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Any replacements are listed further down
[342] viXra:1205.0077 [pdf] submitted on 2012-05-19 16:01:41
Authors: Germán Paz
Comments: 44 Pages.
This paper is a LaTeX document which combines previously posted papers 'Infinitely Many Prime Numbers of the Form ap±b' (viXra:1202.0063 submitted on 2012-02-18 19:13:46; url: http://vixra.org/abs/1202.0063) and 'Solution to One of Landau's Problems' (viXra:1202.0061 submitted on 2012-02-18 21:49:14; url: http://vixra.org/abs/1202.0061) into one paper. The information contained in this paper is the same as the information contained in those two original papers. No new information or results are being added.
ABSTRACT. In this paper it is proved that for every positive integer 'k' there are infinitely many prime numbers of the form n^2+k, which means that there are infinitely many prime numbers of the form n^2+1. In addition to this, in this document it is proved that if 'a' and 'b' are two positive integers which are coprime and also have different parity, then there are infinitely many prime numbers of the form ap+b, where 'p' is a prime number. Moreover, it is also proved that there are infinitely many prime numbers of the form ap-b. In other words, it is proved that the progressions ap+b and ap-b generate infinitely many prime numbers. In particular, all this implies that there are infinitely many prime numbers of the form 2p+1 (since the numbers 2 and 1 are coprime and have different parity), which means that there are infinitely many Sophie Germain Prime Numbers. This paper also proposes an important new conjecture about prime numbers called 'Conjecture C'. If this conjecture is true, then Legendre's Conjecture, Brocard's Conjecture and Andrica's Conjecture are all true, and also some other important results will be true.
Category: Number Theory
[341] viXra:1205.0076 [pdf] submitted on 2012-05-18 16:52:12
Authors: Stephen Crowley
Comments: 8 Pages.
The Riemann zeta function can be written as the Mellin transform of the unit interval map w(x)=⌊x−1⌋(x⌊x−1⌋+x−1) multiplied by s((s+1)/(s-1)). A finite-sum approximation to ζ(s) denoted by ζw(N;s) is examined and an associated function χ(N;s) is found which solves the reflection formula ζw(N;1−s)=χ(N;s)ζw(N;s). The function χ(N;s) is singular at s=0 and the residue at this point changes sign from negative to positive between the values of N = 176 and N = 177. Some rather elegant graphs of the reflection functions χ(N;s) are also provided. The values ζw(N;1−s) are found to be related to the Bernoulli numbers.
Category: Number Theory
[340] viXra:1205.0075 [pdf] submitted on 2012-05-19 00:15:09
Authors: Reuven Tint
Comments: 10 Pages. . Original article written in Russian.
We give second proof of Fermat’s Last theorem without reference to the closure of numerical systems with respect to the operation of superposition.
Category: Number Theory
[339] viXra:1205.0042 [pdf] submitted on 2012-05-07 20:38:15
Authors: chun-xuan jiang
Comments: 98 Pages.
Using Jiang function we prove the new prime theorems(1541)-(1590)
Category: Number Theory
[338] viXra:1205.0019 [pdf] submitted on 2012-05-04 05:53:24
Authors: A.S.N. Misra
Comments: 8 Pages. I include an abstract with this upload as requested and also as the first paragraph of the PDF.
Here we argue that entropy is more fundamentally an analytical than an empirical concept, thus explaining its hitherto puzzling manifestation in the prime number distribution.
We suggest a precise formula for quantifying the presence of entropy in the continuum of positive integers and we use this breakthrough equation (connecting the world of pure mathematics with that of physics) to strongly suggest possible lines of proof of the Riemann hypothesis and the P versus NP problem and also the necessary primality of the number one.
Category: Number Theory
[337] viXra:1204.0099 [pdf] submitted on 2012-04-28 22:26:43
Authors: Carlos Giraldo Ospina
Comments: 2 Pages.
In this paper it is proved that if Fermat's Last Theorem (FLT) is true, then Beal's Conjecture is also true. Since FLT was already proved, then Beal's Conjecture is true.
Category: Number Theory
[336] viXra:1204.0063 [pdf] submitted on 2012-04-15 20:09:07
Authors: chun-xuan jiang
Comments: 15 Pages.
We establish Santilli-Jiang isonumber theory based on the generalization of unit and zero.
Category: Number Theory
[335] viXra:1204.0053 [pdf] submitted on 2012-04-14 22:23:19
Authors: Carlos Giraldo Ospina
Comments: 6 Pages.
Goldbach's Conjecture states that every even integer greater than 2 can be expressed as the sum of two prime numbers. This paper provides a quantitative proof of this famous conjecture.
Category: Number Theory
[334] viXra:1204.0046 [pdf] submitted on 2012-04-12 21:15:39
Authors: chun-xuan jiang
Comments: 4 Pages.
In $ a^2+b^2=c^2$ there are infinitely many primes a and c solutions.The generalized Pythagoren triples:$(a1)^2+(b1)^2+...+(bn)^2=(cn)^2 $ has infinitely many integer solutions.There are infinitely many primes a1=p such that c1,...
cn are all prime.
Category: Number Theory
[333] viXra:1204.0044 [pdf] submitted on 2012-04-11 18:18:01
Authors: Wilber Valgusbitkevyt
Comments: 3 Pages.
This follows from the proof for Goldbach Conjecture.
Category: Number Theory
[332] viXra:1204.0040 [pdf] submitted on 2012-04-10 16:12:48
Authors: Shekman Arieh
Comments: 6 Pages.
Abstract. This article presents the shortest possible proof of Fermat's Last Theorem of any that have ever been published. It might be the one that Fermat had hinted about in his copy of Diophanti's Arithmetic Book.
Category: Number Theory
[331] viXra:1204.0021 [pdf] submitted on 2012-04-05 13:26:51
Authors: Wilber Valgusbitkevyt
Comments: 3 Pages.
For any given even number 2X, there exists prime numbers which can be noted as X – A. If X – A (mod Pq) =/= 2X for all q’s, then X + A is a prime number, and the sum of X – A and X + A is 2X for all even numbers bigger than 4. This is shown with Arbitrary Modular Arithmetic and Fermat’s Infinite Descent Method. Then, it is shown that the number of possible X – A such that guarantees X + A to be a prime number is at least 1 for any X >= 4. In other words, all even numbers can be represented as the sum of two odd prime numbers.
Category: Number Theory
[330] viXra:1204.0010 [pdf] submitted on 2012-04-03 20:05:28
Authors: Wilber Valgusbitkevyt
Comments: 3 Pages.
I used reverse modus ponens. The rest is just a regular 3rd year undergraduate mathematics.
Category: Number Theory
[329] viXra:1204.0007 [pdf] submitted on 2012-04-03 18:28:40
Authors: Carlos Giraldo Ospina
Comments: 4 Pages.
