Does anyone know whether the following average converges or not when N goes to infinity. $$\frac{1}{N^2}\sum _{d=1}^N \log d \sum _{n=1}^{N/d} \frac{\phi(n)}{\log (dn)}$$

I've been reading Mander's & Adleman's paper on "NP-Complete decision problems for binary quadratics" (https://www.sciencedirect.com/science/article/pii/0022000078900442?np=y). …

Hejhal's algorithm [1] was a little gadget invented in the 90's for calculating the Hecke eigenvalues and Fourier coefficients of Maass wave forms. Later, Booker, Strombergsson, an …

Can we ever know the sum of the first $10^{10^{100}}$ digits of $\pi$? Can we calgulate the $n$th digit of $\pi$ when the Kolmogorov-complexity of $n$ is larger than the complexit …

Hamkins showed that his infinite time Turing machine has the power to decide some $\Delta_2^1$ sets. I wonder if some modifications of the machine could be made to reach level $\Si …

Given $a \in \mathbb{Z}$ with $a > 1$. Let $g(x) \in \mathbb{Z}[x]$ be a polynomial with $g(a)=\pm 1$. Let $h(x) \in \mathbb{Z}[x]$ be a polynomial with $h(a)= p$, a prime. Let $g( …

Given a multivariate polynomial $f(x_1,\dots,x_n)$ with integer coefficients, how to find an integer $m$ (if it exists) such that $f(x_1,\dots,x_n) + m$ factors into polynomials of …

For a given a bivariate polynomial $P(x,y)$ with rational coefficients: Q1. How compute such (invertible) substitutions of its variables that would transform the polynomial into a …

We know that if $q=4k+3$ ($q$ is a prime), then $(a+bi)^q=a-bi \pmod q$ for every Gaussian integer $a+bi$. Now consider a composite number $N=4k+3$ that satisfies this condition fo …

Hi, I want to ask a simple question in diophantine systems. I have tried to search under different headings, but was unable to find a suitable answer to my question. I have a set …

It is known how to compute the Fourier expansion of elliptic modular forms using modular symbols, and it is known how to get numerical evaluations of $L$-functions of various type …

Let $N$ be a positive integer. Thanks to the Miller-Rabin test and the work of Agrawal, Kayal and Saxena, these days people have much much faster algorithms for testing whether $N$ …

Suppose that I find a small prime factor $p$ dividing a large number $n$ and I wish to prove that it is the least prime dividing $n$. There are two obvious approaches: either facto …

Let $S$ be a set of all positive rational numbers $x$ such that $2x^2 - 1$ is a square, excluding $x=1$. I am interested in computing as many as possible solutions in $S$ to eithe …

Hello, The question below might be well known, and using different words (I made these up, I'm not a number theorist or specialist in combinatorics) For all integers $n\geq 2$ d …