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Riemann Hypothesis 3 : Extracting regular pattern
Any nontrivial zeros that do not lie along the Critical Line of real part = 0.5 must do so in 2 pairs symmetric about both the Critical Line and the Real Line. Therefore, these zeros would necessarily have real part = 0.5 ? some value less than 0.5.Imagine walking down the Critical Strip and picking up a perpendicular sliver of the strip, from real part = 0 across to 1, then moving it over onto the complex output plane, adjusting its shape and length so that every point mapped according to the Riemann Zeta Function.For a pair of nontrivial zeros to exist here, this sliver of the Critical Strip would have to satisfy 3 requirements: ? First, it would now have to be intersecting itself, meaning 2 input points mapping to the same output point. ? Second, the overlap would have to be that of 2 points that were originally the same distance in the Critical Strip on either side of the point that had real part = 0.5, (e.g. 0.4 and 0.6). ? Third and finally, this perfectly symmetric coincidence would have to land exactly on the zero of the complex output plane.This video shows that instances of the first condition do indeed occur when the loops of the Critical Line are sufficiently small.However, it seems to be the case that the second condition can never be met. The loop that forms to allow for the self-intersection creates a pinch point that appears always to remain in the right half of the Critical Strip sliver, originally with real part ＞ 0.5 up to 1. Furthermore, the right half of the sliver appears always to be shorter than the left half; so, even though intersection points can exist out into the left half (see between Zeta Zeros 34 and 35 for the first time this occurs), the points from the right half would always start out behind and never be able to catch up to their symmetric left half partner, thus always failing to meet the second condition.If so, then the Riemann Hypothesis is true.In fact, in his paper Geometrisches zur Riemannschen Zetafunktion written in 1934, Andreas Speiser showed that, given input from the left half of the Critical Strip, the derivative of the Riemann Zeta Function never producing a zero is equivalent to the Riemann Hypothesis.~The Riemann Zeta Function, as the sum of all positive integer reciprocals each raised to the complex input, ( A+i?B ), only converges for input with real part greater than 1. ζ( A+i?B ) = ( for N = 1 to ∞ ) ∑ { ( 1/N )^( A+i?B ) }, for A ＞ 1The Dirichlet Eta Function, also known as the Alternating Zeta Function, with every 2nd term subtracted rather than added, however, converges for all input with positive real part. η( A+i?B ) = ( for N = 1 to ∞ ) ∑ { ( 1/N )^( A+i?B ) ? ( ?1 )^( N?1 ) }, for A ＞ 0By taking advantage of the odd-term and even-term subsequences in these functions, it is possible to define the Riemann Zeta Function in terms of the Dirichlet Eta Function, thus extending its definition to the remaining positive real portion of its complex input plane, the Critical Strip. ζ( A+i?B ) = η( A+i?B ) / ( 1 ? 2?( 1/2 )^( A+i?B ) ), for A ＞ 0So, this animation depicts not only how discrete paths in the complex output plane taken by this infinite alternating sum??the Eta-defined Zeta Function??converge to specific output values, but also how these paths change and move as the input values (top-left) in the complex input plane move in the positive imaginary direction along the lines of real part A = 0.5+h, with the Critical Strip's minimum and maximum being h = -0.5 and 0.5. path( ζ( 0.5+h + i?B ) ) = ( for N = 1 to ∞ ) ∑ { ( X ? i?Y ) ? ( ?1 )^( N?1 ) } X = [ √(2) ? cos( B?ln(N/2) ) ? ( 2^h ) ? cos( B?ln(N) ) ] / [ N^( 0.5+h ) ? ( √( ? cos( B?ln(2) ) - 2^( 1-h ) - 2^h ) ] Y = [ √(2) ? sin( B?ln(N/2) ) ? ( 2^h ) ? sin( B?ln(N) ) ] / [ N^( 0.5+h ) ? ( √( ? cos( B?ln(2) ) - 2^( 1-h ) - 2^h ) ]For the animation, the number of steps used for N was 512+floor(B/3) which was enough to reach the final twirl in the path that spirals in toward the actual output value for the Zeta Function.The animation also depicts the trail for the line of real part = 0.5 from imaginary part = B fading to B-1, along with the full sliver of points with the same imaginary part = B.Finally, about the path with real part = 0--it does not actually converge to a point given that its steps do not decrease in length. However, it does orbit around the point towards which the rest of its Critical Strip sliver approaches. That being said, being this far along the Critical Strip, it has grown quite large and extends much farther out than the path with real part = 1, thus lying well outside the scale relevant to this video.For example, ζ(0.5 + i?5229.22) is about 0.00415568 + i?0.00252234 (the itty bitty loop) and ζ(1 + i?5229.22) is about 0.605669 + i?0.200235; meanwhile, ζ(0 + i?5229.22) reaches out to about 13.1889 + i?12.8344.
