The holomorphic flow of the Riemann zeta function

The flow of the Riemann zeta function, , is considered, and phase portraits are presented. Attention is given to the characterization of the flow lines in the neighborhood of the first 500 zeros on the critical line. All of these zeros are foci. The majority are sources, but in a small proportion of exceptional cases the zero is a sink. To produce these portraits, the zeta function was evaluated numerically to 12 decimal places, in the region of interest, using the Chebyshev method and using Mathematica.

The phase diagrams suggest new analytic properties of zeta, of which some are proved and others are given in the form of conjectures.

     

Holomorphic Flows on Simply Connected Regions Have No Limit Cycles

The dynamical system or flow = f(z), where f is holomorphic on C, is considered. The behavior of the flow at critical points coincides with the behavior of the linearization when the critical points are non-degenerate: there is no center-focus dichotomy. Periodic orbits about a center have the same period and form an open subset. The flow has no limit cycles in simply connected regions. The advance mapping is holomorphic where the flow is complete. The structure of the separatrices bounding the orbits surrounding a center is determined. Some examples are given including the following: if a quartic polynomial system has four distinct centers, then they are collinear.     

Linear law for the logarithms of the Riemann periods at simple critical zeta zeros

Each simple zero of the Riemann zeta function on the critical line with is a center for the flow of the Riemann xi function with an associated period . It is shown that, as ,

Numerical evaluation leads to the conjecture that this inequality can be replaced by an equality. Assuming the Riemann Hypothesis and a zeta zero separation conjecture for some exponent , we obtain the upper bound . Assuming a weakened form of a conjecture of Gonek, giving a bound for the reciprocal of the derivative of zeta at each zero, we obtain the expected upper bound for the periods so, conditionally, . Indeed, this linear relationship is equivalent to the given weakened conjecture, which implies the zero separation conjecture, provided the exponent is sufficiently large. The frequencies corresponding to the periods relate to natural eigenvalues for the Hilbert-Polya conjecture. They may provide a goal for those seeking a self-adjoint operator related to the Riemann hypothesis.

     

On the Fürstenberg closure of a class of binary recurrences

In this paper, we determine the closure in the full topology over Z of the set , where (un)_n?0 is a nondegenerate binary recurrent sequence with integer coefficients whose characteristic roots are quadratic units. This generalizes the result for the case when un=Fn was the nth Fibonacci number.     

A Note on Reducing the Number of Variables in Integer Programming Problems

A necessary and sufficient condition for identification of dominatedcolumns, which correspond to one type of redundant integer variables,in the matrix of a general Integer Programming problem, isderived. The given condition extends our recent work on eliminatingdominated integer variables in Knapsack problems, and revises arecently published procedure for reducing the number of variables ingeneral Integer Programming problems given in the literature. Areport on computational experiments for one class of large scaleKnapsack problems, illustrating the function of this approach, isincluded.