论文标题
罗宾 - 拉加里亚斯标准的类似假设标准
Analogues of the Robin-Lagarias Criteria for the Riemann Hypothesis
论文作者
论文摘要
Robin的标准指出，RIEMANN假设等于所有整数的$σ（n）<e^γn\ log \ log n $ $ n \ geq 5041 $，其中$σ（n）$是$ n $ and $ n $ and $γ$的划分总和。我们证明，RIEMANN假设等同于$σ（n）<\ frac {e^γ} {2} {2} n \ log \ log \ log \ log \ log \ log n $对于所有奇数数字$ n \ geq 3^4 \ geq 3^4 \ cdot 5^3 \ cdot 5^3 \ cdot 7^2 \ cdot 11 \ cdot 11 \ cdot 11 \ cdots 67 $。 Lagarias对Riemann假设的标准指出，Riemann假设等于$σ（n）<H_n + \ exp {h_n} \ log log {h_n} $所有整数的$ n \ geq 1 $，其中$ n $ h_n $是$ n $ n $ n $ th $ th $ th harmonic harmonic harmonic harmonic harmonic harmonic harmonic harmonic harmonic harmonic harmonic number。我们通过创建一个新的谐波系列$ h^\ prime_n = 2H_n -h__ {2n} $来建立对拉加里亚斯对Riemann假设的标准的类似物，并证明Riemann假设等于$σ（N） \ exp {h^\ prime_n} \ log {h^\ prime_n} $ for hast $ n \ geq 3 $。我们证明了对Robin对奇数无方数字的不平等的类似物。此外，我们发现了一个一般公式，该公式研究了$ n $的主要分解及其在罗宾不平等中的行为的影响。
Robin's criterion states that the Riemann hypothesis is equivalent to $σ(n) < e^γn \log\log n$ for all integers $n \geq 5041$, where $σ(n)$ is the sum of divisors of $n$ and $γ$ is the Euler-Mascheroni constant. We prove that the Riemann hypothesis is equivalent to the statement that $σ(n) < \frac{e^γ}{2} n \log\log n$ for all odd numbers $n \geq 3^4 \cdot 5^3 \cdot 7^2 \cdot 11 \cdots 67$. Lagarias's criterion for the Riemann hypothesis states that the Riemann hypothesis is equivalent to $σ(n) < H_n + \exp{H_n}\log{H_n}$ for all integers $n \geq 1$, where $H_n$ is the $n$th harmonic number. We establish an analogue to Lagarias's criterion for the Riemann hypothesis by creating a new harmonic series $H^\prime_n = 2H_n - H_{2n}$ and demonstrating that the Riemann hypothesis is equivalent to $σ(n) \leq \frac{3n}{\log{n}} + \exp{H^\prime_n}\log{H^\prime_n}$ for all odd $n \geq 3$. We prove stronger analogues to Robin's inequality for odd squarefree numbers. Furthermore, we find a general formula that studies the effect of the prime factorization of $n$ and its behavior in Robin's inequality.
