Պահպանված է:
| Հիմնական հեղինակ: | |
|---|---|
| Ձևաչափ: | Recurso digital |
| Լեզու: | անգլերեն |
| Հրապարակվել է: |
Zenodo
2026
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| Խորագրեր: | |
| Առցանց հասանելիություն: | https://doi.org/10.5281/zenodo.19972792 |
| Ցուցիչներ: |
Ավելացրեք ցուցիչ
Չկան պիտակներ, Եղեք առաջինը, ով նշում է այս գրառումը!
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Բովանդակություն:
- <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>