Explicit averages of square-free supported functions: to the edge of the convolution method

We give a general statement of the convolution method so that one can provide explicit asymptotic estimations for all averages of square-free supported arithmetic functions that have a sufficiently regular order on the prime numbers and observe how the nature of this method gives error term estimations of order $X^{-\delta}$, where $\delta$ belongs to an open real positive set $I$. In order to have a better error estimation, a natural question is whether or not we can achieve an error term of critical order $X^{-\delta_0}$, where $\delta_0$, the critical exponent, is the right hand endpoint of $I$. We reply positively to that question by presenting a new method that improves qualitatively almost all instances of the convolution method under some regularity conditions; now, the asymptotic estimation of averages of well-behaved square-free supported arithmetic functions can be given with its critical exponent and a reasonable explicit error constant. We illustrate this new method by analyzing a particular average related to the work of Ramar\'e--Akhilesh (2017), which leads to notable improvements when imposing non-trivial coprimality conditions.Comment: Updated. Some correction

On a logarithmic sum related to a natural quadratic sieve

We study the sum $\Sigma_q(U)=\sum_{\substack{d,e\leq U\\(de,q)=1}}\frac{\mu(d)\mu(e)}{[d,e]}\log\left(\frac{U}{d}\right)\log\left(\frac{U}{e}\right)$, $U>1$, so that a continuous, monotonic and explicit version of Selberg's sieve can be stated. Thanks to Barban-Vehov (1968), Motohashi (1974) and Graham (1978), it has been long known, but never explicitly, that $\Sigma_1(U)$ is asymptotic to $\log(U)$. In this article, we discover not only that $\Sigma_q(U)\sim\frac{q}{\varphi(q)}\log(U)$ for all $q\in\mathbb{Z}_{>0}$, but also we find a closed-form expression for its secondary order term of $\Sigma_q(U)$, a constant $\mathfrak{s}_q$, which we are able to estimate explicitly when $q=v\in\{1,2\}$. We thus have $\Sigma_v(U)= \frac{v}{\varphi(v)}\log(U)-\mathfrak{s}_v+O_v^*\left(\frac{K_v}{\log(U)}\right)$, for some explicit constant $K_v > 0$, where $\mathfrak{s}_1=0.60731\ldots$ and $\mathfrak{s}_2=1.4728\ldots$. As an application, we show how our result gives an explicit version of the Brun-Titchmarsh theorem within a range.Comment: accepted in Acta Arithmetic

Investing in a More Robust Public Policy Environment in the Middle East

Outlines considerations for funders with respect to the development of human capital and strengthening civil society organizations, including the need for long-term, targeted, and sustained investment as well as funder restraint

Forty Years of the BLS Export and Import Price Indexes: Trends and Competition

[Excerpt] U.S. competitiveness is measured in many different ways. However, two sets of measures that historically have been of interest in helping to assess the strength of the economy are the price trends of U.S. exports and of U.S. imports. The Bureau of Labor Statistics (BLS, the Bureau) first began publication of a limited set of export price indexes in 1971, followed 2 years later by a set of import price indexes. The Bureau developed these series in response to two key concerns, one statistical and the other economic. The statistical issue was over the validity of the import and export unit value indexes published by the Census Bureau. The economic question had to do with the nation’s ability to compete in the increasingly global economy. The publication of detailed export and import price indexes helped address these concerns because those indexes were true price indexes and because they provided a greater level of detail than the analogous indexes previously published by the Census Bureau. The number of published series increased until the Bureau was able to publish the first all-import price index in 1983, followed a year later by the first all-export price index. Although the publication of these series alleviated the need to rely upon the unit value data—indeed, the Census Bureau discontinued publication of its unit value indexes in 1989—concerns over U.S. competitiveness have only grown over time

From a minor to a major profession: Can planning and planning theory meet the challenges of globalisation?

The years 2016–2017 have opened up a dream-world set of opportunities for the planning profession. To what extent are planning education and the global planning profession intrinsically ready to take up these opportunities, and are there prices to be paid