Investigation about a statement equivalent to Riemann Hypothesis (RH)
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910058365845504 |
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| author | Lodone, Giovanni |
| author_facet | Lodone, Giovanni |
| contents | First idea is to compute a quantity like the angular momentum with respect to (0, 0), of an unitary mass of coordinates (<[Xi(s)], =[Xi(s)]) while =[s] is the time, and, <[s] = constant. If we impose that the derivative along <[s], at points <[s] = 1/2 is grater than zero, then, we find exactly a known RH equivalence statement about relative maxima and minima of Xi(1/2 + i=[s]) along critical line. After representing this fictitious angular momentum by Euler Product, and, using PNT as a tool, it can be proved that this positivity condition is granted everywhere at least for Xi(1/2 + i=[s]) 6 = 0. So, if the above equivalence is true, it is found that off-critical line zeros must be excluded for Z(s) function along all critical strip . Further analysis on Euler Product(Lemma 2) has evidenced others shorter ways to same objective. Besides the converging spectrum of prime numbers is highlighted as a by-product. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_11130 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Investigation about a statement equivalent to Riemann Hypothesis (RH) Lodone, Giovanni General Mathematics 11M06, 11M99 First idea is to compute a quantity like the angular momentum with respect to (0, 0), of an unitary mass of coordinates (<[Xi(s)], =[Xi(s)]) while =[s] is the time, and, <[s] = constant. If we impose that the derivative along <[s], at points <[s] = 1/2 is grater than zero, then, we find exactly a known RH equivalence statement about relative maxima and minima of Xi(1/2 + i=[s]) along critical line. After representing this fictitious angular momentum by Euler Product, and, using PNT as a tool, it can be proved that this positivity condition is granted everywhere at least for Xi(1/2 + i=[s]) 6 = 0. So, if the above equivalence is true, it is found that off-critical line zeros must be excluded for Z(s) function along all critical strip . Further analysis on Euler Product(Lemma 2) has evidenced others shorter ways to same objective. Besides the converging spectrum of prime numbers is highlighted as a by-product. |
| title | Investigation about a statement equivalent to Riemann Hypothesis (RH) |
| topic | General Mathematics 11M06, 11M99 |
| url | https://arxiv.org/abs/2412.11130 |