2022/11/20 by Aditya Ghosh, Ghosh, Aditya
Mathematics · #11Mxx #11Yxx #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #History and Theory of Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2211.11042
openalex publication_date 2022/11/20 · openalex created_date 2022/11/29 · openalex updated_date 2026/07/28
We look at approximations ζX of the ζ-function introduced in Gonek's paper (arXiv:0704.3448). We look at how close the approximate zeroes are to the actual zeroes when (i) X is fixed (Section 1)(ii) X varies like t/2π (Section 3.1). We establish a heuristic for estimating these differences, involving values of FX^⋆(t) and its near-constant slopes near zeta-zero ordinates γ. In Section 3.2 we see the slope around the zeroes behaves logarithmically and we calculate a numerical formula for it. In Section 3.3 and 3.4 we scale the differences with the slopes and compare them with models involving 1 or 2 pairs of neighbouring zeta-zeroes. In Section 3.5, we also look at how often these models capture these scaled differences accurately. In Section 4, we look at our methods from a theoretical standpoint. In Section 5, we look at how close the approximate zeroes are to the actual zeroes when (i) X is fixed (ii) X varies like t/2π. The errors seem to behave like powers of log which should be investigated further from a theoretical standpoint.