Riemann Conjecture Proof and Disproof

Transcription

1 From the SelectedWorks of James T Struck Winter March 2, 2018 Riemann Conjecture Proof and Disproof James T Struck Available at:

2 Disproving and Proving the Riemann Conjecture Using 4 New Prime Number Lines to meet the suggestions of Riemann s 1859 article title On the Number of Primes Less Than a Given Magnitude, Line Thickness and Zeros have real parts other than ½ and some real parts ½ too, Discussion of the Zeta Function and Dirichlet Function By James T. Struck BA, BS, AA, MLIS Georg Friedrich Riemann lived from September 17, 1826 until 1866 and he was especially interested in the Christian faith. (Accessed from on 11/3/2017) He died apparently while he was saying the Lord s Prayer with his wife of TB during a journey to Italy. Bernhard Riemann has been associated with the Riemann conjecture for about 150 years since his authorship of the article English title (English title: "On the Number of Primes Less Than a Given Magnitude") in If the article in question is about the number of primes less than a given magnitude, then we can solve the conjecture by consideration of primes less than a given magnitude. I approach the conjecture then from the perspective of what was his original article about- primes less than a given magnitude! I show here 4 new prime number lines that have numbers off the Riemann line to show that prime numbers do not need to be connected so much to Riemann. I invented the

3 1. Abscissa Prime number line with prime numbers off the Riemann line. With this prime number line, we have proof of the conjecture as there is ½ at (½, 0) along the abscissa prime number line and there is other than ½ along the line as well disproving the conjecture too. This prime number line gives just as much comprehension and understanding of prime numbers less than a given magnitude as his 1859 article conjectures about than the issue of the zeta function and the Riemann line.

4 2. Ordinate prime number line with prime numbers off the Riemann line. We can see here proof of the conjecture as there is a zero at ½ or (0, ½) as above with the abscissa prime number line and disproof of the conjecture as zeros do not occur at ½ as well. 3. Positive Abscissa and Ordinate Prime number line with numbers off the Riemann line. Notice how all these prime numbers less than a given magnitude do not have an x or y intercept or a zero at ½ but rather an x or y intercept at 0. If the line were very thick however it would have intersection with the x intercept at ½ being a type of proof of the conjecture that prime number lines can intersect with 0 at 1/2. The rest of the line not intersecting with the x axis is a type of disproof. There does not need to be intersection with the x axis or y axis either. Recall we do not know that Riemann did not want this

5 abscissa ordinate number prime number x, y axis line to be considered or discussed really in his conjecture as the title allows or permits consideration of the subject. As there is not a zero any place else besides at 0, we can my abscissa ordinate prime number line as a disproof of the Riemann conjecture as the zero is not at ½.

6 4. Negative and Positive Abscissa and Ordinate Prime number line with numbers off the Riemann prime number line. See how the prime number lines here never intersect with the x axis or zero at the number ½ which would be a type of disproof of the conjecture. But if the lines were very thick there would be intersection with the number ½ at the x or y intercept or x or y axis as a type of proof of the Riemann conjecture.

7 The connection between Riemann and prime numbers can be corrected by emphasizing that prime numbers do not need to appear on the Riemann line. As a type of disproof of the Riemann conjecture, I show that 4 new prime number lines can easily be constructed and drawn which do not have any link or association with the zeta function. Riemann was concerned with prime numbers less than a certain magnitude he stated in the article title in question. These 4 prime number lines as a type of disproof show that we can look at prime numbers from the perspective of where the primes lie on a graph rather than in relation to the zeta function. As a type of disproof, we do not need to consider the zeta function, but rather can look at other representations of prime numbers shown here. Prime numbers do not need to have zeros at ½, but if the number lines were very thick, they would intersect with the x or y axis at ½. Thickness as a proof and a disproof. We can consider the thickness of a prime number line. As we can make a prime number line very thick, we can include numbers that do not have real or imaginary parts of ½. Take my negative and positive abscissa and ordinate prime number line. If we make those prime number lines very thick, these lines would not include only solutions of ½. Other numbers would be on such prime number lines besides ½. But if very thick as a type of proof, all 4 new prime number lines can have zeros at ½.

8 Even with Riemann s Line, Thick Lines would include numbers that are not 1/2 And that goes for Riemann s line as well that zeros do not actually lie at 1/2. Make Riemann s line very thick and we have solutions that do not have imaginary or real part of ½. The real part of the Riemann line has zeros at points like ½, 14, 21, 25, 30 rather than just at ½ In addition, even with the Riemann equation and line, we can imagine zeros as the primes on my abscissa graph of prime numbers that are not 1/2. We can put Prime numbers on the graph that are not on Riemann s line such as negative and positive prime number line. We can take numbers that are zeros on Riemann s line that do not have real part ½. Just look at the chart

