A Reformulation of the Riemann Hypothesis in Terms of Continuity of the Limit Function of a Certain Ratio of Partial Sums of a Series for the Dirichlet Eta Function

Details A Reformulation of the Riemann Hypothesis in Terms of Continuity of the Limit Function of a Certain Ratio of Partial Sums of a Series for the Dirichlet Eta Function A Reformulation of the Riemann Hypothesis in Terms of Continuity of the Limit Function of a Certain Ratio of Partial Sums of a Series for the Dirichlet Eta Function

2009-07-14

arxiv.org/abs/0907.2426v5

Mathematics

General Mathematics

22 pages, 9 figures. Major restructuring. Emphasis on continuity of L(s). New Corollaries. Added Fig. 7 to 9 to aid understand

Scientific paper

For any $s \in \mathbb{C}$ with $\Re(s)>0$, denote by $S_n(s)$ the $n^{th}$ partial sum of the alternating Dirichlet series $1-2^{-s}+3^{-s}-... $ . We first show that $S_n(s)\neq 0$ for all $n$ greater than some index $N(s)$ . Denoting by $D={s \in \mathbb{C}: 0< \Re(s) < {1/2}}$ the open left half of the critical strip, define for all $s\in D$ and $n>N(s)$ the ratio $P_n(s) = S_n(1-s) / S_n(s)$ . We then prove that the limit $L(s)=\lim_{N(s)

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