Introduction 1. Discrete group actions 2. Invariant differential operators 3. Automorphic forms and L-functions for SL(2,Z) 4. Existence of Maass forms 5. Maass forms and Whittaker functions for… Expand

We explore the subsequence of primes with prime subscripts, (qn), and derive its density and estimates for its counting function. We obtain bounds for the weighted gaps between elements of the… Expand

For a prime $p$ and an integer $a \in \Z$ we obtain nontrivial upper bounds on the number of solutions to the congruence $x^x \equiv a \pmod p$, $1 \le x \le p-1$. We use these estimates to estimate… Expand

The gcd-sum is an arithmetic function defined as the sum of the gcd’s of the first n integers with n : g(n) = P n=1 (i,n). The function arises in deriving asymptotic estimates for a lattice point… Expand

A proof is given that the improper Riemann integral of ζ(s, a) with respect to the real parameter a, taken over the interval (0, 1], vanishes for all complex s with ℜ(s) < 1. The integral does not… Expand

Abstract A topology on $\mathbb{Z}$ , which gives a nice proof that the set of prime integers is infinite, is characterised and examined. It is found to be homeomorphic to $\mathbb{Q}$ , with a… Expand

There are a surprising number of advantages to Lisp, especially in cases where a mixture of symbolic and numeric processing is needed, and it is compared to Fortran in terms of the speed of efficiency of generated code and the structure and convenience of the language.Expand

Using a result of Bordelles, we derive the second term and improved error expres- sions for the partial sums of the Dirichlet series of the gcd-sum function, for all real values of the parameter.