arXiv subjects
Publications and source records attributed to Michael Revers.
In this paper, we present an improved explicit subconvexity result for the Riemann zeta function $\zeta\left( s\right)$ along the critical line $s=1/2+it$, given by Hiary, Patel and Yang in 2024. This new bound is derived by combining a refined, explicit version of the van der Corput method together with computational calculations.
We denote by $\pi\left( x\right) $ the usual prime counting function and let $li\left( x\right) $ the logarithmic integral of $x$. In 1966, R.S. Lehman came up with a new approach and an effective method for finding an upper bound where it is assured that a sign change occurs for $\pi\left( x\right) -li\left( x\right) $ for some value $x$ not higher than this given bound. In this paper we provide further improvements on the error terms including an improvement upon Lehman's famous error term $S_{3}$ in his original paper. We are now able to eliminate the lower condition for the size-length $\eta$ completely. For further numerical computations this enables us to establish sharper results on the positions for the sign changes. We illustrate with some numerical computations on the lowest known crossover regions near $10^{316}$ and we discuss numerically on potential crossover regions below this value.
In this paper, we discuss asymptotic relations for the approximation of $\left\vert x\right\vert ^{\alpha},\alpha>0$ in $L_{\infty}\left[ -1,1\right] $ by Lagrange interpolation polynomials based on the zeros of the Chebyshev polynomials of first kind. The limiting process reveals an entire function of exponential type for which we can present an explicit formula. As a consequence, we further deduce an asymptotic relation for the Approximation error when $\alpha\rightarrow\infty$. Finally, we present connections of our results together with some recent work of Ganzburg [5] and Lubinsky [10], by presenting numerical results, indicating a possible constructive way towards a representation for the Bernstein constants.