Integral Representation for Riemann-Siegel Z(t) function

2024/06/27 by Juan Arias de Reyna, de Reyna, Juan Arias
Mathematics · #Advanced Algebra and Geometry #Analytic Number Theory Research #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2406.18968

Abstract

We apply Poisson formula for a strip to give a representation of Z(t) by means of an integral. F(t)=∫_-∞^∞ (h(x)ζ(4+ix))/(7\coshπ(x-t)/(7)) dx, Z(t)=\frac\Re F(t)(\frac14+t²)^\frac12((25)/(4)+t²)^\frac12. After that we get the estimate Z(t)=((t)/(2π))^\frac74\Re\e^iϑ(t)H(t)\+O(t^-3/4), with H(t)=∫_-∞^∞((t)/(2π))^ix/2(ζ(4+it+ix))/(7\cosh(πx/7)) dx=((t)/(2π))^-\frac74∑_n=1^∞ \frac1n^\frac12+it\frac21+((t)/(2πn²))^-7/2. We explain how the study of this function can lead to information about the zeros of the zeta function on the critical line.

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