Goldbach's Conjecture states that every even integer greater than 2 can be expressed as the sum of two prime numbers. This paper provides a qualitative proof of this famous conjecture.
Category: Number Theory
[328] viXra:1203.0093 [pdf] submitted on 2012-03-24 14:26:06
Authors: Reuven Tint
Comments: 7 Pages. Original article written in Russian.
We give a proof of the solvability in a natural numbers for Fermat’s Last theorem and the equations has not been found earlier, significantly different from known, and allow us to obtain infinite set of solutions in natural numbers, and examples.
Category: Number Theory
[327] viXra:1203.0082 [pdf] submitted on 2012-03-21 23:21:01
Authors: Choe Ryong Gil
Comments: 37 pages
In this paper we obtain a certain estimate for the difference between the consecutive prime numbers. For this purpose, we consider an opimization problem of an exponential function including the prime numbers. The estimate obtained from this paper is a new result for the distribution of the prime numbers.
Category: Number Theory
[326] viXra:1203.0075 [pdf] submitted on 2012-03-19 10:44:15
Authors: Louis D. Grey
Comments: 3 Pages.
It is a well known result that for a prime of the form 4k+3, there are more quadratic residues than non-residues in the interval (1,(p-1)/2). Using elementary methods,we provide an asymptotic estimate for the number of residues in the interval.
Category: Number Theory
[325] viXra:1203.0073 [pdf] submitted on 2012-03-19 11:47:19
Authors: Louis D. Grey
Comments: Pages. Remove word "louis" after title of paper
We prove a special case of C.L. Siegel's theorem regarding the class number of binary quadratic forms with fundamental discriminant -D<0.
Category: Number Theory
[324] viXra:1203.0064 [pdf] submitted on 2012-03-17 12:38:17
Authors: Ricardo G. Barca
Comments: 24 Pages.
Let a (a greater equal than 5042) be an even number such that a is not twice
a prime. Let { p_1, p_2, p_3, ..., p_k } be the ordered set of k primes
less than the square root of a. Every natural number n < a can be
associated with a k-tuple, the elements of which are the remainders of
dividing n by p_1, p_2, p_3, ..., p_k. These k-tuples are classified in
two types: prohibited or permitted, according to the type of the
remainders that form the k-tuple. We prove that if p is a prime number less
than a, therefore if p is not congruent with p_h, for every p_h in { p_1, p_2,
p_3, ..., p_k }, then a-p is a prime or a-p=1. In that case, a permitted
k-tuple is associated with p. We use this fact to prove that every even
number a > 5042 is the sum of two odd primes.
Category: Number Theory
[323] viXra:1203.0060 [pdf] submitted on 2012-03-16 02:45:52
Authors: chun-xuan jiang
Comments: 90 Pages. It is the very important paper for number theory
Using Jiang function we prove the new prime theorems(1441)-(1480).
Category: Number Theory
[322] viXra:1203.0050 [pdf] submitted on 2012-03-14 19:21:00
Authors: chun-xuan jiang
Comments: 90 Pages.
using Jiang function we prove the new prime theorems(1491)-��1540��
Category: Number Theory
[321] viXra:1203.0019 [pdf] submitted on 2012-03-06 03:36:04
Authors: chun-xuan jiang
Comments: 5 Pages.
All eyes are n the Riemann zeta and L-functions, which are false.please read this paper
Category: Number Theory
[320] viXra:1202.0086 [pdf] submitted on 2012-02-28 19:42:12
Authors: Song Wen-Miao
Comments: 3 Pages. Chenese
Controversy between Jiang proof and Wiles proof for Fermat last theorem in China
Category: Number Theory
[319] viXra:1202.0085 [pdf] submitted on 2012-02-28 04:04:30
Authors: Predrag Terzich
Comments: 4 Pages.
We explore some properties of generalized Fermat primes of the form : F_n(2q)=(2q)^(2^n)+1 , where n > 1 and q is an odd prime number .
Category: Number Theory
[318] viXra:1202.0079 [pdf] submitted on 2012-02-26 22:36:06
Authors: Stephen Crowley
Comments: 38 Pages.
The harmonic sawtooth map w(x) of the unit interval onto itself is defined. It is shown that its fixed points {x: w(x) = x} are enumerated by the n-th derivatives of a Meijer-G function and Lerch transcendent, serving as exponential and ordinary generating functions respectively, and involving the golden ratio in their parameters. The appropriately scaled Mellin transform of w(x) is an analytic continuation of the Riemann zeta function ζ(s) valid ∀−Re(s) not in N. The series expansion of the inverse scaling function which makes the Mellin transform of w(x) equal to the zeta function has coefficients enumerating the Large Schröder Numbers S_n, defined as the number of perfect matchings in a triangular grid of n squares and expressible as a hypergeometric function. The Mellin and Laplace transforms of the individual component functions of the infinite sums and their roots are compared. The Lambert W function answers the inverse question for the first component function covering the interval (1/2, 1), associated with the equation 2^−s + s − 1 = 0, the roots of which are enumerated by ρ^Mw_1(m) = (ln(2)+W(m,-ln(√2))/ln(2). The difference between f(m) = 2*floor((m+2)/ln(2))-4 and a(m) = {x: πx+Im(−ln(e^-ρ^Mw_1(m)))=Im(ρ^Mw_1(m))} is conjectured to take only values 0 or 1, that is, f (m) − a(m) ∈ {0, 1}. The successive differences b(m + 1) − b(m) of the indices b(m) = {m: f (m) − a(m) = 1} appears to follow a pattern where each value is apparently always one of the three numbers 9, 52, or 61, and is possibly related to the number of allowable lattice paths, the least common multiple of which is lcm(9, 52, 61) = 28548 = 134 − 13. The Gauss map h(x) is recalled so that its fixed points and Mellin transform can be contrasted to those of w(x). The geometric counting function NLw(x) = floor(sqrt(2x+1)/2-1/2) of the fractal string Lw associated to the lengths of the harmonic sawtooth map components {w_n(x)}n=1..∞ happens to coincide with the counting function for the number of Pythagorean triangles of the form {(a, b, b + 1): (b + 1)<=x}. The volume of the inner tubular neighborhood of the boundary of the map ∂Lw with radius ε is shown to have the particuarly simple closed-form VLw(ε) = (4εv(ε)^2 − 4εv(ε) + 1)/2v(ε) where v(ε) = floor( (ε + sqrt( ε^2 + ε ) ) / 2ε ) . Also, the Minkowski content of Lw is shown to be MLw = 2 and the Minkowski dimension to be DLw = 2 and thus not invertible. The geometric zeta function, which is the Mellin transform of the geometric counting function NLw(x), is calculated and shown to have a rather unusual closed form involving a finite sum of Riemann zeta functions and binomioal coefficients. Some definitions from the theory of fractal strings and membranes are also recalled.
