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Continuous crop circles drawn by Riemann's zeta function
- Source :
- Journal of Number Theory. 229:199-217
- Publication Year :
- 2021
- Publisher :
- Elsevier BV, 2021.
-
Abstract
- Let η ( s ) = ∑ n = 1 ∞ ( − 1 ) n + 1 n − s be the alternating zeta function. For a real number τ we define certain complex numbers b M , m ( τ ) and consider finite Dirichlet series υ M ( τ , s ) = ∑ m = 1 M b M , m ( τ ) m − s and η N ( τ , s ) = ∑ M = 1 N υ M ( τ , s ) . Computations demonstrate some remarkable properties of these finite Dirichlet series, but nothing was supported by a proof so far. First, numerical data show that η N ( τ , s ) approximates η ( s ) with high accuracy for s in the vicinity of 1 / 2 + i τ ; this allows one to surmise that (*) η ( s ) = ∑ M = 1 ∞ υ M ( τ , s ) . Moreover, it looks that lim M → ∞ m υ M ( τ , 1 − σ + i t ) ‾ m υ M ( τ , σ + i t ) = η ( σ + i t ) m η ( 1 − σ + i t ) ‾ m ; in other words, the individual summands in expected expansion ( ⁎ ) satisfy with an increasing accuracy a counterpart of the classical functional equation. Let ϒ M ( τ , σ + i t ) = υ M ( τ , σ + i t ) / η ( σ + i t ) . When M, τ, and either σ or t are fixed, and the fourth parameter varies, the plot of ϒ M ( τ , σ + i t ) on the complex plane contains numerous almost ideally circular arcs with geometrical parameters closely related to the non-trivial zeros of the zeta function.
- Subjects :
- Algebra and Number Theory
010102 general mathematics
010103 numerical & computational mathematics
01 natural sciences
Riemann zeta function
Riemann hypothesis
symbols.namesake
symbols
Functional equation (L-function)
0101 mathematics
Complex plane
Complex number
Dirichlet series
Mathematics
Real number
Mathematical physics
Subjects
Details
- ISSN :
- 0022314X
- Volume :
- 229
- Database :
- OpenAIRE
- Journal :
- Journal of Number Theory
- Accession number :
- edsair.doi...........b87691fc42e8a395ead859dd63a36c58