• Corpus ID: 119139621

Zeros of Lattice Sums: 2. A Geometry for the Generalised Riemann Hypothesis

@article{McPhedran2016ZerosOL,
  title={Zeros of Lattice Sums: 2. A Geometry for the Generalised Riemann Hypothesis},
  author={Ross C. McPhedran},
  journal={arXiv: Mathematical Physics},
  year={2016}
}
  • R. McPhedran
  • Published 12 February 2016
  • Mathematics
  • arXiv: Mathematical Physics
The location of zeros of the basic double sum over the square lattice is studied. This sum can be represented in terms of the product of the Riemann zeta function and the Dirichlet beta function, so that the assertion that all its non-trivial zeros lie on the critical line is a particular case of the Generalised Riemann Hypothesis (GRH). It is shown that a new necessary and sufficient condition for this special case of the GRH to hold is that a particular set of equimodular and equiargument… 
1 Citations

Zeros of Lattice Sums: 3. Reduction of the Generalised Riemann Hypothesis to Specific Geometries

The location of zeros of the basic double sum over the square lattice is studied. This sum can be represented in terms of the product of the Riemann zeta function and the Dirichlet beta function, so

References

SHOWING 1-10 OF 18 REFERENCES

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

Zeros of two-dimensional sums of the Epstein zeta type over rectangular lattices of the type investigated by Hejhal and Bombieri in 1987 are considered, and in particular a sum first studied by

The Riemann Hypothesis for Angular Lattice Sums

We present further results on a class of sums which involve complex powers of the distance to points in a two-dimensional square lattice and trigonometric functions of their angle, supplementing

The Riemann Hypothesis for Symmetrised Combinations of Zeta Functions

This paper studies combinations of the Riemann zeta function, based on one defined by P.R. Taylor, which was shown by him to have all its zeros on the critical line. With a rescaled complex argument,

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

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

Statistical properties of the zeros of zeta functions-beyond the Riemann case

We investigate the statistical distribution of the zeros of Dirichlet L-functions both analytically and numerically. Using the Hardy-Littlewood conjecture about the distribution of primes we show

Statistical properties of the zeros of zeta functions-beyond the Riemann case

We investigate the statistical distribution of the zeros of Dirichlet L-functions both analytically and numerically. Using the Hardy-Littlewood conjecture about the distribution of primes we show

Distributive and analytic properties of lattice sums

We use sums over Bessel functions of the first kind to derive a convenient form of the Poisson summation identity relating sums over direct lattices in two dimensions to sums over reciprocal

The Theory of the Riemann Zeta-Function

The Riemann zeta-function embodies both additive and multiplicative structures in a single function, making it our most important tool in the study of prime numbers. This volume studies all aspects

The Riemann zeta function

DORIN GHISA Riemann Zeta function is one of the most studied transcendental functions, having in view its many applications in number theory,algebra, complex analysis, statistics, as well as in