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Yes, the conjecture is true -- $\zeta_\theta(s)$ extends to a meromorphic function on $\mathbb{C}$ for all real quadratic irrationals $\theta$ with poles at $s=2$, $s=1$ and in the infinite set of ver …

No, convergence does not hold for all irrational $x$. I posted a full answer to the question at math.stackexchange and I'll summarize the result here. There are uncountably many values of $x$ for whi …

Searching including the key phrases "Hardy-Littlewood circle method" and "singular series", as suggested by the other answers, turned up some interesting references which shed light on the question an …

Let $f\in\mathbb{Z}[X_1,\ldots,X_n]$ be a Diophantine equation which, for the purposes of this question, I will assume is homogeneous and nonsingular on $\mathbb{R}^n\setminus\{0\}$ (so that $\nabla f …

It is not difficult to calculate upper bounds on $s(n)$ from bounds on the prime counting function $\pi(n)$. Just use integration by parts, $$ s(n) = \int_0^n x\,d\pi(x) = n\pi(n) - \int_0^n\pi(x)\,dx …

To expand on the answer in my comment, the proportion of integers $a$ for which $4a+1,4a+2,4a+3$ are all squarefree is $\prod_{p\not=2}(1-3/p^2)$, with the product taken over all odd primes $p$. As th …

I'm not an expert here, but reading J.S.Milne's notes on elliptic curves (available from his website), I noticed the following statement. ASIDE 2.4 There is a heuristic explanation for Mordell’s …

No. In fact, given a finite set of primes $S=\lbrace p_1,p_2,\ldots,p_n\rbrace$, we can find a finite extension $K$ of $\mathbb{Q}$ in which elements of $S$ decompose in any way that we like. That is, …

To answer Question 1: Yes, there do exist integrally closed abstract number rings with infinite class group. By factorization of ideals, for $R$ to be an abstract number ring it is enough that it is …

The following question is an attempt to find a lower bound for the value of a polynomial at integer points. It is something that I originally thought about while trying to understand how it would be p …

After thinking about this problem for a bit using rather a naive approach, looking at regions where f grows faster than quadratically (as mentioned in Qiaochu's attempt), it certainly appears that obt …

I don't know of a paper proving the result, but I can prove it for you now. In fact, the methods in the paper you link generalize to an arbitrary base $g\gt2$. The authors of the paper don't seem to t …

This response is in answer to David's further question about whether it is possible to bound the rate at which SN/N tends to zero, as he was wanting to use Weyl's inequality to do. This is not possibl …

I made the following attempt to prove the result using Weyl's inequality, which failed. It only manages to show that $\liminf S_N/N\to0$. However, as mentioned in Benoît's answer, with a little bit mo …

The statement was shown to be false by J. Bourgain in a paper published in 1988 (Almost Sure Convergence and Bounded Entropy, doi:10.1007/BF02765022). Well before either Harman's 1997 book and the 200 …