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Professor Tao, I do not know whether this answer is in the least useful but will post it anyway! I'm not sure but perhaps one approach is via Linnik's theorem that the least prime, say, $p(r,q)$ in …

That's a very interesting problem. If one restricts to initial point $(x,1)$, its like a generalization of the question of points $(x,y) \in (\mathbb{F}_p)^2$ so that $xy\equiv 1 \bmod p$ with $(x,y)$ …

In Theorem 2 of their paper, "Pseudoprime values of the Fibonacci sequence, polynomials and the Euler function", Indagationes Mathematicae, Volume 17, Issue 4, pp. 611--625, Luca and Shparlinski show …

Let $f(x_1,\dots,x_n) \in \mathbb{Z}[x_1,\dots,x_n]$ be a polynomial for which the set of integers not represented by it is infinite. I'm curious about cases in which there exists an integer $c$ such …

EDIT: Gerhard Paseman has given some wonderful answers to this question below. Thank you. This is an attempt to revisit this to hopefully make the question more rigorous with some notation and try to …

For a class of concrete examples with at least asymptotically more than $n/2$ zeros on the unit circle, the Fekete polynomials, which were just mentioned recently by Franz Lemmermeyer at this question …

Hi there, I hope I'm not duplicating anything with what I will write. Here goes: If one considers the Bateman-Horn conjecture, it predicts that $$ \sum_{n \leq x}\Lambda(f(n)) \sim \prod_p\left(\frac{ …

Hi, I refer to formula (8) in Chapter 1 of H. Davenport, Multiplicative Number Theory, Third Edition, Springer (2000), which says that for primes $q\equiv 3 \bmod 4$: $$ L\left(\left(\frac{\cdot}{q}\ …

Mr Helms, This is the $n=1$ case. Your formula gives $e_{1,q}=q$. Say we want to study how often prime $q=q_k$ divides $\prod_{p \leq x}(p-1)$. Maybe write this product as $$ \left(\prod_{i=1}^m\prod …

Not claiming to be a real number theorist or anything of that sort...Just wondering, is one possible approach to Mark Sapir's strong conjecture (assuming an answer isn't already known) to calculate $$ …

Perhaps this might be another perspective on this problem. In an answer to a question I had previously asked on Math Overflow, "A generalized Möbius function?", the following paper of Addison was cite …

Hello, I too would be very curious about this bound and will think further about this. The nice thing about the $u_i(y)$, as I recall my advisor having told me, is that the estimates for geometric su …

Hi, I don't know if this question is appropriate for Math Overflow but I was wondering if there is anything known about the following: Let $$ S(\alpha) = \sum_{n \leq x}\Lambda(n)e(n\alpha). $$ Then …

There are a number of generalizations of the Möbius function out there, which can be found by Google. But I'd just like to know if anything has been said about this: For $k \geq 2$, $k \in \mathbb{Z} …

Hi, In Remark 2 of "Zeros of Fekete Polynomials", (http://arxiv.org/PS_cache/math/pdf/9906/9906214v1.pdf), Conrey et. al. give $$\sup_{|z|=1}|f(z)| \ll p^{0.5} \log p.$$ But the Mahler measure of $f …