Patrick Brown
Euler showed that if an odd perfect number $N$ exists, it must consist of two parts $N=q^k n^2$, with $q$ prime, $q \equiv k \equiv 1 \pmod{4}$, and gcd$(q,n)=1$. Dris conjectured that $q^k < n$. We first show that $q<n$ for all odd perfect numbers. Afterwards, we show $q^k < n$ holds in many cases.
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The Euler's form of odd perfect numbers, if any, is $n=π^αN^2$, where $π$ is prime, $(π,N)=1$ and $π\equiv α\equiv 1 \pmod{4}$. Dris conjecture states that $N>π^α$. We find that $N^2>\frac{1}{2}π^γ$, with $γ=max\{ω(n)-1,α\}$; $ω(n)\geq 9$ is the number of distinct prime factors of $n$.
Dris conjectured in his masters thesis that the inequality $q^k < n$ always holds, if $N = {q^k}{n^2}$ is an odd perfect number with special prime $q$. In this note, we initially show that either of the two conditions $n < q^k$ or $σ(q)/n < σ(n)/q$ holds. This is achieved by first proving that $σ(q)/n \neq σ(n)/q^k$, where $σ(x)$ is the sum of the divisors of $x$. Using this analysis, we further show that the condition $q < n < q^k$ holds in four out of a total of six cases. Finally, we prove that $n < q^k$, and that this holds unconditionally. This finding disproves both the Dris Conjecture a…
If $N={q^k}{n^2}$ is an odd perfect number given in Eulerian form, then Sorli's conjecture predicts that $k=ν_{q}(N)=1$. In this article, we give some further results related to this conjecture and those contained in the papers \cite{Dris} and \cite{Dris2}. (withdrawn because of a crucial gap in Theorem 1 [see https://arxiv.org/pdf/1309.0906.pdf for what is currently provable in this regard], as well as elementary mistakes in the numerical bounds from pages 2 to 4)
We investigate the implications of a curious biconditional involving divisors of odd perfect numbers, if Dris conjecture that $q^k < n$ holds, where $q^k n^2$ is an odd perfect number with Euler prime $q$. We then show that this biconditional holds unconditionally. Lastly, we prove that the inequality $q<n$ holds unconditionally.
Disproof of a conjecture by Rademacher on partial fractions
在他的著作《解析数论专题》中，Rademacher认为生成函数的分区至多可以分成$N$部分，并对它的部分分式分解的系数给出了一定的限制。我们进行了一个渐近分析，反驳了这个猜想，从而证实了最近的观察Sills和Zeilberger（差分方程和应用杂志19，2013），谁给了有力的数值证据反对这个猜想。
Partial Resolution of the Erdös-Straus, Sierpinski, and Generalized Erdös-Straus Conjectures Using New Analytical Formulas
本文通过新的分析公式部分解决了埃德尔斯-斯特劳斯、西尔伯尼克及广义猜想，并探讨了其进一步证明的可能性。
On the partial fraction decomposition of the restricted partition generating function
我们提供了新的公式的系数在部分分式分解的限制分区生成函数。这些技术使我们能够部分解决最近的猜想Sills和Zeilberger。我们还介绍了即将开展的工作，给出了决议Rademacher的猜想，这些系数的渐近。
On the Littlewood conjecture in simultaneous Diophantine approximation
对于任意给定的具有有界部分幂元的实数$α$，我们构造了许多具有有界部分幂元的实数$β$，其中对$(α, β)$满足Littlewood猜想的强形式.我们的证明是初等的，并依赖于连分数的基本理论。
We introduce a new conjecture on products of two distinct primes that would provide a partial answer to a conjecture of McIntosh. Also, $\binom{2p-1}{p-1}-1$ is written in terms of a polynomial in prime $p$ over the integers and we discuss one way this form may be useful.
An Improvement To Zaremba's Conjecture
证明了存在一个密度为1的子集$\dd \subset \N$，使得每个$n \in \dd$是一个部分分母以5为界的有限连分数的分母.
We present an infinite family of Borwein type $+ - - $ conjectures. The expressions in the conjecture are related to multiple basic hypergeometric series with Macdonald polynomial argument.