Category: Number Theory
[317] viXra:1202.0067 [pdf] submitted on 2012-02-19 15:28:34
Authors: Carlos Giraldo Ospina
Comments: 2 Pages.
In this paper Legendre’s Conjecture and Brocard’s Conjecture are proved by determining the amount of prime numbers that are located between N^2 and (N+1)^2.
Category: Number Theory
[316] viXra:1202.0066 [pdf] submitted on 2012-02-20 00:21:01
Authors: Stephen Crowley
Comments: 24 Pages. This is my first submission, please be kind :)
The Riemann zeta function at integer arguments can be written as an infinite sum of certain hypergeometric functions. An analytic continuation formula for these hypergeometric functions exists and is used to derive some novel infinite sums which allow the zeta function at integer arguments n to be written as a weighted infinite sum of the zeta function evaluated at at the “previous integer” n − 1. The notion of fractal strings is related to the Gauss map which arises in the study of continued fractions, and another chaotic map is also introduced called the “Harmonic sawtooth” whose Mellin transform is the zeta function. Some number theoretic definitions are also recalled for kicks.
Category: Number Theory
[315] viXra:1202.0063 [pdf] submitted on 2012-02-18 19:13:46
Authors: Germán Paz
Comments: 32 Pages, 15 pages of tables. On the tables that appear from pages 8 to 22, the numbers
that are located on the third column and have commas should rather have dots, since this work is
written in English. This detail does not change results at all.
In this paper it is proved that if 'a' and 'b' are two positive integers which are coprime and also have
different parity, then there are infinitely many prime numbers of the form ap + b (where 'p' is a prime
number) and infinitely many prime numbers of the form ap - b. In particular, all this proves that there
are infinitely many prime numbers of the form 2p + 1, which proves there are infinitely many Sophie
Germain Prime Numbers. This document also contains Lic. Carlos Giraldo Ospina's solution to the
Polignac's Conjecture and to the Twin Prime Conjecture, which is one of Landau's Problems. Previous
papers (written in Spanish language) were reviewed and approved by Lic. C. G. Ospina and versions of
those papers were posted by this person on his own website and on ABCdatos. You may search for the
papers' titles on the internet. You may also visit the websites that are mentioned in this paper.
This work was submitted to the Journal of Number Theory.
Category: Number Theory
[314] viXra:1202.0061 [pdf] submitted on 2012-02-18 21:49:14
Authors: Germán Paz
Comments: 19 Pages. Secondary email: germanpaz@frro.utn.edu.ar. This paper was submitted to the Journal of Number Theory.
In this paper it is proved that for every positive integer 'k' there are infinitely many prime numbers of the form n^2+k. As a result, it is proved that there are infinitely many prime numbers of the form n^2+1. Therefore, one of Landau’s Problems is now solved.
This document also proposes a new and important conjecture about prime numbers called 'Conjecture C'. If this conjecture is true, then Legendre’s Conjecture, Brocard’s Conjecture and Andrica’s Conjecture are all true, and also some other important results will be true.
Previous papers (written in Spanish language) were reviewed and approved by Carlos Giraldo Ospina (Lic. Matemáticas, USC, Cali, Colombia). This person posted versions of these papers at his own personal website and at ABCdatos. You may search for those papers on the internet. You may also visit the websites that are mentioned in this paper.
Category: Number Theory
[313] viXra:1202.0056 [pdf] submitted on 2012-02-17 10:43:30
Authors: Chun-Xuan Jiang
Comments: 6 Pages.
Using cmplex hyperbolic functions and complex trionometric functions ,we reapear the Fermat marvelus proofs for Fermat last theorem
Category: Number Theory
[312] viXra:1202.0055 [pdf] submitted on 2012-02-16 22:54:53
Authors: Choe Ryong Gil
Comments: 12 pages, 1 schema
The Riemann hypothesis is one of the most important unsolved problems in the modern mathematics. The Riemann hypothesis is closely related with the distribution of prime numbers. The Robin inequality is one of the famous criterions for the Riemann hypothesis. The Robin inequality is related with the sum of divisors function. In this paper we prove that the Robin inequality holds unconditionally. The main idea is to prove a certain inequality on the sum of divisors function, whch is equivalent to the Robin inequality.
Category: Number Theory
[311] viXra:1202.0048 [pdf] submitted on 2012-02-13 19:07:59
Authors: Choe Ryong Gil
Comments: 7 pages
The Riemann hypothesis is well known. The Riemann hypothesis is related with many problems of the analytical number theory. And there have been found some propositions equivalent to one. In particular, the Robin inequality is one of the most famous criterions for the Riemann hypothesis. In this paper we would show one inequality on the sum of divisors function. This inequality is weaker than the Robin inequality, but equivalent to one.
Category: Number Theory
[310] viXra:1202.0030 [pdf] submitted on 2012-02-11 06:07:45
Authors: Choe Ryong Gil
Comments: 58 pages, two tables
The Robin inequality is well known. This inequality is one of the most important and famous criterion for the Riemann hypothesis, but it is unsolved completely yet. In this paper we prove one inequality on the sum of divisors function. This inequality is closely related with the Robin inequality. The main idea is to introduce a new concept, which would be called a sigma-index of the natural number.
Category: Number Theory
[309] viXra:1202.0029 [pdf] submitted on 2012-02-11 07:12:18
Authors: Predrag Terzich
Comments: 5 Pages.
We present deterministic primality test for Fermat
numbers . Essentially this test is similar to the Lucas-Lehmer primality test for Mersenne numbers.
Category: Number Theory
[308] viXra:1202.0024 [pdf] submitted on 2012-02-10 06:54:39
Authors: Predrag Terzich
Comments: 4 Pages.
We present a theorem concerning the form of Mersenne
numbers . We also discuss a closed form expression which links prime numbers and natural logarithms .
Category: Number Theory
[307] viXra:1201.0048 [pdf] submitted on 2012-01-10 17:45:26
Authors: Marco Ripà
Comments: The paper is in Italian, 4 pages long. It is related to Graham number. Traditional Copyright "all rights reserved"
In this paper we present a super-fast hyperoperator plus a method to create a new hierarchy of hyperoperators. For comparison, applying it to a base n=2, the result will be far larger than Graham's number.
Finally we show a very large number based on Graham's one.
Category: Number Theory
[306] viXra:1201.0012 [pdf] submitted on 2012-01-05 00:58:17
Authors: Chun-Xuan Jiang
Comments: 3 Pages.
Usina Jiang function we study prime distribution in Pythagporean triples(1)
Category: Number Theory
[84] viXra:1204.0099 [pdf] replaced on 2012-05-06 18:51:05
Authors: Carlos Giraldo Ospina
Comments: 4 Pages.
