论文标题
Newman对分区功能的猜想Modulo Integer至少两个不同的素数除外
Newman's conjecture for the partition function modulo integers with at least two distinct prime divisors
论文作者
论文摘要
令$ m $为正整数，$ p（n）$是正整数$ n $的分区数量。纽曼的猜想断言，对于每个整数$ r $，都有无限的正整数$ n $，因此\ [p（n）\ equiv r \ pmod {m}。 \]对于一个正整数$ d $，让$ b_ {d} $是一组正整数$ m $，使得$ m $的prime除数为$ d $。在本文中，我们证明，对于每个正整数$ d $，纽曼猜想在$ b_ {d} $中持有的正整数$ m $的密度为$ 1 $。此外，我们研究了Newman的猜想的类似物，该猜想对Nebentypus的$γ_0（N）$上的弱塑形模块化形式，这适用于$ t $ core-core-core-core-core分区和具有$ h $ colors的广义Frobenius分区。
Let $M$ be a positive integer and $p(n)$ be the number of partitions of a positive integer $n$. Newman's Conjecture asserts that for each integer $r$, there are infinitely many positive integers $n$ such that \[ p(n)\equiv r \pmod{M}. \] For a positive integer $d$, let $B_{d}$ be the set of positive integers $M$ such that the number of prime divisors of $M$ is $d$. In this paper, we prove that for each positive integer $d$, the density of the set of positive integers $M$ for which Newman's Conjecture holds in $B_{d}$ is $1$. Furthermore, we study an analogue of Newman's Conjecture for weakly holomorphic modular forms on $Γ_0(N)$ with nebentypus, and this applies to $t$-core partitions and generalized Frobenius partitions with $h$-colors.
