Remainder in Modified Mertens Formula and Ramanujan Inequality

RH-Dependent Estimates of Remainder in Modified Mertens Formula

Assuming the validity of Riemann Hypothesis (RH), we derive the explicit bilateral estimates ("narrow passage") of the remainder in the modified Mertens asymptotic formula for the sums of primes' reciprocals. These results are reversable, thus yielding some new criteria for RH.

关于整数最大格罗旺数在给定最大素因子和余数下的格朗沃尔数最大值

On Maximal Values of Gronwall Numbers for Integers with Given Greatest Prime Factor and Remainder in Modified Mertens Formula

研究整数最大格罗旺数与修改后的梅特恩公式中素数倒数和余数之间的关系，不依赖黎曼假设，提出新的极限关系。

Generalized Mertens sums

Extending a classical estimate of Mertens for the sum of the reciprocals of the first primes, we provide an explicit remainder formula for products of an arbitrary, but fixed, number of primes.

关于拉马努金一个公式的注记

Remarks on a Formula of Ramanujan

在Mertens猜想平均形式和黎曼zeta函数零点纵坐标有理线性无关的假设下，分析了拉马努金著名公式各项的精细结构。

Factorization Theorems for Relatively Prime Divisor Sums, GCD Sums and Generalized Ramanujan Sums

We generalize recent matrix-based factorization theorems for Lambert series generating functions generating the coefficients $(f \ast 1)(n)$ for some arithmetic function $f$. Our new factorization theorems provide analogs to these established expansions generating sums of the form $\sum_{d: (d,n)=1} f(d)$ (type I) and the Anderson-Apostol sums $\sum_{d|(m,n)} f(d) g(n/d)$ (type II) for any arithmetic functions $f$ and $g$. Our treatment of the type II sums includes a matrix-based factorization method relating the partition function $p(n)$ to arbitrary arithmetic functions $f$. We also conclude…

椭圆点附近 j-函数的反演公式

Inversion Formulas for the $j$-function Around Elliptic Points

本文在已有关于模 j-函数在椭圆点 i 和 ρ 附近泰勒级数猜想证明的基础上，利用 Gauss 超几何函数与 Ramanujan 椭圆函数理论中签名 4 和 6 的情形，进一步给出了 j-函数在 i 和 ρ 附近的反演公式。