Arithmetic properties of the Ramanujan function

We study some arithmetic properties of the Ramanujan function $\tau(n)$, such as the largest prime divisor $P(\tau(n))$ and the number of distinct prime divisors $\omega(\tau(n))$ of $\tau(n)$ for various sequences of $n$. In particular, we show that \hbox{$P(\tau(n)) \geq (\log n)^{33/31 + o(1)}$} for infinitely many $n$, and \begin{equation*} P(\tau(p)\tau(p^2)\tau(p^3)) > (1+o(1))\frac{\log\log p\log\log\log p} {\log\log\log\log p} \end{equation*} for every prime $p$ with \hbox{$\tau(p)\neq 0$}.Comment: 8 page

The pp-Center Problem in Tree Networks Revisited

We present two improved algorithms for weighted discrete $p$-center problem for tree networks with $n$ vertices. One of our proposed algorithms runs in $O(n \log n + p \log^2 n \log(n/p))$ time. For all values of $p$, our algorithm thus runs as fast as or faster than the most efficient $O(n\log^2 n)$ time algorithm obtained by applying Cole's speed-up technique [cole1987] to the algorithm due to Megiddo and Tamir [megiddo1983], which has remained unchallenged for nearly 30 years. Our other algorithm, which is more practical, runs in $O(n \log n + p^2 \log^2(n/p))$ time, and when $p=O(\sqrt{n})$ it is faster than Megiddo and Tamir's $O(n \log^2n \log\log n)$ time algorithm [megiddo1983]

Faster polynomial multiplication over finite fields

Let p be a prime, and let M_p(n) denote the bit complexity of multiplying two polynomials in F_p[X] of degree less than n. For n large compared to p, we establish the bound M_p(n) = O(n log n 8^(log^* n) log p), where log^* is the iterated logarithm. This is the first known F\"urer-type complexity bound for F_p[X], and improves on the previously best known bound M_p(n) = O(n log n log log n log p)

The master function and applications

In this paper we introduce a function that is neither additive nor multiplicative, and is somewhat akin to the Von Mangoldt function. As an application we show that \begin{align}\sum \limits_{p\leq x/2}\frac{\pi(p)}{p}\geq (1+o(1))\log \log x\nonumber \end{align}as $x\longrightarrow \infty$, and \begin{align}\sum \limits_{p\leq x/2}\theta(x/p)\bigg(\frac{\log x}{\log p}-1\bigg)^{-1}\ll x\log \log x \nonumber \end{align} where $p$ runs over the primes.Comment: 5 page