Since the 4765th loop is much too small in the video, here is a 12 second clip at 3 scales: ? The value after the equals sign refers to when the real part is 0.5, and the dots are from 121 input values that were equally spaced across the Critical Strip (black refers to the "left half"; white the "right half"). Notice how the dots never cross their symmetric partner.
Quotewoah ... apparently it is an open question whether zeta has any double zeros.Yep! In fact, that?s pretty much the Riemann Hypothesis. If there end up being any double zeros, then it?s false; but if there aren?t any, then it?s true.So, I completely misunderstood this comment on first reading. I got stuck thinking of double as a reference to a hypothetical zero off of the critical line and its partner reflected across the critical line. However, a double zero of a function actually refers to an input value that evaluates to zero both for the function and for the derivative of the function. A triple zero would also evaluate to zero for the 2nd derivative, and, in general, a multiple zero of multiplicity M refers to an input value that evaluates to zero for the function and for its first M?1 derivatives. Also, zeros with multiplicity 1 are called simple zeros.That being said, while none of the Riemann Zeta function zeros have been shown to be multiple, there is no proof that all of them are simple. So, yeh, it is indeed still an open question, but, contrary to my initial response, it is different from the open question about zeros off of the critical line that the Riemann Hypothesis addresses.
woah ... apparently it is an open question whether zeta has any double zeros.
In mathematics, the Riemann sphere, named after Bernhard Riemann, [1] is a model of the extended complex plane (also called the closed complex plane): the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers plus a value ∞ for infinity. With the Riemann model, the point ∞ is near to very large numbers, just as the point 0 is near to very small numbers.The extended complex numbers are useful in complex analysis because they allow for division by zero in some circumstances, in a way that makes expressions such as 1/0 = ∞ well-behaved. For example, any rational function on the complex plane can be extended to a holomorphic function on the Riemann sphere, with the poles of the rational function mapping to infinity. More generally, any meromorphic function can be thought of as a holomorphic function whose codomain is the Riemann sphere.
A peek into the world of Riemann surfaces, and how complex analysis is algebra in disguise.This video is a quick-and-dirty introduction to Riemann Surfaces. But as with any quick introduction, there are many details that I gloss over. To learn these details rigorously, I've listed a few resources down below.(a) Complex Analysis To learn complex analysis, I really like the book "Visual Complex Functions: An Introduction with Phase Portraits" by Elias Wegert. It explains the whole subject using domain coloring front and center. Another one of my favorite books is "A Friendly Approach To Complex Analysis" by Amol Sasane and Sara Maad Sasane. I think it motivates all the concepts really well and is very thoroughly explained. (b) Riemann Surfaces and Algebraic Curves A beginner-friendly resource to learn this is "A Guide to Plane Algebraic Curves" by Keith Kendig. It starts off elementary with lots of pictures and visual intuition. Later on in the book, it talks about Riemann surfaces. A more advanced graduate book is "Algebraic Curves and Riemann Surfaces" by Rick Miranda. 00:00-00:54 Intro00:55-04:30 Complex Functions4:31-5:53 Riemann Sphere5:54-6:50 Sponsored Message6:51-11:06 Complex Torus11:07-11:50 Riemann Surfaces12:11-13:53 Riemann's Existence Theorem
How you would tell a high-school student or an interested lay person about the Riemann Hypothesis?Dr. Barry Mazur, Gerhard Gade University Professor of Mathematics at Harvard University, gave a talk on Primes, based on his book-in-progress with William Stein on the Riemann Hypothesis. Date: April 25, 2014 (11:00 AM PDT - 12:00 PM PDT)
Sir Michael Francis Atiyah: "The Riemann Hypothesis"Monday September 24, 2018 9:45Abstract:The Riemann Hypothesis is a famous unsolved problem dating from 1859. I will present a simple proof using a radically new approach. It is based on work of von Neumann (1936), Hirzebruch (1954) and Dirac (1928).
https://en.wikipedia.org/wiki/Riemann_sphereQuoteIn mathematics, the Riemann sphere, named after Bernhard Riemann, [1] is a model of the extended complex plane (also called the closed complex plane): the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers plus a value ∞ for infinity. With the Riemann model, the point ∞ is near to very large numbers, just as the point 0 is near to very small numbers.The extended complex numbers are useful in complex analysis because they allow for division by zero in some circumstances, in a way that makes expressions such as 1/0 = ∞ well-behaved. For example, any rational function on the complex plane can be extended to a holomorphic function on the Riemann sphere, with the poles of the rational function mapping to infinity. More generally, any meromorphic function can be thought of as a holomorphic function whose codomain is the Riemann sphere.Inspired by Riemann's sphere, the extended domain of Zeta function through analytic continuation using functional equation can be remapped to a similar sphere, which I call Zeta sphere.The dotted black curve shows the location of trivial zeros. While dotted yellow curve shows the location of non-trivial zeros
This can be shown by combining Y function with S function which have been shifted to the left by a half.https://www.wolframalpha.com/input?i=plot+re%28log%28%28zeta%280.5%2Be+d%2B9999i%29-zeta+%280.5-e+d%2B9999i%29%29+%2F+%28zeta%280.5%2Bd%2B9999i%29-zeta+%280.5-d%2B9999i%29%29%29+%29+from+-0.2+to+0.2
The behavior of V function should be convincing enough to conclude that zeta function can't be 0 when d is not 0. In other words, there is no anomalous zero of Zeta function, implying that Riemann's hypothesis is true.