A partial result towards the Chowla--Milnor conjecture
本文研究Hurwitzζ函数线性无关性，证明对于固定整数k≥2，特定ζ值的有理数线性 span 维度下界为logq的正比例，扩展了已知结果。
Partial Heights, Entire Curves, and the Geometric Bombieri-Lang Conjecture
本文提出利用部分高度与 Brody 引理从无穷有理点序列构造全纯曲线的新方法，并在部分高度非退化猜想下证明了特征 0 函数域上具有到阿贝尔簇有限态射的双曲射影簇的几何 Bombieri-Lang 猜想。
Rademacher's infinite partial fraction conjecture is (almost certainly) false
在他的书\emph{Topics in Analytic Number Theory}，汉斯拉德马赫阐明，限制某些序列的系数出现在普通的部分分式分解的生成函数分区的整数到最多$N$部分存在和平等的特定价值观，他指定。尽管开放了近40年，但在证明或反驳该猜想方面几乎没有取得进展，部分原因可能是实际计算所讨论的系数的困难。在本文中，我们提供了一个快速的算法来计算Rademacher系数，大量的数据，直接公式的某些集合的Rademacher系数，和压倒性的证据对真理的猜想。虽然Rademacher系数序列的极限不存在（序列振荡并达到任意大的正值和负值），但对于序列中的某些（可预测的）指数，序列确实非常接近Rademacher的极限。
A few conjectures about the multiple zeta values
多重zeta值（MZV）是一组实数，具有有理数上的代数的美丽结构。它们可能与当今数学中最重要的猜想有关，即黎曼假设。在本文中，我们将显示部分的解决方案，五个著名的MZV，我们部分证明使用线性系和shuffles产品。我们看到，要部分解决其中四个问题，我们只需要线性代数。
The spectral set conjecture and multiplicative properties of roots of polynomials
我们得到了新的部分结果支持谱集猜想在1维。
We prove that for any prime $p$ there is a divisible by $p$ number $q = O(p^{30})$ such that for a certain positive integer $a$ coprime with $q$ the ratio $a/q$ has bounded partial quotients. In the other direction we show that there is an absolute constant $C>0$ such that for any prime $p$ exist divisible by $p$ number $q = O(p^{C})$ and a number $a$, $a$ coprime with $q$ such that all partial quotients of the ratio $a/q$ are bounded by two.
Partial-Fraction Decompositions and Harmonic Number Identities
利用部分分式方法研究了有理函数的分解问题。我们将建立几个关于调和数和广义Aesthetic数的显著恒等式，包括与关于Aesthetic数的Beukers猜想相关的二项式调和数恒等式。
Which Partial Sums of the Taylor Series for $e$ are Convergents to $e$? (and a Link to the Primes 2, 5, 13, 37, 463), II
这是我们早期论文的扩展版本。设泰勒级数$e = \sum_{r=0}^{\infty} 1/r!$的第$n$次部分和为$A_n/n!$，$p_k/q_k$为$e$的简单连分式的第$k$次收敛.使用最近的措施，$e$的非理性，我们证明了我们的猜想，只有两个部分和的弱版本是converterable到$e$。证明了关于因子$q_k$和因子幂的一个相关结果。我们还发现了$A_n$和素数2、5、13、37、463之间令人惊讶的联系。在附录中，我们给出了一个条件证明的猜想，假设第二个猜想，我们对零的$A_n$和$q_k$模幂2。表支持这零猜想，我们讨论了2进重新制定它。
We prove in particular that for any sufficiently large prime $p$ there is $1\le a<p$ such that all partial quotients of $a/p$ are bounded by $O(\log p/\log \log p)$. For composite denominators a similar result is obtained. This improves the well--known Korobov bound concerning Zaremba's conjecture from the theory of continued fractions.
Recently, Li obtained an asymptotic formula for a certain partial sum involving coefficients for the polynomial in the First Borwein conjecture. As a consequence, he showed the positivity of this sum. His result was based on a sieving principle discovered by himself and Wan. In fact, Li points out in his paper that his method can be generalized to prove an asymptotic formula for a general partial sum involving coefficients for any prime $p>3$. In this work, we extend Li's method to obtain asymptotic formula for several partial sums of coefficients of a very general polynomial. We find that in …
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