Residue Classes Having Tardy Totients

We show, in an effective way, that there exists a sequence of congruence classes $a_k\pmod {m_k}$ such that the minimal solution $n=n_k$ of the congruence $\phi(n)\equiv a_k\pmod {m_k}$ exists and satisfies $\log n_k/\log m_k\to\infty$ as $k\to\infty$. Here, $\phi(n)$ is the Euler function. This answers a question raised in \cite{FS}. We also show that every congruence class containing an even integer contains infinitely many values of the Carmichael function $\lambda(n)$ and the least such $n$ satisfies $n\ll m^{13}$.Comment: 14 page

On Waring's problem: two squares and three biquadrates

We investigate sums of mixed powers involving two squares and three biquadrates. In particular, subject to the truth of the Generalised Riemann Hypothesis and the Elliott-Halberstam Conjecture, we show that all large natural numbers n with 8 not dividing n, n not congruent to 2 modulo 3, and n not congruent to 14 modulo 16, are the sum of 2 squares and 3 biquadrates.Comment: to appear in Mathematik

Products in Residue Classes

We consider a problem of P. Erdos, A. M. Odlyzko and A. Sarkozy about the representation of residue classes modulo m by products of two not too large primes. While it seems that even the Extended Riemann Hypothesis is not powerful enough to achieve the expected results, here we obtain some unconditional results ``on average'' over moduli m and residue classes modulo m and somewhat stronger results when the average is restricted to prime moduli m = p. We also consider the analogous question wherein the primes are replaced by easier sequences so, quite naturally, we obtain much stronger results.Comment: 18 page

Open Access, Public Access: Policies, Implementation, Developments, and the Future of U.S.-Published Research

In February, the White House Office of Science and Technology Policy (OSTP) released a memo directing each US funding agency with over $100 million in annual research expenditure to develop a plan to support public access to the results of research funded by the federal government, including results published in scholarly journals. How has the OSTP memo impacted scholarly publishing so far? What exactly has been achieved so far, and what will it mean for the future of U.S.-published research? This interactive session features a panel of speakers who will be discussing the recent developments and emerging issues from the librarian, funder, researcher, and publisher perspectives. Among the questions to be addressed are: What is the US position on implementing open access? What steps have been taken to implement OSTP objectives? What role do institutional repositories play in open access? How will funder’s open access policies impact universities and researchers? What role are publishers playing in the implementation of open access? How will open access content be linked and measured? Will it affect usage