On Short Exponential Sums Involving Fourier Coefficients of Holomorphic Cusp Forms

By an exponential sum of the Fourier coefficients of a holomorphic cusp form we mean the sum which is formed by first taking the Fourier series of the said form,then cutting the beginning and the tail away and considering the remaining sum on the real axis. For simplicity’s sake, typically the coefficients are normalized. However, this isn’t so important as the normalization can be done and removed simply by using partial summation. We improve the approximate functional equation for the exponential sums of the Fourier coefficients of the holomorphic cusp forms by giving an explicit upper bound for the error term appearing in the equation. The approximate functional equation is originally due to Jutila [9] and a crucial tool for transforming sums into shorter sums. This transformation changes the point of the real axis on which the sum is to be considered. We also improve known upper bounds for the size estimates of the exponential sums. For very short sums we do not obtain any better estimates than the very easy estimate obtained by multiplying the upper bound estimate for a Fourier coefficient (they are bounded by the divisor function as Deligne [2] showed) by the number of coefficients. This estimate is extremely rough as no possible cancellation is taken into account. However, with small sums, it is unclear whether there happens any remarkable amounts of cancellation.

Expanding a Global Company Management System to An Integrated Coverage of Several Standards

This Master’s thesis examines the implementation of management system standard requirements as integrated in the organization. The aim is to determine how requirements from management system standards ISO 14001:2015 and ISO 9001:2015 can be integrated and implemented into the existing ISO 9001:2008 compliant management system. Research was executed as action research by utilizing an operating model about the integrated use of management system standards created by the International Organization for Standardization. Phases of the operating model were applied to the target organization. The similarity and integration potential of relevant standards were assessed by using comparative matrices. Allocation of the requirements and conformity assessment of the processes was executed by gap analysis. The main results indicate that the requirements of the relevant standards are principally equivalent or have the same kind of purpose. The results also show the most important processes of the target organization in terms of requirement compliance, as well as the requirements which affect the process the most. Prioritizing the compliance achievement of the most important processes and implementation of those requirements that have the most effect create an opportunity for organizations to implement the integrated requirements effectively.