In this paper it is proved that if Fermat's Last Theorem (FLT) is true, then Beal's Conjecture is also true. Since FLT was already proved, then Beal's Conjecture is true.
Category: Number Theory
[83] viXra:1204.0053 [pdf] replaced on 2012-05-03 19:23:08
Authors: Carlos Giraldo Ospina
Comments: 7 Pages.
Goldbach's Conjecture states that every even integer greater than 2 can be expressed as the sum of two prime numbers. This paper provides a quantitative proof of this famous conjecture.
Category: Number Theory
[82] viXra:1204.0044 [pdf] replaced on 2012-04-12 10:43:16
Authors: Wilber Valgusbitkevyt
Comments: 2 Pages. Finished solving it.
The proof for Goldbach's Conjecture is applicable here.
Category: Number Theory
[81] viXra:1204.0021 [pdf] replaced on 2012-04-12 09:03:04
Authors: Wilber Valgusbitkevyt
Comments: 2 Pages. Rewrote after Latex tutorials
For all even numbers 2X >= 4, if X >= 4, there exists a prime number X - A always such that X + A is also a prime number. For X = 2 and 3, 2X is 2 + 2 and 3 + 3. Hence, Goldbach Conjecture is true.
Category: Number Theory
[80] viXra:1204.0021 [pdf] replaced on 2012-04-11 18:16:23
Authors: Wilber Valgusbitkevyt
Comments: 2 Pages. Used Latex to write cleanly.
For all even numbers 2X >= 4, if X >= 4, there exists a prime number X - A always such that X + A is also a prime number. For X = 2 and 3, 2X is 2 + 2 and 3 + 3. Hence, Goldbach Conjecture is true.
Category: Number Theory
[79] viXra:1203.0093 [pdf] replaced on 2012-03-30 14:52:51
Authors: Reuven Tint
Comments: 14 Pages. Original article written in Russian.
We give a proof of the solvability in a natural numbers for Fermat’s Last theorem and the equations has not been found earlier, significantly different from known, and allow us to obtain infinite set of solutions in natural numbers, and examples.
Category: Number Theory
[78] viXra:1203.0064 [pdf] replaced on 2012-04-05 08:25:08
Authors: Ricardo G. Barca
Comments: 23 Pages.
Let a (a greater equal than 5042) be an even number such that a is not twice
a prime. Let { p_1, p_2, p_3, ..., p_k } be the ordered set of k primes
less than the square root of a. Every natural number n < a can be
associated with a k-tuple, the elements of which are the remainders of
dividing n by p_1, p_2, p_3, ..., p_k. These k-tuples are classified in
two types: prohibited or permitted, according to the type of the
remainders that form the k-tuple. We prove that if p is a prime number less
than a, therefore if p is not congruent with p_h, for every p_h in { p_1, p_2,
p_3, ..., p_k }, then a-p is a prime or a-p=1. In that case, a permitted
k-tuple is associated with p. We use this fact to prove that every even
number a > 5042 is the sum of two odd primes.
Category: Number Theory
[77] viXra:1203.0019 [pdf] replaced on 2012-03-06 19:53:44
Authors: chun-xuan jiang
Comments: 5 Pages.
All eyes are on the Riemann hypothesis,zeta and L-functions ,which are false ,please read this paper.
Category: Number Theory
[76] viXra:1202.0086 [pdf] replaced on 2012-03-03 23:45:52
Authors: Song Wen Miao
Comments: 3 Pages. CHINESE
ALL EYES ARE ON FERMAT LAST THEOREM WHO PROVED FORST
Category: Number Theory
[75] viXra:1202.0079 [pdf] replaced on 2012-05-18 13:09:21
Authors: Stephen Crowley
Comments: 39 Pages.
The harmonic sawtooth map w(x) of the unit interval onto itself is defined. It is shown that its fixed points {x: w(x) = x} are enumerated by the n-th derivatives of a Meijer-G function and Lerch transcendent, serving as exponential and ordinary generating functions respectively, and involving the golden ratio in their parameters. The appropriately scaled Mellin transform of w(x) is an analytic continuation of the Riemann zeta function ζ(s) valid ∀−Re(s) not in N. The series expansion of the inverse scaling function which makes the Mellin transform of w(x) equal to the zeta function has coefficients enumerating the Large Schröder Numbers S_n, defined as the number of perfect matchings in a triangular grid of n squares and expressible as a hypergeometric function. A finite-sum approximation to ζ(s) denoted by ζw(N;s) is examined and an associated function χ(N;s) is found which solves the reflection formula ζw(N;1−s)=χ(N;s)ζw(N;s). The function χ(N;s) is singular at s = 0 and the residue at this point changes sign from negative to positive between the values of N = 176 and N = 177. Some rather elegant graphs of the reflection functions χ(N;s) are also provided. The values ζw(1-N;s) are found to be related to the Bernoulli numbers. The Mellin and Laplace transforms of the individual component functions of the infinite sums and their roots are compared. The Gauss map h(x) is recalled so that its fixed points and Mellin transform can be contrasted to those of w(x). The geometric counting function NLw(x) = floor(sqrt(2x+1)/2-1/2) of the fractal string Lw associated to the lengths of the harmonic sawtooth map components {w_n(x)}n=1..∞ happens to coincide with the counting function for the number of Pythagorean triangles of the form {(a,b,b+1):(b+1)<=x}. The volume of the inner tubular neighborhood of the boundary of the map ∂Lw with radius ε is shown to have the particuarly simple closed-form VLw(ε) = (4εv(ε)^2−4εv(ε)+1)/2v(ε) where v(ε) = floor((ε+sqrt(ε^2+ε))/2ε). Also, the Minkowski content of Lw is shown to be MLw = 2 and the Minkowski dimension to be DLw = 1/2 and thus not invertible. The geometric zeta function, which is the Mellin transform of the geometric counting function NLw(x), is calculated and shown to have a rather unusual closed form involving a finite sum of Riemann zeta functions and binomioal coefficients. Some definitions from the theory of fractal strings and membranes are also recalled.
Category: Number Theory
[74] viXra:1202.0079 [pdf] replaced on 2012-05-15 14:18:48
Authors: Stephen Crowley
Comments: 39 Pages.
