A Gamma Distribution Hypothesis for Prime $k$-tuples
Abstract
We conjecture average counting functions for prime $k$-tuples based on a gamma distribution hypothesis for prime powers. The conjecture is closely related to the Hardy-Littlewood conjecture for $k$-tuples but yields better estimates. Possessing average counting functions along with their corresponding exact counting functions allows to implicitly define pertinent $k$-tuple zeta functions. The $k$-tuple zeta functions in turn allow construction of $k$-tuple analogs of explicit formulae. If the zeros of the (implicitly defined) $k$-tuple zeta can be determined, the explicit formulae should yield a (dis)proof of the $k$-tuple analog of the prime number theorem.
- Publication:
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arXiv e-prints
- Pub Date:
- June 2014
- DOI:
- 10.48550/arXiv.1406.6289
- arXiv:
- arXiv:1406.6289
- Bibcode:
- 2014arXiv1406.6289L
- Keywords:
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- Mathematics - Number Theory
- E-Print:
- arXiv admin note: substantial text overlap with arXiv:1307.0754