A Branch and Bound Solution for the Contrapositive Proof of the Riemann Hypothesis
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Book Details
Author(s)
J. W. Helkenberg
Publisher
J. W. Helkenberg
ISBN / ASIN
B004CRSTDC
ISBN-13
978B004CRSTD8
Marketplace
France 🇫🇷
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Description
Abstract:
IFF the Riemann Zeta function (p) is an algorithm f.(p) whose analytic continuation results with a sequence of prime number occurrences resembling a list in a spreadsheet (q = the enumeration of Sloan’s A002410), WHERE [f.(p) = (q)], WHERE said spreadsheet is closed under some enumerable row limit (x), AND WHERE (x) is the total number of both prime AND composite row locations [(x) – (q) = (s)], AND WHERE (s) is the total number of composite row locations [(s) = (x) - (q)], AND WHERE [(q)/(x) = a/b = prime counting function], AND WHERE (q)0,;
AND WHERE (¬ p) when taken as an algorithm f.(¬ p) produces a sequence, or list in a spreadsheet, of composite number locations (¬ q) arrayed in a column of a spreadsheet, said spreadsheet being closed under the same limit (x), AND WHERE [(x) - (¬ q) = (q)], AND WHERE each row (x, x+1, x+n) must contain a member that is either in the sequence of (q) or (¬ q);
THEN the algorithm f.(¬ p) is contradistinct to the algorithm f.(p).
An algorithm is demonstrated for f.( ¬ p); it is proven that the domain of the positive counting numbers (x, x+1, x + n ) can be modeled as a tree structure using modular arithmetic; the distribution of composite “ states†[(¬ q < (x)] is discovered to be inherently of the Branch and Bound type. The f.( ¬ p) is shown to reduce to 300 Fibonacci-like sequence generator functions [300 power series = f.(¬ p)]; these sequences generate fractals (are Fibonacci-like in the generator function). A Perl Program is provided for running the sequence generator functions. A decision procedure for proving Riemann Zeta is shown: Reducing the continuum of positive counting numbers into 24 tree structures (digital root last digit preserving modular sequences), sub-sequence generators are described that fully enumerate (q) by the relation [(x) – (¬ q) = (q)] (for all values of x > 1). The zeros of the Zeta Function correspond with those leaf nodes (sequence given by [f.(p) = (q)]) that have neither parent nor child branches for the tree under consideration (See Sloan’s A181732 for a partial trace). The Prime Number Theorem is also given an exact bound, as the prime number spectrum decomposes into an arbitrarily scalable (branch and bound) fractal; from symmetries underlying the composite state distribution functions [f.( ¬ p)] the exact prime counting function becomes inherent.
IFF the Riemann Zeta function (p) is an algorithm f.(p) whose analytic continuation results with a sequence of prime number occurrences resembling a list in a spreadsheet (q = the enumeration of Sloan’s A002410), WHERE [f.(p) = (q)], WHERE said spreadsheet is closed under some enumerable row limit (x), AND WHERE (x) is the total number of both prime AND composite row locations [(x) – (q) = (s)], AND WHERE (s) is the total number of composite row locations [(s) = (x) - (q)], AND WHERE [(q)/(x) = a/b = prime counting function], AND WHERE (q)0,;
AND WHERE (¬ p) when taken as an algorithm f.(¬ p) produces a sequence, or list in a spreadsheet, of composite number locations (¬ q) arrayed in a column of a spreadsheet, said spreadsheet being closed under the same limit (x), AND WHERE [(x) - (¬ q) = (q)], AND WHERE each row (x, x+1, x+n) must contain a member that is either in the sequence of (q) or (¬ q);
THEN the algorithm f.(¬ p) is contradistinct to the algorithm f.(p).
An algorithm is demonstrated for f.( ¬ p); it is proven that the domain of the positive counting numbers (x, x+1, x + n ) can be modeled as a tree structure using modular arithmetic; the distribution of composite “ states†[(¬ q < (x)] is discovered to be inherently of the Branch and Bound type. The f.( ¬ p) is shown to reduce to 300 Fibonacci-like sequence generator functions [300 power series = f.(¬ p)]; these sequences generate fractals (are Fibonacci-like in the generator function). A Perl Program is provided for running the sequence generator functions. A decision procedure for proving Riemann Zeta is shown: Reducing the continuum of positive counting numbers into 24 tree structures (digital root last digit preserving modular sequences), sub-sequence generators are described that fully enumerate (q) by the relation [(x) – (¬ q) = (q)] (for all values of x > 1). The zeros of the Zeta Function correspond with those leaf nodes (sequence given by [f.(p) = (q)]) that have neither parent nor child branches for the tree under consideration (See Sloan’s A181732 for a partial trace). The Prime Number Theorem is also given an exact bound, as the prime number spectrum decomposes into an arbitrarily scalable (branch and bound) fractal; from symmetries underlying the composite state distribution functions [f.( ¬ p)] the exact prime counting function becomes inherent.
