Numerical study of the derivative of the Riemann zeta function at zeros

The derivative of the Riemann zeta function was computed numerically on several large sets of zeros at large heights. Comparisons to known and conjectured asymptotics are presented.Comment: 13 pages, 5 figures; minor typos fixe

The zeta function on the critical line: Numerical evidence for moments and random matrix theory models

Results of extensive computations of moments of the Riemann zeta function on the critical line are presented. Calculated values are compared with predictions motivated by random matrix theory. The results can help in deciding between those and competing predictions. It is shown that for high moments and at large heights, the variability of moment values over adjacent intervals is substantial, even when those intervals are long, as long as a block containing 10^9 zeros near zero number 10^23. More than anything else, the variability illustrates the limits of what one can learn about the zeta function from numerical evidence. It is shown the rate of decline of extreme values of the moments is modelled relatively well by power laws. Also, some long range correlations in the values of the second moment, as well as asymptotic oscillations in the values of the shifted fourth moment, are found. The computations described here relied on several representations of the zeta function. The numerical comparison of their effectiveness that is presented is of independent interest, for future large scale computations.Comment: 31 pages, 10 figures, 19 table

Exact asymptotics of monomer-dimer model on rectangular semi-infinite lattices

By using the asymptotic theory of Pemantle and Wilson, exact asymptotic expansions of the free energy of the monomer-dimer model on rectangular $n \times \infty$ lattices in terms of dimer density are obtained for small values of $n$, at both high and low dimer density limits. In the high dimer density limit, the theoretical results confirm the dependence of the free energy on the parity of $n$, a result obtained previously by computational methods. In the low dimer density limit, the free energy on a cylinder $n \times \infty$ lattice strip has exactly the same first $n$ terms in the series expansion as that of infinite $\infty \times \infty$ lattice.Comment: 9 pages, 6 table

On the existence of optimum cyclic burst-correcting codes

It is shown that for each integer b >= 1 infinitely many optimum cyclic b-burst-correcting codes exist, i.e., codes whose length n, redundancy r, and burst-correcting capability b, satisfy n = 2^{r-b+1} - 1. Some optimum codes for b = 3, 4, and 5 are also studied in detail

On the Properties of a Tree-Structured Server Process

Let X0 be a nonnegative integer-valued random variable and let an independent copy of X0 be assigned to each leaf of a binary tree of depth k. If X0 and X0′ are adjacent leaves, let X1=(X0−1)++(X0′−1)+ be assigned to the parent node. In general, if Xj and Xj′ are assigned to adjacent nodes at level j = 0,⋯, k − 1, then Xj and Xj′ are, in turn, independent and the value assigned to their parent node is then Xj+1=(Xj−1)++(Xj′−1)+. We ask what is the behavior of Xk as k→∞. We give sufficient conditions for Xk→∞ and for Xk→0 and ask whether these are the only nontrivial possibilities. The problem is of interest because it asks for the asymptotics of a nonlinear transform which has an expansive term (the + in the sense of addition) and a contractive term (the + in the sense of positive part)

The Asymptotic Number of Irreducible Partitions

A partition of [1, n] = {1,..., n} is called irreducible if no proper subinterval of [1, n] is a union of blocks. We determine the asymptotic relationship between the numbers of irreducible partitions, partitions without singleton blocks, and all partitions when the block sizes must lie in some specified set

Scale invariant correlations and the distribution of prime numbers

Negative correlations in the distribution of prime numbers are found to display a scale invariance. This occurs in conjunction with a nonstationary behavior. We compare the prime number series to a type of fractional Brownian motion which incorporates both the scale invariance and the nonstationary behavior. Interesting discrepancies remain. The scale invariance also appears to imply the Riemann hypothesis and we study the use of the former as a test of the latter.Comment: 13 pages, 8 figures, version to appear in J. Phys.