9 The real part (red) and imaginary part (blue) of the Riemann zeta function along the critical line Re(s) = 1/2. The first non-trivial zeros can be seen at Im(s) = ±14.135, ± and ± Source-Slonzor - Own work. Made with Mathematica using the following code: Show[Plot[{Re[Zeta[1/2+I x]], Im[Zeta[1/2+I x]]}, {x,-30, 30},AxesLabel->{"x"}, PlotStyle- >{Red, Blue}, Ticks->{Table[4x-28,{x,0,14}]}, ImageSize->{800,600}], Graphics[Text[Style[\[DoubleStruckCapitalR][\[Zeta][ I x + "1/2"]],14,Red,Background - >White],{-22,2.6} ]], Graphics[Text[Style[\[GothicCapitalI][\[Zeta][ I x + "1/2"]],14,Blue,Background ->White],{-14,2.6} ]]] Graph of real (red) and imaginary (blue) parts of the critical line Re(z)=1/2 of the Riemann zeta function. Accessed from Wikipedia.org on 11/4/2017 The conjecture is disproved in a way as we are still able to show zeros that do not have real parts of ½. Consider the statement above first non-trivial zeros can be seen at

10 Im(s) = ±14.135, ± and ± Well then those do not have real part ½ as they actually have real parts linked to 14, 21 and 25 and 30. Even if we take numbers along the critical line, we can see that many of those ½ solutions are not zeros; 1/2 solutions do not need to be zeros. As a partial disproof, we can choose many numbers along the critical line for the zeta function ½ that are not zeros. The chart above shows that; points along the critical line do not have real part ½ actually. As a type of proof, however, you can see a zero or an x intercept that is right about at (1/2, 0). So along the Riemann line, you can find 0 s that lie at about (½, 0). My 4 New prime number line inventions show that the connection we have had between prime numbers and the Riemann line does not need to be emphasized and we can see the Riemann conjecture as disproved as we do not need to emphasize the Riemann line to understand prime numbers. Prime numbers do not need to have real part ½. Prime numbers do not need to be expressed by the zeta function; we show 4 other presentation methods for prime numbers. Thank you for your consideration of my disproof of the Riemann conjecture using thickness, 4 new prime number lines and solutions do not have real part 1/2. I appreciate your time and consideration very much!

11 If we consider the subject of Bernhard Riemann s article title (English title: "On the Number of Primes Less Than a Given Magnitude") in 1859, my 4 graphs of prime numbers are a simpler approach than the zeta function as a type of disproof of representation of primes. If we wanted to meet the requirements of Bernhard Riemann s title, we can just show prime numbers graphed in different ways Abscissa graph Ordinate graph Positive abscissa and ordinate graph Negative and positive abscissa and ordinate graphs The real part of these prime numbers meeting the requirement of Riemann s title of his article, do not have to have real parts ½ as a type of disproof. My abscissa prime number graph can have real part (1/2, 0) My ordinate prime number graph has real part (0, ½) Both these graphs show that real part of a prime number graph can be 1/2, but the other 2 charts show that the real part of 2 prime number graphs does not have to be ½. In addition, even considering the zeta function along the value 1/2, we see that real parts are not really ½ and do not need to be ½. Why? Because to be zeros of a graph, they need to zero out the function. To zero out a function or be an x intercept, one

12 does not need to have a real part ½. A real part (0, -1.4) is more of a zero on the y axis of the above graph disproving the conjecture of Riemann. A real part of (½, 0), (14, 0) (21,0), ( 25, 0) (30, 0) is more of a zero on the x axis proving for the ½ point but disproving for the other points the conjecture of Riemann from Many other prime number representations can be made besides the Riemann line in relation to the zeta function. 4 prime number line graphs are presented here showing a type of disproof of the Riemann conjecture as many of the primes do not intersect with axes at ½. Providing a partial proof of the Riemann conjecture however, some of the zeros can be at ½ too. We perhaps have been following the wrong path in looking at the Riemann conjecture by forgetting the article title. Riemann wanted us as he makes clear in his title to talk about and discuss On the Number of Primes Less Than a Given Magnitude. We did that here focused on prime numbers, so we can be happy that our look at his title helped us understand prime numbers a bit better than talking about the real part of ½ in relation to the zeta function. Riemann s conjecture can be both proved by showing x axis and y axis intersects can be at ½, and by showing that many other real number x axis and y axis intersects are also real too! Riemann conjecture both proved and disproved! Discussion of the Zeta Function and Dirichlet Function 1. Disproof Dirichlet goes to infinity, so that would be more than ½. See here from Wikipedia.org accessed on 11/8/2017 that Riemann goes to infinity The Riemann zeta function or Euler Riemann zeta function, ζ(s), is a function of a complex variable s that analytically continues the sum of the Dirichlet series

13 2. He also knew that all nontrivial zeros are symmetric with respect to the critical line x = 1 /2. Riemann conjectured that all of the nontrivial zeros are on the critical line, a conjecture that subsequently became known as the Riemann hypothesis. We can put a non-trivial zero away from the critical line disproving the conjecture. As we can put a non-trivial zero someone else, we do not need to place so much mysticism into Riemann s critical line but can see reality in the far wider Universe of numbers. 3. Disproof- We Can understand primes better by looking at more than the real number ½ 4. Proof- The critical strip on the Slonzor chart above hits (1/2.0), so that would serve as some kind of proof 5. Proof- The critical Strip (0, -1/2) if line thick 6. Disproof- The real part will always be more than ½ as there are always new primes away from the critical strip 7. Disproof. The zeta and Dirichlet function are for more than primes alone, so that would be a type of disproof. Riemann is using Dirichlet and zeta to a critical strip area when the zeta function and Dirichlet are not supposed to be limited to the critical strip or to primes 8. James T. Struck BA, BS, AA, MLIS P.O. Box 61 Evanston IL 60204