The harmonic sawtooth map w(x) of the unit interval onto itself is defined. It is shown that its fixed points {x: w(x) = x} are enumerated by the n-th derivatives of a Meijer-G function and Lerch transcendent, serving as exponential and ordinary generating functions respectively, and involving the golden ratio in their parameters. The appropriately scaled Mellin transform of w(x) is an analytic continuation of the Riemann zeta function ζ(s) valid ∀−Re(s) not in N. The series expansion of the inverse scaling function which makes the Mellin transform of w(x) equal to the zeta function has coefficients enumerating the Large Schröder Numbers S_n, defined as the number of perfect matchings in a triangular grid of n squares and expressible as a hypergeometric function. A finite-sum approximation to ζ(s) denoted by ζw(N;s) is examined and an associated function χ(N;s) is found which solves the reflection formula ζw(N;1−s)=χ(N;s)ζw(N;s). The function χ(N;s) is singular at s = 0 and the residue at this point changes sign from negative to positive between the values of N = 176 and N = 177. Some rather elegant graphs of the reflection functions χ(N;s) are also provided. The values ζw(1-N;s) are found to be related to the Bernoulli numbers. The Mellin and Laplace transforms of the individual component functions of the infinite sums and their roots are compared. The Gauss map h(x) is recalled so that its fixed points and Mellin transform can be contrasted to those of w(x). The geometric counting function NLw(x) = floor(sqrt(2x+1)/2-1/2) of the fractal string Lw associated to the lengths of the harmonic sawtooth map components {w_n(x)}n=1..∞ happens to coincide with the counting function for the number of Pythagorean triangles of the form {(a,b,b+1):(b+1)<=x}. The volume of the inner tubular neighborhood of the boundary of the map ∂Lw with radius ε is shown to have the particuarly simple closed-form VLw(ε) = (4εv(ε)^2−4εv(ε)+1)/2v(ε) where v(ε) = floor((ε+sqrt(ε^2+ε))/2ε). Also, the Minkowski content of Lw is shown to be MLw = 2 and the Minkowski dimension to be DLw = 1/2 and thus not invertible. The geometric zeta function, which is the Mellin transform of the geometric counting function NLw(x), is calculated and shown to have a rather unusual closed form involving a finite sum of Riemann zeta functions and binomioal coefficients. Some definitions from the theory of fractal strings and membranes are also recalled.
Category: Number Theory
[73] viXra:1202.0079 [pdf] replaced on 2012-05-10 21:59:51
Authors: Stephen Crowley
Comments: 39 Pages.
The harmonic sawtooth map w(x) of the unit interval onto itself is defined. It is shown that its fixed points {x: w(x) = x} are enumerated by the n-th derivatives of a Meijer-G function and Lerch transcendent, serving as exponential and ordinary generating functions respectively, and involving the golden ratio in their parameters. The appropriately scaled Mellin transform of w(x) is an analytic continuation of the Riemann zeta function ζ(s) valid ∀−Re(s) not in N. The series expansion of the inverse scaling function which makes the Mellin transform of w(x) equal to the zeta function has coefficients enumerating the Large Schröder Numbers S_n, defined as the number of perfect matchings in a triangular grid of n squares and expressible as a hypergeometric function. A finite-sum approximation to ζ(s) denoted by ζw(N;s) is examined and an associated function χ(N;s) is found which solves the reflection formula ζw(N;1−s)=χ(N;s)ζw(N;s). The function χ(N;s) is singular at s = 0 and the residue at this point changes sign from negative to positive between the values of N = 176 and N = 177. Some rather elegant graphs of the reflection functions χ(N;s) are also provided. The Mellin and Laplace transforms of the individual component functions of the infinite sums and their roots are compared. The Gauss map h(x) is recalled so that its fixed points and Mellin transform can be contrasted to those of w(x). The geometric counting function NLw(x) = floor(sqrt(2x+1)/2-1/2) of the fractal string Lw associated to the lengths of the harmonic sawtooth map components {w_n(x)}n=1..∞ happens to coincide with the counting function for the number of Pythagorean triangles of the form {(a,b,b+1):(b+1)<=x}. The volume of the inner tubular neighborhood of the boundary of the map ∂Lw with radius ε is shown to have the particuarly simple closed-form VLw(ε) = (4εv(ε)^2−4εv(ε)+1)/2v(ε) where v(ε) = floor((ε+sqrt(ε^2+ε))/2ε). Also, the Minkowski content of Lw is shown to be MLw = 2 and the Minkowski dimension to be DLw = 1/2 and thus not invertible. The geometric zeta function, which is the Mellin transform of the geometric counting function NLw(x), is calculated and shown to have a rather unusual closed form involving a finite sum of Riemann zeta functions and binomioal coefficients. Some definitions from the theory of fractal strings and membranes are also recalled.
Category: Number Theory
[72] viXra:1202.0079 [pdf] replaced on 2012-05-04 15:57:56
Authors: Stephen Crowley
Comments: 38 Pages.
The harmonic sawtooth map w(x) of the unit interval onto itself is defined. It is shown that its fixed points {x: w(x) = x} are enumerated by the n-th derivatives of a Meijer-G function and Lerch transcendent, serving as exponential and ordinary generating functions respectively, and involving the golden ratio in their parameters. The appropriately scaled Mellin transform of w(x) is an analytic continuation of the Riemann zeta function ζ(s) valid ∀−Re(s) not in N. The series expansion of the inverse scaling function which makes the Mellin transform of w(x) equal to the zeta function has coefficients enumerating the Large Schröder Numbers S_n, defined as the number of perfect matchings in a triangular grid of n squares and expressible as a hypergeometric function. A finite-sum approximation to ζ(s) denoted by ζw(N;s) is examined and an associated function χ(N;s) is found which solves the reflection formula ζw(N;1−s)=χ(N;s)ζw(N;s). The function χ(N;s) is singular at s = 0 and the residue at this point changes sign from negative to positive between the values of N = 176 and N = 177. Some rather elegant graphs of the reflection functions χ(N;s) are also provided. The Mellin and Laplace transforms of the individual component functions of the infinite sums and their roots are compared. The Gauss map h(x) is recalled so that its fixed points and Mellin transform can be contrasted to those of w(x). The geometric counting function NLw(x) = floor(sqrt(2x+1)/2-1/2) of the fractal string Lw associated to the lengths of the harmonic sawtooth map components {w_n(x)}n=1..∞ happens to coincide with the counting function for the number of Pythagorean triangles of the form {(a,b,b+1):(b+1)<=x}. The volume of the inner tubular neighborhood of the boundary of the map ∂Lw with radius ε is shown to have the particuarly simple closed-form VLw(ε) = (4εv(ε)^2−4εv(ε)+1)/2v(ε) where v(ε) = floor((ε+sqrt(ε^2+ε))/2ε). Also, the Minkowski content of Lw is shown to be MLw = 2 and the Minkowski dimension to be DLw = 1/2 and thus not invertible. The geometric zeta function, which is the Mellin transform of the geometric counting function NLw(x), is calculated and shown to have a rather unusual closed form involving a finite sum of Riemann zeta functions and binomioal coefficients. Some definitions from the theory of fractal strings and membranes are also recalled.
Category: Number Theory
[71] viXra:1202.0079 [pdf] replaced on 2012-04-20 16:56:04
Authors: Stephen Crowley
Comments: 38 Pages.
