematical Monthly: Vol. 78, No. 6, pp. 639-644.

Positive integers expressible as a sum of three squares in essentially only one way

Abstract The set S consisting of those positive integers n which are uniquely expressible in the form n = a2 + b2 + c2, a ≧ b ≧ c ≧ 0 , is considered. Since n ∈ S if and only if 4n ∈ S, we may restrict attention to those n not divisible by 4. Classical formulas and the theorem that there are only finitely many imaginary quadratic fields with given class number imply that there are only finitely many n ∈ S with n = 0 (mod 4). More specifically, from the existing knowledge of all the imaginary quadratic fields with odd discriminant and class number 1 or 2 it is readily deduced that there are precisely twelve positive integers n such that n ∈ S and n ≡ 3 (mod 8). To determine those n ∈ S such that n ≡ 1, 2, 5, 6 (mod 8) requires the determination of the imaginary quadratic fields with even discriminant and class number 1, 2, or 4. While the latter information is known empirically, it has not been proved that the known list of 33 such fields is complete. If it is complete, then our arguments show that there are exactly 21 positive integers n such that n ∈ S and n ≡ 1, 2, 5, 6 (mod 8).

On prime representing polynomials

A heuristic method is presented to determine the number of primesp ≤x, represented by an irreducible polynomialf(n), without non-trivial fixed factor (f(y)<∈Z[y]; n∈Z. The method is applied to two specific polynomials and the results are compared with those of the heuristic approach of Hardy and Littlewood.

The Circle Method

We mentioned in Chapter 1 that the number r s (n) of solutions of the Diophantine equation

Alternative Methods for Evaluating r s ( n )

\sum\limits_{{k = 1}}^s {x_i^2} = n

Representations of Integers as Sums of Nonvanishing Squares

(12.1) is the coefficient of x n in the Taylor expansion of the function \( 1 + \sum\nolimits_{{n = 1}}^{\infty } {{r_s}(n){x^n}} \). Here, as in Chapter 8, we write θ(x) for θ3(1;x) and we shall suppress the first entry, which will always be z = 1. From (12.1); it follows, by Cauchy’s theorem, that