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The generating function is $$ \sum_{n\geq 0} \text{sq}(n) z^n = \prod_{k\geq 1} (1+z^{k^2}). $$ Using complex integration you can use this to get an asymptotic formula for $\text{sq}(n)$. This involv …

I assume that you choose $(A,\alpha)=(B, \alpha)=1$, for otherwise the inverses don't exist. As $A, B$ run over all integers $\leq\alpha$ which are coprime to $\alpha$, $(AB)^{-1}$ attains each resid …

Your question is equivalent to $$ dm - s(dm) \geq pdn -(p-1)s(dn)+(p-1)j $$ where $s$ is the sum of digits to base $p$. Since $s$ is rather small, for large $n$ we have that $m$ is pretty close to $pn …

Suppose that $G_i=C_2^i$, and let $S_i$ be a minimal generating set of $\widehat{G_i}$. Since all characters are rational, $S_i$ is stable, but we have $|H_{S_i}|=|G_i|=2^i$, $|S_i|=i\asymp\log|G_i|$. …

Let $A$ be the set of integers $n$ which are divisible by a prime number $p>n^\theta$, $\frac{1}{2}<\theta<1$. Then for a prime number $q<x^{1/3}$ we have \begin{eqnarray*} \#\{n\leq x, n\in A, q|n\} …

The maximal cycle length of a permutation follows the same statistic as the maximal prime divisor of an integer (suitable scaled): The probability that the largest cycle of a random permutation in $S_ …

All approaches to Goldbach's problem give not only the existence of solutions, but lower bounds for the number of solutions. Using the circle method one can show that the asymptotic formula $$ \#\{(p …

Suppose there are prime numbers $p, q, r$, such that $2^k p+3^kq=5^k r$. Then the left hand side of the second inequality becomes $k+1$, while the right hand side is 5. Thus, in this case, the second …

In general no non-trivial bound exists. Suppose e.g. that $a=1$. Then for all $k$ in the range of summation we have $\frac{ka}{q}\in[0, q^{-\delta}]$, thus $e(\frac{ka}{q})=1+\mathcal{O}(q^{-\delta})$ …

No. There are various results which give counterexamples. For example, Rankin's construction of large prime gaps boils down to the fact that if $p_1, \ldots, p_k$ denote all prime numbers below $x$, …

Let $\pi_k(x)$ be the number of integers $\leq x$ with exactly $k$ prime factors. In the range $k<e\log\log x$, an asymptotic formula for $\pi_k(x)$ was given by Sathe (J. Indian Math. Soc. (N.S.) 17, …

If $x_0<x$ satisfies that $[x_0, x_0+\log x]$ contains $\log\log x$ primes, then for a parameter $r$ we have that this interval contains $\binom{\log\log x}{r}$ different $r$-tuples $p, p+d_1, p+d_2, …

Unless $3^P$ is very close to $2^{P+Q}$, the right hand side will be smaller than 1. Hence the linear form $(P+Q)\log 2 - P\log 3$ is exceptionally small, and you should be able to obtain effective up …

Assume that the Riemann hypothesis for the non-principal $L$-series $\pmod{3}$ is false, say, this series has a zero $\rho=\sigma+i\gamma$ with $\sigma>1/2$. Then Turan and Knapowski have shown that b …

No, such a result would be a major breakthrough regarding our knowledge on odd perfect numbers. A few years ago there was some confusion, since due to careless reading and citing of the article "Eve …