The harmonic sawtooth map w(x) of the unit interval onto itself is defined. It is shown that its fixed points {x: w(x) = x} are enumerated by the n-th derivatives of a Meijer-G function and Lerch transcendent, serving as exponential and ordinary generating functions respectively, and involving the golden ratio in their parameters. The appropriately scaled Mellin transform of w(x) is an analytic continuation of the Riemann zeta function ζ(s) valid ∀−Re(s) not in N. The series expansion of the inverse scaling function which makes the Mellin transform of w(x) equal to the zeta function has coefficients enumerating the Large Schröder Numbers S_n, defined as the number of perfect matchings in a triangular grid of n squares and expressible as a hypergeometric function. A finite-sum approximation to ζ(s) denoted by ζw(N;s) is examined and an associated function χ(N;s) is found which solves the reflection formula ζw(N;1−s)=χ(N;s)ζw(N;s). The function χ(N;s) is singular at s = 0 and the residue at this point changes sign from negative to positive between the values of N = 176 and N = 177. Some rather elegant graphs of the reflection functions χ(N;s) are also provided. The Mellin and Laplace transforms of the individual component functions of the infinite sums and their roots are compared. The Gauss map h(x) is recalled so that its fixed points and Mellin transform can be contrasted to those of w(x). The geometric counting function NLw(x) = floor(sqrt(2x+1)/2-1/2) of the fractal string Lw associated to the lengths of the harmonic sawtooth map components {w_n(x)}n=1..∞ happens to coincide with the counting function for the number of Pythagorean triangles of the form {(a,b,b+1):(b+1)<=x}. The volume of the inner tubular neighborhood of the boundary of the map ∂Lw with radius ε is shown to have the particuarly simple closed-form VLw(ε) = (4εv(ε)^2−4εv(ε)+1)/2v(ε) where v(ε) = floor((ε+sqrt(ε^2+ε))/2ε). Also, the Minkowski content of Lw is shown to be MLw = 2 and the Minkowski dimension to be DLw = 1/2 and thus not invertible. The geometric zeta function, which is the Mellin transform of the geometric counting function NLw(x), is calculated and shown to have a rather unusual closed form involving a finite sum of Riemann zeta functions and binomioal coefficients. Some definitions from the theory of fractal strings and membranes are also recalled.
Category: Number Theory
[70] viXra:1202.0079 [pdf] replaced on 2012-04-15 11:27:21
Authors: Stephen Crowley
Comments: 38 Pages.
The harmonic sawtooth map w(x) of the unit interval onto itself is defined. It is shown that its fixed points {x: w(x) = x} are enumerated by the n-th derivatives of a Meijer-G function and Lerch transcendent, serving as exponential and ordinary generating functions respectively, and involving the golden ratio in their parameters. The appropriately scaled Mellin transform of w(x) is an analytic continuation of the Riemann zeta function ζ(s) valid ∀−Re(s) not in N. The series expansion of the inverse scaling function which makes the Mellin transform of w(x) equal to the zeta function has coefficients enumerating the Large Schröder Numbers S_n, defined as the number of perfect matchings in a triangular grid of n squares and expressible as a hypergeometric function. A finite-sum approximation to ζ(s) denoted by ζw(N;s) is examined and an associated function χ(N;s) is found which solves the reflection formula ζw(N;1−s)=χ(N;s)ζw(N;s). The function χ(N;s) is singular at s = 0 and the residue at this point changes sign from negative to positive between the values of N = 176 and N = 177. Some rather elegant graphs of the reflection functions χ(N;s) are also provided. The Mellin and Laplace transforms of the individual component functions of the infinite sums and their roots are compared. The Gauss map h(x) is recalled so that its fixed points and Mellin transform can be contrasted to those of w(x). The geometric counting function NLw(x) = floor(sqrt(2x+1)/2-1/2) of the fractal string Lw associated to the lengths of the harmonic sawtooth map components {w_n(x)}n=1..∞ happens to coincide with the counting function for the number of Pythagorean triangles of the form {(a,b,b+1):(b+1)<=x}. The volume of the inner tubular neighborhood of the boundary of the map ∂Lw with radius ε is shown to have the particuarly simple closed-form VLw(ε) = (4εv(ε)^2−4εv(ε)+1)/2v(ε) where v(ε) = floor((ε+sqrt(ε^2+ε))/2ε). Also, the Minkowski content of Lw is shown to be MLw = 2 and the Minkowski dimension to be DLw = 1/2 and thus not invertible. The geometric zeta function, which is the Mellin transform of the geometric counting function NLw(x), is calculated and shown to have a rather unusual closed form involving a finite sum of Riemann zeta functions and binomioal coefficients. Some definitions from the theory of fractal strings and membranes are also recalled.
Category: Number Theory
[69] viXra:1202.0079 [pdf] replaced on 2012-04-02 14:52:25
Authors: Stephen Crowley
Comments: 38 Pages.
The harmonic sawtooth map w(x) of the unit interval onto itself is defined. It is shown that its fixed points {x: w(x) = x} are enumerated by the n-th derivatives of a Meijer-G function and Lerch transcendent, serving as exponential and ordinary generating functions respectively, and involving the golden ratio in their parameters. The appropriately scaled Mellin transform of w(x) is an analytic continuation of the Riemann zeta function ζ(s) valid ∀−Re(s) not in N. The series expansion of the inverse scaling function which makes the Mellin transform of w(x) equal to the zeta function has coefficients enumerating the Large Schröder Numbers S_n, defined as the number of perfect matchings in a triangular grid of n squares and expressible as a hypergeometric function. A finite-sum approximation to ζ(s) denoted by ζw(N;s) is examined and an associated function χ(N;s) is found which solves the reflection formula ζw(N;1−s)=χ(N;s)ζw(N;s). The function χ(N;s) is singular at s = 0 and the residue at this point changes sign from negative to positive between the values of N = 176 and N = 177. Some rather elegant graphs of the reflection functions χ(N;s) are also provided. The Mellin and Laplace transforms of the individual component functions of the infinite sums and their roots are compared. The Gauss map h(x) is recalled so that its fixed points and Mellin transform can be contrasted to those of w(x). The geometric counting function NLw(x) = floor(sqrt(2x+1)/2-1/2) of the fractal string Lw associated to the lengths of the harmonic sawtooth map components {w_n(x)}n=1..∞ happens to coincide with the counting function for the number of Pythagorean triangles of the form {(a,b,b+1):(b+1)<=x}. The volume of the inner tubular neighborhood of the boundary of the map ∂Lw with radius ε is shown to have the particuarly simple closed-form VLw(ε) = (4εv(ε)^2−4εv(ε)+1)/2v(ε) where v(ε) = floor((ε+sqrt(ε^2+ε))/2ε). Also, the Minkowski content of Lw is shown to be MLw = 2 and the Minkowski dimension to be DLw = 1/2 and thus not invertible. The geometric zeta function, which is the Mellin transform of the geometric counting function NLw(x), is calculated and shown to have a rather unusual closed form involving a finite sum of Riemann zeta functions and binomioal coefficients. Some definitions from the theory of fractal strings and membranes are also recalled.
