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Niven, Fermat's theorem for matrices, Duke Math. J. 15 (1948) 823–826, MR0026672 (10,183e), proved, for any positive integers $a$ and $r$, $${\rm lcm}[x-1,x^2-1,\dots,x^r-1]_{x=a}={\rm lcm}[a-1,a^2-1, …

Michael Drmota and Christian Mauduit [EDIT: and Joel Rivat], The sum-of-digits function of polynomial sequences, J. Lond. Math. Soc. (2) 84 (2011), no. 1, 81–102, MR2819691 (2012f:11193) Theorem 1: …

It is known that $\sum_{x<n\le x+h(x)}\tau(n)$ is asymptotic to $h(x)\log x$ for $h(x)=x^{\theta}$ with $\theta>131/416$, and it is conjectured that the asymptotic holds for $h(x)\gg x^{\epsilon}$. Se …

D Karagulyan, On certain aspects of the Mobius randomness principle, writes (Remark 1, page 9), "We remark, that the result proved above contradicts with what is claimed in [Reference 1]. There it is …

Bary-Soroker and Kozma, Irreducible polynomials of bounded height, https://arxiv.org/abs/1710.05165 write, concerning monic polynomials, "one fixes $n=\deg f$, and the coefficients are i.i.d. random v …

J Wu, Chen's double sieve, Goldbach's conjecture, and the twin prime problem, Acta Arith 114 (2004) 215-273, MR 2005e:11128, bounds the number of twin primes above by $2aCx/\log^2x$, with $C=\prod p( …

3-smooth numbers are tabulated at http://oeis.org/A003586 and there you will also find the asymptotic, $${1\over\sqrt6}e^{\sqrt{2(\log2\log3)n}}$$ (attributed to Benoit Cloitre) as well as several lin …

This is proved by Kevin Brown. The numbers are tabulated, and many links and references given, at the Online Encyclopedia of Integer Sequences.

Stopple, A Primer of Analytic Number Theory, proves a theorem which looks something like the one under discussion. On page 248, he has $$\pi(x)=R(x)+\sum_{\rho}R(x^{\rho})+\sum_{n=1}^{\infty}{\mu(n)\o …

Are you just trying to show $s(n)$ is unbounded? and do you insist on a non-trivial technique? Let $n=m!$; then $s(n)>\sum_{p\lt m}{p-1\over p^2}=\sum_{p\lt m}{1\over p}+O(1)$ and of course the sum di …

The sine function has little to do with this; you get $\epsilon_n=1$ if $n\theta/2\pmod1$ is in $(0,1/2)$, $-1$ if it's in $(1/2,1)$. Now you can probably apply bounds for the discrepancy of the seque …

I beleive this is done in Keith Conrad's paper, On Weil's proof of the bound for Kloosterman sums, J. Number Theory 97 (2002), no. 2, 439–446, MR1942969 (2003j:11087).

Not quite an answer, but maybe it points to one: Ivan Niven, Uniform distribution of sequences of integers, Compositio Mathematica 16 (1964) 158-160, defines a sequence $A=(a_1,a_2,\dots)$ of intege …

If $p/q$ and $r/s$ are consecutive convergents of the continued fraction for $x$, then $(tp+ur)/(tq+us)$ is a fairly good approximation to $x$, and you may be able to choose small numbers $t$ and $u$ …

To answer your "lighter note" question, of course we want to find exact formulas. It's only when we can't find an exact (and useful) formula that we settle for asymptotic formulas, and if we can't eve …