Title Analysis of the prime zeta function zeroes placement patterns
Translation of Title Pirminių skaičių dzeta funkcijos nulių išsidėstymo tyrimas.
Authors Tuzcu, Yunus Emre
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Pages 33
Keywords [eng] pirminių skaičių dzeta funkcija, nulių pasiskirstymas, analizinis pratęsimas, atsitiktinių matricų teorija, intervalų statistikos, GUA hipotezė prime zeta function, zeros distribution, analytic continuation, random matrix theory, spacing statistics, GUE conjecture
Abstract [eng] The prime zeta function, P (s) = ∑_{p∈P} p^{−s}, encodes fundamental information about the distribution of prime numbers. Although its analytic continuation obtained via the Möbius inversion formula is well understood, the structure and placement patterns of its zeros remain underexplored. This thesis investigates the distribution of zeros of the prime zeta function in the complex plane, combining theoretical analysis with computational methods. The computational framework employs global optimization using the differential evolution algorithm to locate candidate zeros, which are subsequently verified by analyzing the intersections of the real and imaginary surfaces of the prime zeta function. Analysis of 10318 zeros of the prime zeta function in the region (σ,t) ∈ (0.1, 1.65)×(0, 104) reveals several notable patterns. The distribution of real parts exhibits multimodality, with prominent peaks near σ ≈ 0.1 and σ ≈ 0.6. The zero counting function N (T ) exhibits T log T -type growth, reminiscent of the Riemann-von Mangoldt formula for the Riemann zeta function. Furthermore, the spacing-ratio statistics show behavior similar to predictions from the Gaussian Unitary Ensemble (GUE). Together, these findings suggest that the high-lying zeros of the prime zeta function may asymptotically follow GUE-type statistics analogous to those conjectured for the zeros of the Riemann zeta function.
Dissertation Institution Vilniaus universitetas.
Type Master thesis
Language English
Publication date 2026