Category: Number Theory
[68] viXra:1202.0079 [pdf] replaced on 2012-03-30 18:13:20
Authors: Stephen Crowley
Comments: 36 Pages.
The harmonic sawtooth map w(x) of the unit interval onto itself is defined. It is shown that its fixed points {x: w(x) = x} are enumerated by the n-th derivatives of a Meijer-G function and Lerch transcendent, serving as exponential and ordinary generating functions respectively, and involving the golden ratio in their parameters. The appropriately scaled Mellin transform of w(x) is an analytic continuation of the Riemann zeta function ζ(s) valid ∀−Re(s) not in N. The series expansion of the inverse scaling function which makes the Mellin transform of w(x) equal to the zeta function has coefficients enumerating the Large Schröder Numbers S_n, defined as the number of perfect matchings in a triangular grid of n squares and expressible as a hypergeometric function. A finite-sum approximation to ζ(s) denoted by ζw(N;s) is examined and an associated function χ(N;s) is found which solves the reflection formula ζw(N;1−s)=χ(N;s)ζw(N;s). The function χ(N;s) is singular at s = 0 and the residue at this point changes sign from negative to positive between the values of N = 176 and N = 177, thus indicating some sort of phase transition at this level of approximation. The Mellin and Laplace transforms of the individual component functions of the infinite sums and their roots are compared. The Gauss map h(x) is recalled so that its fixed points and Mellin transform can be contrasted to those of w(x). The geometric counting function NLw(x) = floor(sqrt(2x+1)/2-1/2) of the fractal string Lw associated to the lengths of the harmonic sawtooth map components {w_n(x)}n=1..∞ happens to coincide with the counting function for the number of Pythagorean triangles of the form {(a,b,b+1):(b+1)<=x}. The volume of the inner tubular neighborhood of the boundary of the map ∂Lw with radius ε is shown to have the particuarly simple closed-form VLw(ε) = (4εv(ε)^2−4εv(ε)+1)/2v(ε) where v(ε) = floor((ε+sqrt(ε^2+ε))/2ε). Also, the Minkowski content of Lw is shown to be MLw = 2 and the Minkowski dimension to be DLw = 1/2 and thus not invertible. The geometric zeta function, which is the Mellin transform of the geometric counting function NLw(x), is calculated and shown to have a rather unusual closed form involving a finite sum of Riemann zeta functions and binomioal coefficients. Some definitions from the theory of fractal strings and membranes are also recalled.
Category: Number Theory
[67] viXra:1202.0079 [pdf] replaced on 2012-03-30 14:42:21
Authors: Stephen Crowley
Comments: 36 Pages. new section: finite reflection formula 1.2.2
The harmonic sawtooth map w(x) of the unit interval onto itself is defined. It is shown that its fixed points {x: w(x) = x} are enumerated by the n-th derivatives of a Meijer-G function and Lerch transcendent, serving as exponential and ordinary generating functions respectively, and involving the golden ratio in their parameters. The appropriately scaled Mellin transform of w(x) is an analytic continuation of the Riemann zeta function ζ(s) valid ∀−Re(s) not in N. The series expansion of the inverse scaling function which makes the Mellin transform of w(x) equal to the zeta function has coefficients enumerating the Large Schröder Numbers S_n, defined as the number of perfect matchings in a triangular grid of n squares and expressible as a hypergeometric function. A finite-sum approximation to ζ(s) denoted by ζw(N;s) is examined and an associated function χ(N;s) is found which solves the reflection formula ζw(N;1−s)=χ(N;s)ζw(N;s). The function χ(N;s) is singular at s = 0 and changes sign from negative to positive between the values of N = 176 and N = 177. The Mellin and Laplace transforms of the individual component functions of the infinite sums and their roots are compared. The Gauss map h(x) is recalled so that its fixed points and Mellin transform can be contrasted to those of w(x). The geometric counting function NLw(x) = floor(sqrt(2x+1)/2-1/2) of the fractal string Lw associated to the lengths of the harmonic sawtooth map components {w_n(x)}n=1..∞ happens to coincide with the counting function for the number of Pythagorean triangles of the form {(a,b,b+1):(b+1)<=x}. The volume of the inner tubular neighborhood of the boundary of the map ∂Lw with radius ε is shown to have the particuarly simple closed-form VLw(ε) = (4εv(ε)^2−4εv(ε)+1)/2v(ε) where v(ε) = floor((ε+sqrt(ε^2+ε))/2ε). Also, the Minkowski content of Lw is shown to be MLw = 2 and the Minkowski dimension to be DLw = 1/2 and thus not invertible. The geometric zeta function, which is the Mellin transform of the geometric counting function NLw(x), is calculated and shown to have a rather unusual closed form involving a finite sum of Riemann zeta functions and binomioal coefficients. Some definitions from the theory of fractal strings and membranes are also recalled.
Category: Number Theory
[66] viXra:1202.0079 [pdf] replaced on 2012-03-01 21:00:49
Authors: Stephen Crowley
Comments: 35 Pages.
The harmonic sawtooth map w(x) of the unit interval onto itself is defined. It is shown that
its fixed points {x: w(x) = x} are enumerated by the n-th derivatives of a Meijer-G function
and Lerch transcendent, serving as exponential and ordinary generating functions respectively,
and involving the golden ratio in their parameters. The appropriately scaled Mellin transform
of w(x) is an analytic continuation of the Riemann zeta function ζ(s)
valid ∀−Re(s) not in N. The series expansion of the inverse scaling function
which makes the Mellin transform of w(x) equal to the zeta function has coefficients
enumerating the Large Schröder Numbers S_n, defined as the number of perfect matchings
in a triangular grid of n squares and expressible as a hypergeometric function. The Mellin
and Laplace transforms of the individual component functions of the infinite sums and their
roots are compared. The Gauss map h(x) is recalled so that its fixed points and Mellin
transform can be contrasted to those of w(x). The geometric counting
function NLw(x) = floor(sqrt(2x+1)/2-1/2) of the fractal string Lw associated to the
lengths of the harmonic sawtooth map components {w_n(x)}n=1..∞ happens to coincide
with the counting function for the number of Pythagorean triangles of the
form {(a, b, b + 1): (b + 1)<=x}. The volume of the inner tubular neighborhood of the
boundary of the map ∂Lw with radius ε is shown to have the particuarly simple
closed-form VLw(ε) = (4εv(ε)^2 − 4εv(ε) + 1)/2v(ε)
where v(ε) = floor( (ε + sqrt( ε^2 + ε ) ) / 2ε ) . Also, the
Minkowski content of Lw is shown to be MLw = 2 and the Minkowski dimension to be DLw = 1/2
and thus not invertible. The geometric zeta function, which is the Mellin transform of the
geometric counting function NLw(x), is calculated and shown to have a rather unusual closed
form involving a finite sum of Riemann zeta functions and binomioal coefficients. Some
definitions from the theory of fractal strings and membranes are also recalled.
