On the Taylor coefficients of the composition of two analytic functions.
Generalised prime systems with periodic integer counting function
An arithmetical mapping and applications to Ω-results for the Riemann zeta function
Determinants of multiplicative Toeplitz matrices
On the zeros of a class of arithmetical entire functions
Flows of Mellin transforms with periodic integrator
We study Mellin transforms for which is periodic with period in order to investigate ‘flows’ of such functions to Riemann’s and the possibility of proving the Riemann Hypothesis with such an approach. We show that, excepting the trivial case where , the supremum of the real parts of the zeros of any such function is at least . We investigate a particular flow of such functions which converges locally uniformly to as , and show that they exhibit features similar to . For...
An example in Beurling's theory of generalised primes
We prove some connections between the growth of a function and its Mellin transform and apply these to study an explicit example in the theory of Beurling primes. The example has its generalised Chebyshev function given by [x]-1, and associated zeta function ζ₀(s) given via , where ζ is Riemann’s zeta function. We study the behaviour of the corresponding Beurling integer counting function N(x), producing O- and Ω- results for the ’error’ term. These are strongly influenced by the size of ζ(s) near...
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