For the sake of the argument, let's assume that there exist anomalous Zeta zeros, where s is not on the critical line and not on the real line either. Here are the implications:ζ(0.5+d+c i ) = 0ζ(0.5-d+c i ) = 0ζ(0.5+d+c i)-ζ(0.5-d+c i) = 0when d ≠ 0This will make the V function to blow up because its denominator becomes 0, except when the numerator is also 0.ζ(0.5+d eb+c i)-ζ(0.5-d eb+c i) = 0ζ(0.5+d eb+c i) = ζ(0.5-d eb+c i) for all finite b
https://en.wikipedia.org/wiki/On_the_Number_of_Primes_Less_Than_a_Given_Magnitude"Ueber die Anzahl der Primzahlen unter einer gegebenen Gr?sse" (usual English translation: "On the Number of Primes Less Than a Given Magnitude") is a seminal 9-page paper by Bernhard Riemann published in the November 1859 edition of the Monatsberichte der K?niglich Preu?ischen Akademie der Wissenschaften zu Berlin.OverviewThis paper studies the prime-counting function using analytic methods. Although it is the only paper Riemann ever published on number theory, it contains ideas which influenced thousands of researchers during the late 19th century and up to the present day. The paper consists primarily of definitions, heuristic arguments, sketches of proofs, and the application of powerful analytic methods; all of these have become essential concepts and tools of modern analytic number theory.Among the conjectures made:The Riemann hypothesis, that all (nontrivial) zeros of ζ(s) have real part 1/2. Riemann states this in terms of the roots of the related ξ function,... es ist sehr wahrscheinlich, dass alle Wurzeln reell sind. Hiervon w?re allerdings ein strenger Beweis zu w?nschen; ich habe indess die Aufsuchung desselben nach einigen fl?chtigen vergeblichen Versuchen vorl?ufig bei Seite gelassen, da er f?r den n?chsten Zweck meiner Untersuchung entbehrlich schien.That is,it is very probable that all roots are real. One would, however, wish for a strict proof of this; I have, though, after some fleeting futile attempts, provisionally put aside the search for such, as it appears unnecessary for the next objective of my investigation.(He was discussing a version of the zeta function, modified so that its roots are real rather than on the critical line.)
The plots below show that b determines the intersection between V function and the critical line, regardless the value of c. The last plot shows that the statement is still true even when a non-trivial zero is involved.
V(s) = log( { ζ(0.5+d eb+c i)-ζ(0.5-d eb+c i) } / { ζ(0.5+d+c i)-ζ(0.5-d+c i) } )For the sake of the argument, let's assume that there exist anomalous Zeta zeros, where s is not on the critical line and not on the real line either. Here are the implications:ζ(0.5+d+c i ) = 0ζ(0.5-d+c i ) = 0ζ(0.5+d+c i)-ζ(0.5-d+c i) = 0when d ≠ 0This will make the V function to blow up because its denominator becomes 0, except when the numerator is also 0.ζ(0.5+d eb+c i)-ζ(0.5-d eb+c i) = 0ζ(0.5+d eb+c i) = ζ(0.5-d eb+c i) for all finite b
Quote from: hamdani yusuf on 19/05/2024 22:09:19https://www.wolframalpha.com/input?i=plot+%28log%28%28zeta%280%2B+s+i%29%29+%2F+%28Zeta%281%2B+s+i%29%29%29%29+from+0+to+30This is a variation of backslash (aka S) function, but plotted against the imaginary part, instead of the real part of s.https://www.wolframalpha.com/input?i=plot+re+%28log%28%28zeta%280.499%2B+s+i%29%29+%2F+%28Zeta%281-0.499+%2B+s+i%29%29%29%29+from+-90+to+90https://www.wolframalpha.com/input?i=plot+re+%28log%28%28zeta%280.499%2B+s+i%29%29+%2F+%28Zeta%281-0.499+%2B+s+i%29%29%29%29+from+-9+to+9https://www.wolframalpha.com/input?i=plot+re+%28log%28%28zeta%280.49999%2B+s+i%29%29+%2F+%28Zeta%281-0.49999+%2B+s+i%29%29%29%29+from+-9+to+9Plotted symmetrically, it looks like the letter V. So, I'll just call it V function.
https://www.wolframalpha.com/input?i=plot+%28log%28%28zeta%280%2B+s+i%29%29+%2F+%28Zeta%281%2B+s+i%29%29%29%29+from+0+to+30This is a variation of backslash (aka S) function, but plotted against the imaginary part, instead of the real part of s.