Let $P^+(n)$ denote the largest prime factor of $n$. One of Erd\H{o}s and Tur\'an's conjectures asserts that the asymptotic density of integers $n$ satisfying $P^+(n)<P^+(n+1)$ is 1/2. In this paper, we prove that this density is larger than 0.280, which improves the previous result 0.2017 by L\"u and Wang (2025). We also prove that there exists a positive density of $n$ such that $P^+(n)<P^+(n+1)<x^{41/107+\varepsilon}$. Define $T_c(x):=\#\{p\leq x:P^+(p-1)\geq p^c\}$. For $1/2<c<1$, we also show that \begin{align*} \mathop{\lim \sup}_{x\rightarrow\infty}\frac{T_c(x)}{\pi(x)}\leq \min\left(-\frac{7}{2}\log c,\frac{1-\delta}{2c}\right), \end{align*} where $\delta=\delta(c)>0$.
Let $p(n)$ denote the least prime factor of $n$ and $L_C(x)=Cx^{1/2}(\log x)^2$ with $C>0$. The sum of $p(n)/n$ over composite $n$ lying in the short interval $[x,\,x+L_C(x)]$, a question raised by Erd\H{o}s and Graham, is studied. (i) The constant $c=8$ in the mean asymptotic is estimated \[ S(x)=\sum_{{n<x,\ n\ \text{composite}}}\frac{p(n)}{n}=\frac{c\,x^{1/2}}{(\log x)^2}\Bigl(1+O\Bigl(\frac1{\log x}\Bigr)\Bigr) +O\Bigl(\frac{x^{1/3}}{\log x}\Bigr). \] (ii)For every fixed $C>0$, the window sums $\mu_C(x):=\sum_{x\le n\le x+L_C(x)}p(n)/n$ over composites have mean $4C$: $\frac{1}{X}\sum_{x\le X}\mu_C(x)=4C+O_C(1/\log X)$, and second moment $\frac{1}{X}\sum_{x\le X}(\mu_C(x)-4C)^2=O_C((\log X)^{-2})$. In particular $\mu_C(x)=4C+o(1)$ for almost all $x$. (iii)Under a weak Cram\'{e}r-type hypothesis on primes in intervals of length $(\log y)^{2+o(1)}$, the estimate $\mu_C(x)=4C+O_C(1/\log x)$ holds uniformly in $x$, giving an affirmative answer to the Erd\H{o}s--Graham question. Unconditionally, the uniform statement remains open; proving the uniform statement unconditionally would require resolving short-interval prime estimates at scale $(\log y)^2$.
Let $P^+(n)$ be the largest prime factor of $n$, and let $\chi=\chi_{-4}$ be $1$ on primes $1\pmod4$ and $-1$ on primes $3\pmod4$. For fixed $k\ge2$ we study \[ D_k(x)=\sum_{\substack{n\le x\\ \Omega(n)=k}}\chi(P^+(n)). \] Thus $D_k(x)$ compares the two residue classes according to the largest prime factor of integers having exactly $k$ prime factors, counted with multiplicity. Assuming RH for $\zeta(s)$ and $L(s,\chi)=\beta(s)$, we prove a pointwise explicit formula. The main term is a fixed negative contribution plus an absolutely convergent oscillating sum over the zeros $\rho=\frac12+i\gamma$ of $L(s,\chi)$. The ordering of the prime factors produces $k$ Perron denominators, and the principal coefficient of a zero is $O_k((1+|\gamma|)^{-k})$. We then show that the total size of all zero terms is strictly smaller than the fixed contribution. Hence $D_k(x)<0$ for all sufficiently large $x$. The analogous problem without fixing $k$ is still open.
