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For an application in statistical group theory, we need explicit upper and lower bounds that an expert in number theory (I am not one) may know how to prove. Question 1: What are "good" bounds $f_1(x …

An answer to Question 2 follows from the lemma below by letting $y\to\infty$. Lemma. Suppose $x,y$ are real numbers with $12\leqslant x\leqslant y$ and $p$ denotes a prime. Then $$\sum_{x<p\leqslant y …

Fix integers $n\geqslant1$ and $k\geqslant 0$. For an integer $i$, the $k$-fold derivative of $x^i$ can be denoted by $i^{\underline{k}}x^{i-k}$ where $i^{\underline{k}}$ means $i(i-1)\cdots(i-k+1)$ i …

Question: Does anyone know a reference to the following lemmas involving two partitions? (The proofs are not hard, and may well be previously recorded, but where?) First some notation. Let $r$ be a po …

Since $G$ is a finite abelian group we have $\widehat{G}\cong G$. For $\chi\in\widehat{G}$ to be Galois-stable, or just stable under complex conjugation, its order must be at most 2. However $\widehat …

Your first problem has a simple solution. Suppose $p$ is a prime and $(n!)_p$ is the $p$-part of $n!$. Dirichlet proved $(n!)_p=p^k$ where $k = (n-s_p(n))/(p-1)$ and $s_p(n)$ is the sum of the base-$p …

A problem in group theory (indices of imprimitive groups) gives rise to the following conjectures in number theory. Suppose a positive integer $n$ has binary and ternary expansions $n=\sum_{k\geqslant …

Let $\sigma_p(m,n)$ (resp. $\pi_p(m,n)$) denote the sum of the carries when adding (resp. multiplying) the numbers $m=\sum_{k\ge0}m_kp^k$ and $n=\sum_{k\ge0}n_kp^k$ using base-$p$ arithmetic where $m_ …

I was asked (offline) for a proof of the formula for the sum of the carries $\pi_p(a,b)$ when multiplying $a$ and $b$ in base $p$. Proof. Multiplying the base-$p$ expansions $a=\sum_{k\ge0}a_kp^k$ an …

There are (at least) two explicit formulas. First, $D(n)$ can be shown to be a multiplicative function (this means $\gcd(m,n)=1$ implies $D(mn)=D(m)D(n)$), and the values of $D(p^k)$ for all primes $p …

Does anyone know a reference to the following results, which I can prove, but I suspect may be known. Let $R(n)$ denote the number of primitive $n$th roots of unity with positive real part, and $L(n)$ …

This question is not posed well. Let $F={\mathbb F}_q$ where $q$ is a prime-power. A polynomial $f\in F[x]$ gives rise to a function $\phi(f)\colon F\to F$ via evaluation. Moreover, $\phi\colon F[x]\ …