On the Riemann property of angular lattice sums and the one-dimensional limit of two-dimensional lattice sums

Abstract

We consider a general class of two-dimensional lattice sums consisting of complex powers s of inverse quadratic functions. We consider two cases, one where the quadratic function is negative definite and another more restricted case where it is positive definite. In the former, we use a representation due to H. Kober, and consider the limit u →∞, where the lattice becomes ever more elongated along one period direction (the one-dimensional limit). In the latter, we use an explicit evaluation of the sum due to Zucker and Robertson. In either case, we show that the one-dimensional limit of the sum is given in terms of ζ (2 s ) if Re( s )>1/2 and either ζ (2 s −1) or ζ (2−2 s ) if Re( s )<1/2. In either case, this leads to a Riemann property of these sums in the one-dimensional limit: their zeros must lie on the critical line Re( s )=1/2. We also comment on a class of sums that involve complex powers of the distance to points in a two-dimensional square lattice and trigonometric functions of their angle. We show that certain of these sums can have their zeros on the critical line but not in a neighbourhood of it; others are identically zero on it, while still others have no zeros on it.

The Riemann hypothesis and the zero distribution of angular lattice sums

“…We recall the definition from McPhedran et al (2008), hereafter referred to as (I), of two sets of angular lattice sums for the square array…”

“…We begin by briefly describing the key properties of the angular sums C(1, 4m; s), such as their functional equation and an exponentially convergent representation that enables their numerical evaluation everywhere in the complex s plane. More comprehensive accounts of these properties may be found in previous papers (McPhedran et al 2004(McPhedran et al , 2008(McPhedran et al , 2010). We then give numerical data on the distributions of zeros, which motivated the analytical results derived in §4.…”

“…We then give numerical data on the distributions of zeros, which motivated the analytical results derived in §4. These are obtained by studying the properties of the quotient function D 4 (2, 2m; s) of C(1, 4m; s) and C(0, 1; s), whereas our previous investigations (McPhedran et al 2008) had focused on their product function D 3 (2, 2m; s). As we will see, the quotient function is well adapted to the comparison of the distributions of zeros of C(1, 4m; s) and C(0, 1; s).…”

Singular Euler-Maclaurin expansion on multidimensional lattices

“…HSEM operator coefficients. For a square lattice in d = 2 dimensions, a simple approach for generating the HSEM operator coefficients is available that avoids derivatives of Epstein zeta functions and uses efficient summation formulas that have been found in the analysis of the Riemann hypothesis in higher dimensions [24]. For a two-dimensional square lattice, it has been shown that [34, Eq.…”

Zeros of Lattice Sums: 1. Zeros off the Critical Line

“…Epstein zeta functions take the form of a double sum McPhedran and coworkers [9][10][11] considered a set of double sums incorporating a trigonometric function of p 1 and p 2 in the numerator, with the denominator (p 2 1 + p 2 2 ) s . They presented some numerical evidence that a particular group of sums, varying trigonometrically as cos(4θ), had all zeros on the critical line, with gaps between the zeros behaving in the manner expected of Dirichlet L functions.…”