We prove the $p$-Frobenius problems proposed as Conjectures 7.1 and 7.5 developed by T. Komatsu and R. Pandey (Bull. Korean Math. Soc. 2025;62:1397--1409.) for two families of consecutive odd integers. For integers $r,L,n\ge0$, the bounded restricted partition function $p_{\le r}^{(\le L)}(\le n)$ counts partitions of $n$ into at most $r$ parts, each at most $L$. Thus the bounded restricted partition functions $p_{\le 3}^{(\le a)}(\le s)$and $p_{\le 3}^{(\le a+1)}(\le s)$ play central roles in the proofs. Their generating functions are Gaussian polynomials, whose symmetry and unimodality provide a common tool for treating both families.
Let $E_n$ denote the number of alternating permutations of $\{1,\dots,n\}$, equivalently characterized by $\sum_{n\ge0}E_nz^n/n!=\sec z+\tan z$. For every $q\ge1$, the sequence $(E_n\bmod q)_{n\ge0}$ is eventually periodic; let $d(q)$ and $s(q)$ denote its minimal eventual period and preperiod. For every odd prime $p$, Knuth and Buckholtz proved $d(p)=\operatorname{lcm}(p-1,4)$ together with \[ d(p^r)\mid p^{r-1}d(p), \qquad s(p^r)\le r, \] and Ramassamy conjectured that both bounds are attained for every $r\ge1$. In this paper, we introduce an algebraic frequency expansion for the Euler numbers over $S_r=(\mathbb Z/p^r\mathbb Z)[x]/(x^2+1)$. Using Hurwitz series, the Euler sequence is represented algebraically as a finite combination of formal exponential modes, in a manner reminiscent of Fourier analysis. Using this expansion, we prove \[ d(p^r)=p^{r-1}d(p) \qquad \text{for every odd prime $p$ and every $r\ge1$}, \] thereby establishing Ramassamy's period conjecture. We also disprove the preperiod conjecture by proving \[ s(5^5)\le4<5. \] Finally, we prove that $5^5$ is the smallest odd prime power for which $s(p^r)\ne r$, and based on our findings we conjecture \[ s(p^r)\ge r-2 \] for every odd prime $p$ and every $r\ge2$.
Cameron and Erd\H{o}s asked if the number of sets free of arithmetic progressions of length $k$ is $2^{r_k(n)(1+o(1))}$, where $r_k(n)$ is the maximum cardinality of a $k$-AP-free subset of $\{1, \dots, n\}$. Balogh, Liu and Sharifzadeh made significant progress on this question showing that it is $2^{O(r_k(n))}$ for an infinite sequence of $n$. We improve their result in two ways. On the one hand, we prove that, for $k\geq 5$, the number of $k$-AP-free sets in $[n]$ is $2^{r_k(n)(1+o(1))}$ for an infinite sequence of $n$, solving the question of Cameron and Erd\H{o}s for infinitely many values. On the other hand, we also prove that for $k \geq 3$ and all $n$ the number of $k$-AP-free sets in $[n]$ is $2^{O(r_k(n))}$. These results are in fact special cases of a general framework that we develop to count families of sets excluding certain arithmetic patterns, which applies as long as the corresponding extremal threshold satisfies certain Behrend-type lower bounds. As further examples, we get analogous results for solution sets to almost all systems of linear equations as well as counting versions of the multidimensional Szemer\'edi theorem.
Patrick Morris, Miquel Ortega, Juanjo Ru'e· 0 citations
We provide an explicit algebraic construction showing that, uniformly for integers $3 \leq s \leq t$, as $t \to \infty$, \[ R( s,t ) \geq t^{(1-o(1)) \log s / \log(\log s + 1) }. \] For large fixed $s$, this improves the dependence on $s$ in the general off-diagonal construction of Alon and Pudl\'ak. In particular, $R(33, t) \geq t^{2.1-o(1)}$, to our knowledge, the first explicit construction showing $R(s, t) \geq t^c$ for some fixed $s$ and some $c>2$. In the diagonal case, it improves the leading constant in the exponent of the classical Frankl--Wilson bound from $1/4$ to $1$, while being almost as simple to describe.