2024/07/08 by de Reyna, Juan Arias
#65D30 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Number Theory (math.NT) #Numerical Analysis (math.NA) #Primary 11M06 #Secondary 41A60
paper · doi:10.48550/arxiv.2407.05719
In arXiv:2406.0243 two real functions g(x,t) and f(x,t) are defined, so that the Riemann-Siegel Z function is given as Z(t)=\mathopRe\\fracu(t)e(πi)/(8)\frac12+it∫0^∞ g(x,t)ei f(x,t) dt\, where u(t) is a real function of order t-1/4 when t→+∞. The function g(x,t) is indefinitely differentiable and tends to 0 as well as all its derivatives when x→0+ or x→+∞. Since, furthermore, for t→+∞ the function f(x,t) tends to +∞ we may expect that the integral depends essentially on the behavior of g(x,t) at the extremes. As Polya in an analogous situation we consider the substitution of ψ(x) by a simpler similar function. A simple function with this behavior is ψ0(x):=2π(1+\tfrac14x-5/2)e^-πx-\fracπ4x. Therefore, we define J0(t) replacing in the definition of J(t) the function ψ(x) by the simpler ψ0(x). J0(t)=2π∫0^∞ (1+\tfrac14x-\frac52)e^-πx-\fracπ4x(1-ix)\frac12(\frac12+it) dx. The resulting Z0(t) disappoints us Z0(t)\asymp \mathopRe\(2)/(√π)exp\i((t)/(2)log(t)/(2π)-(t)/(2)-\fracπ8)\+\frac2(2πt)1/4exp(πi√((t)/(2π)) )\, t→+∞. However, the integral J0(t) is interesting as a technical challenge. And still we have the possibility to get a better result improving ψ0(x). This is a preliminary version, and we set it as a challenge: to compute and study this integral.