Robin’s criterion states that the Riemann hypothesis is equivalent to σ(n) < eγnloglog n for all integers n ≥ 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) < eγ 2 nloglog n for all odd numbers n ≥ 34 ⋅ 53 ⋅ 72 ⋅ 11⋯67. Lagarias’s criterion for the Riemann hypothesis states that the Riemann hypothesis is equivalent to σ(n) < Hn +exp Hnlog Hn for all integers n ≥ 1, where Hn is the nth harmonic number. We establish an analogue to Lagarias’s criterion for the Riemann hypothesis by creating a new harmonic series Hn′ = 2H n − H2n and demonstrating that the Riemann hypothesis is equivalent to σ(n) ≤ 3n log n +exp Hn′log H n′ for all odd n ≥ 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.

中文翻译：

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黎曼假设的 Robin-Lagarias 准则的类似物

罗宾的准则指出，黎曼假设等价于σ(n) < eγn日志日志 n对于所有整数n ≥ 5041， 在哪里σ(n)是除数的总和n和γ是欧拉-马斯切罗尼常数。我们证明了黎曼假设等价于以下陈述σ(n) < eγ 2 n日志日志 n对于所有奇数n ≥ 34 ⋅ 53 ⋅ 72 ⋅ 11⋯67. 拉加里亚斯对黎曼假设的标准指出，黎曼假设等价于σ(n) < Hn +经验 Hn日志 Hn对于所有整数n ≥ 1， 在哪里Hn是个n次谐波数。我们通过创建一个新的调和级数，为黎曼假设建立了一个类似于拉加里亚斯标准的类比Hn' = 2H n - H2n并证明黎曼假设等价于σ(n) ≤ 3n 日志 n +经验 Hn'日志 H n'对于所有奇怪的n ≥ 3. 我们证明了对于奇数平方数的 Robin 不等式更强的类似物。此外，我们找到了一个研究素数分解效果的通用公式n及其在 Robin 不等式中的行为。