## Abel's Lemma and Dirichlet's Test Incorrectly Determine that a Trigonometric Version of the Dirichlet Series $\zeta(s)=\sum N^{-S}$ is Convergent Throughout the Critical Strip at $t\ne0$

**Authors:** Ayal Sharon

Euler's formula is used to derive a trigonometric version of the Dirichlet series $\zeta(s)=\sum n^{-s}$, which is divergent in the half-plane $\sigma \le 1$, wherein $s \in \mathbb{C}$ and $s=\sigma +it$. Abel's lemma and Dirichlet's test incorrectly hold that trigonometric $\zeta(s)$ is convergent in the critical strip $0<\sigma \le 1$ at $t\ne0$, because they fail to consider a divergent monotonically decreasing series (e.g. the harmonic series) in combination with a bounded oscillating function having an increasing period duration (e.g. $f(t, n) = \sin(t \cdot \ln(n))$).

**Comments:** 18 Pages.

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### Submission history

[v1] 2018-02-19 17:56:17

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