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2 fixed the error in the first equation

The sigma function

$$\sigma_{1}({p_i}^{\alpha_i}) = \Sigma_{i=0}^{\alpha_i}{{p_i}^{\alpha_i}}$$displaystyle\sum_{j=0}^{\alpha_i}{{p_i}^j}$$satisfies the inequalities$$\sigma_{1}({p_i}^{\alpha_i}) \gt (\alpha_i + 1)(\sqrt{p_i})^{\alpha_i}\sigma_{1}({p_i}^{\alpha_i}) \gt 1 + \alpha_i(\sqrt{p_i})^{1 + \alpha_i}$$for prime p_i and \alpha_i \ge 1. The "proof" uses the Arithmetic Mean-Geometric Mean Inequality. As a particular application of this result, Sorli's Conjecture implies the OPN Conjecture. 1 [made Community Wiki] The sigma function$$\sigma_{1}({p_i}^{\alpha_i}) = \Sigma_{i=0}^{\alpha_i}{{p_i}^{\alpha_i}}$$satisfies the inequalities$$\sigma_{1}({p_i}^{\alpha_i}) \gt (\alpha_i + 1)(\sqrt{p_i})^{\alpha_i}\sigma_{1}({p_i}^{\alpha_i}) \gt 1 + \alpha_i(\sqrt{p_i})^{1 + \alpha_i}

for prime $p_i$ and $\alpha_i \ge 1$.

The "proof" uses the Arithmetic Mean-Geometric Mean Inequality.

As a particular application of this result, Sorli's Conjecture implies the OPN Conjecture.