Category: Number Theory
[65] viXra:1202.0066 [pdf] replaced on 2012-02-24 18:01:30
Authors: Stephen Crowley
Comments: 25 Pages.
The Riemann zeta function at integer arguments can be written as an infinite sum of certain hypergeometric functions and more generally the same can be done with polylogarithms, for which several zeta functions are a special case. An analytic continuation formula for these hypergeometric functions exists and is used to derive some infinite sums which allow the zeta function at integer arguments n to be written as a weighted infinite sum of hypergeometric functions at n − 1. The form might be considered to be a shift operator for the Riemann zeta function which leads to the curious values ζF(0) = I_0(2) − 1 and ζF(1) = Ei(1) − γ which involve a Bessel function of the first kind and an exponential integral respectively and differ from the values ζ(0) = −1/2 and ζ(1) = ∞ given by the usual method of continuation. Interpreting these “hypergeometrically continued” values of the zeta constants in terms of reciprocal common factor probability we have ζF(0)^-1 ~ 78.15% and ζF(1)^-1 ~ 75.88% which contrasts with the standard known values for sensible cases like ζ(2)^-1 ~ 60.79% and ζ(3)^-1 ~ 83.19%. The combinatorial definitions of the Stirling numbers of the second kind, and the 2-restricted Stirling numbers of the second kind are recalled because they appear in the differential equatlon satisfied by the hypergeometric representation of the polylogarithm. The notion of fractal strings is related to the (chaotic) Gauss map of the unit interval which arises in the study of continued fractions, and another chaotic map is also introduced called the “Harmonic sawtooth” whose Mellin transform is the (appropritately scaled) Riemann zeta function. These maps are within the family of what might be called “deterministic chaos”. Some number theoretic definitions are also recalled.
Category: Number Theory
[64] viXra:1202.0066 [pdf] replaced on 2012-02-21 23:56:42
Authors: Stephen Crowley
Comments: 24 Pages.
The Riemann zeta function at integer arguments can be written as an infinite sum of certain hypergeometric functions. An analytic continuation formula for these hypergeometric functions exists and is used to derive some novel (at least in the authors opinion) infinite sums which allow the zeta function at integer arguments n to be written as a weighted infinite sum of hypergeometric functions evaluated at the “previous integer” n − 1. The form might be considered to be a shift operator for the Riemann zeta function which leads to the curious values ζ F (0) = I0(2) − 1 and ζ F (1) = Ei(1) − γ which differ from the values ζ(0) = − 2 and ζ(1) = ∞. The notion of fractal strings is related to the (chaotic) Gauss map of the unit interval which arises in the study of continued fractions, and another chaotic map is also introduced called the “Harmonic sawtooth” whose Mellin transform is the (appropritately scaled) Riemann zeta function. These maps are within the family of what might be called “deterministic chaos”. Some number theoretic definitions are also recalled just for kicks.
Category: Number Theory
[63] viXra:1202.0055 [pdf] replaced on 2012-03-23 11:28:14
Authors: Choe Ryong Gil
Comments: 12 pages, 1 schema
The Riemann hypothesis is one of the most important unsolved problems in the modern mathematics. The Riemann hypothesis is closely related with the distribution of prime numbers. The Robin inequality is one of the famous criterions for the Riemann hypothesis. The Robin inequality is related with the sum of divisors function. In this paper we prove that the Robin inequality holds unconditionally. The main idea is to prove a certain inequality on the sum of divisors function, whch is equivalent to the Robin inequality.
Category: Number Theory
[62] viXra:1202.0048 [pdf] replaced on 2012-03-25 07:24:41
Authors: Choe Ryong Gil
Comments: 7 pages
The Riemann hypothesis is well known. The Riemann hypothesis is related with many problems of the analytical number theory. And there have been found some propositions equivalent to one. In particular, the Robin inequality is one of the most famous criterions for the Riemann hypothesis. In this paper we would show one inequality on the sum of divisors function. This inequality is weaker than the Robin inequality, but equivalent to one.
Category: Number Theory
[61] viXra:1202.0030 [pdf] replaced on 2012-03-22 11:12:48
Authors: Choe Ryong Gil
Comments: 25 pages, 2 tables
The Robin inequality is well known. This inequality is one of the most important and famous criterion for the Riemann hypothesis, but it is unsolved completely yet. In this paper we prove one inequality on the sum of divisors function. This inequality is closely related with the Robin inequality. The main idea is to introduce a new concept, which would be called a sigma-index of the natural number.
Category: Number Theory
[60] viXra:1202.0030 [pdf] replaced on 2012-02-23 21:17:23
Authors: Choe Ryong Gil
Comments: 58 pages, 2 tables
The Robin inequality is well known. This inequality is one of the most important and famous criterion for the Riemann hypothesis, but it is unsolved completely yet. In this paper we prove one inequality on the sum of divisors function. This inequality is closely related with the Robin inequality. The main idea is to introduce a new concept, which would be called a sigma-index of the natural number.
Category: Number Theory
[59] viXra:1202.0029 [pdf] replaced on 2012-02-15 02:03:24
Authors: Predrag Terzich
Comments: 5 Pages.
We present deterministic primality test for Fermat numbers . Essentially this test is similar to the Lucas-Lehmer primality test for Mersenne numbers
Category: Number Theory
[58] viXra:1201.0048 [pdf] replaced on 2012-01-14 13:08:28
Authors: Marco Ripà
Comments: The paper is in Italian, 4 pages long. It is related to Graham number. Traditional Copyright "all rights reserved"
In this paper we present a super-fast hyperoperator plus a method to create a new hierarchy of hyperoperators. For comparison, applying it to a base n=2, the result will be far larger than Graham's number. Finally we show a very large number based on Graham's one.
Category: Number Theory
[57] viXra:1201.0012 [pdf] replaced on 2012-01-09 18:23:45
Authors: chun-xan jiang
Comments: 3 Pages.
using fiang function we study the prime distribution in pythagorean triples
Category: Number Theory
[56] viXra:1201.0012 [pdf] replaced on 2012-01-09 08:34:36
Authors: chun-xan jiang
Comments: 3 Pages.
using jiang function we study the prime distribution triples
Category: Number Theory
[55] viXra:1201.0012 [pdf] replaced on 2012-01-08 22:09:25
Authors: chun-xan jiang
Comments: 3 Pages.
using jiang function we study the prime distribution in pythagorean triples
Category: Number Theory