Away from the real axis, the second moment partial indefinite R integral ζ(σ + I ∗ t)ζ(σ − I ∗ t)dt of Im(s) behaviour using the end tapered finite Riemann Zeta Dirichlet Series approximation. release_fvuy6amuojccrllepknj7ogl5m

Abstract

The square mean value indefinite integral of the Riemann Zeta function, ζ(σ + I ∗ t)ζ(σ − I ∗ t)dt = |ζ(σ+I∗t)|^2 dt is a partial indefinite integral with respect to t. In this paper the partial indefinite integral is usefully approximated by end tapered Riemann Zeta Dirichlet Series sums, away from the real axis, using the second quiescent region of the Series sum. Below the critical line, the low t trend growth of |ζ(σ+I ∗t)| 2 dt is usefully approximated by co-opting the known leading asymptotic growth term of the definite integral ζ(2 ∗ σ) ∗ t + ζ(2∗σ−1)∗Γ(2∗σ−1)∗sin(π∗σ)∗ t^(2−2∗σ) . For the upper portion of the critical strip 1/2 < σ < 1, the trend growth of |ζ(σ + I ∗ t)| dt is reasonably approximated by adding an offset term ζ(2 ∗ σ) ∗ t +ζ(2∗σ−1)∗Γ(2∗σ−1)∗sin(π∗σ)∗ t^(2−2∗σ) − (ζ(2 ∗ σ) + ζ(2∗σ−1)∗Γ(2∗σ−1)∗sin(π∗σ)+ π(1 − σ)) .
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