A Number-Theoretic Window on the Prime/Composite Landscape
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| Natura: | Recurso digital |
| Lingua: | inglese |
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Zenodo
2026
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| _version_ | 1866901792913096704 |
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| author | Wilhelmsen, Arno |
| author_facet | Wilhelmsen, Arno |
| contents | <p class="p1">We introduce a coordinate decomposition of primitive Gaussian integers z= m+ni∈Z[i] using s= Re(z) + Im(z) and t= (Re(z)−Im(z))/s, which opens a window on the segment √x<s≤√2x. Within this window, every odd scarries a signed quantity e(s,x) measuring the discrepancy between a discrete coprime count and its continuous expectation. Prime s contribute systematically positive values, composite s systematically negative ones, and their totals match to high precision: at x = 10^<span class="s1">17</span>, the prime sum +1,975,343 and the composite sum −1,975,408 yield a residual of −65. We prove an exact formula and equidistribution for prime s (σ<span class="s1">p </span>= 0.31366..., veried to x = 10^<span class="s1">27</span>), derive a Möbius variance formula for composite s (σ<span class="s1">c </span>∼(ln s)<span class="s1">(ln 2)/2</span>, conrmed to x= 10^<span class="s1">17</span>), and identify a SternBrocot critical line that separates coprime pairs from non-coprime gaps.</p> <p class="p2"> </p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19972792 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | A Number-Theoretic Window on the Prime/Composite Landscape Wilhelmsen, Arno Prime numbers composite numbers divisor distribution numbertheory Mathematical analysis <p class="p1">We introduce a coordinate decomposition of primitive Gaussian integers z= m+ni∈Z[i] using s= Re(z) + Im(z) and t= (Re(z)−Im(z))/s, which opens a window on the segment √x<s≤√2x. Within this window, every odd scarries a signed quantity e(s,x) measuring the discrepancy between a discrete coprime count and its continuous expectation. Prime s contribute systematically positive values, composite s systematically negative ones, and their totals match to high precision: at x = 10^<span class="s1">17</span>, the prime sum +1,975,343 and the composite sum −1,975,408 yield a residual of −65. We prove an exact formula and equidistribution for prime s (σ<span class="s1">p </span>= 0.31366..., veried to x = 10^<span class="s1">27</span>), derive a Möbius variance formula for composite s (σ<span class="s1">c </span>∼(ln s)<span class="s1">(ln 2)/2</span>, conrmed to x= 10^<span class="s1">17</span>), and identify a SternBrocot critical line that separates coprime pairs from non-coprime gaps.</p> <p class="p2"> </p> |
| title | A Number-Theoretic Window on the Prime/Composite Landscape |
| topic | Prime numbers composite numbers divisor distribution numbertheory Mathematical analysis |
| url | https://doi.org/10.5281/zenodo.19972792 |