Search Results

This answer summarizes the above discussion: Extend $p \mapsto \left( \frac{n}{p} \right)$ to a multiplicative function $\chi$ on the positive integers. By quadratic reciprocity, $\chi$ is periodic mo …

I'll write $T_f$ for the set of primes $p$ such that $f(x)$ is a bijection $\mathbb{F}_p \to \mathbb{F}_p$. I claim that $T_f$ is always either finite or else $\# \{ p \in T_f : p \leq y \} \sim c \tf …

This is right. Primitive Pythagorean triples are parametrized as $(u^2-v^2, 2uv, u^2+v^2)$ with $\mathrm{GCD}(u,v) = 1$ and $u+v \equiv 1 \bmod 2$. To have $0 < a,b < c \leq R^2$, we must have $0 < v …

Let $$t_P = \sum_{p < P} \log \left| \frac{1}{1-p^{-s}} \right|$$ with $s=\sigma+it$, $\sigma \in (0,1)$ and $t$ a nonzero real. The point of this answer is to show that the $t_P$ jump around a great …

If you believe Bunyakovsky's conjecture, there are infinitely many primes of the form $x^3+2$. For such a prime $p$, consider the matrix $$\begin{bmatrix} 1&x&x^2 \\ x&x^2&-2 \\ x^2&-2&-2x \\ \end{bma …

The following might make you happy: I claim that there are elementary proofs that $$\sum_{p \leq N} \frac{\log p}{p} = \log N + O(1)\ \mbox{and}$$ $$\sum_{p \leq N,\ p \equiv 1 \bmod 4} \frac{\log p}{ …

Here is another, more Chebyshev like, approach. It is possible that this is the proof Terry was sketching and I just didn't understand it. I think this should generalize to AP's for any modulus, but I …

Let $d \pi$ be the measure on $\mathbb{R}_{>0}$ which is a delta function of size $1$ at the primes. In other words, $\int_{y}^x d \pi = \pi(x) - \pi(y)$. The prime number theorem is that $\pi(x)$ is …

I can give a closed formula for the pushforward of $SU(3)$ Haar measure under trace. I don't understand the number theoretic issues well enough to know whether I want $SU(3)$ or $U(3)$, though. I'l …

I thought I'd write out the bounds we can get from the pigeonhole principle. Let's say that I am working with the primes $\leq P$, with polynomials of degree $D$ and I want the largest coefficient to …

Theorem 6 in Chapter XV, $\S$5 of Lang Algebraic Number Theory is a result of the sort you want. Lang formulates it using ideles, but he gives an application in more classical language in Example 3 on …

I'm not an analytic number theorist, so take this all with many grains of salt. Let $S(x,y,p)$ be the set of integers in the interval $(x,x+y)$ which are NOT divisible by $p$. The argument you are i …

These are just obvious observations that are too long to fit in a comment. Define $g(z) = \sum x_i z^{p_i}$. The goal is that, for $\omega$ a primitive $N$-th root of unity, we have $$g(\omega) g(\o …

You probably know this, but: Set $s(n) = \mu(1) + \cdots + \mu(n)$. Suppose, for the sake of contradiction, that $s(n) = O(n^{1/2 - \epsilon})$. Then $$\sum s(n) \left( n^{-s} - (n+1)^{-s} \right)$$ …

Didn't Euler prove that $\sum 1/p$ diverges? This proves that $\pi(N)$ is infinitely often larger than $N^{1-\epsilon}$: If there were only $N^{1-\epsilon}$ primes less than $N$, then there are at mos …