0 Members and 23 Guests are viewing this topic.
In this video, I walk through the derivation of an extension of the factorial function that works for any number: fractional, irrational, and even complex! This turns out to be a very important function, known as the gamma function, which has many surprising connections, one of which I explore in the last chapter of the video.Chapters:0:00 Introduction1:38 A few Disclaimers3:58 The Recursive Formula6:50 The Super Recursive Formula8:45 A minor setback10:28 Logarithms15:21 Deriving the Solution19:26 Our Constraints20:25 History and Conventions22:16 The Miracle25:44 The End
In this video, we build on my last two videos by exploring connections between the gamma function (the extended factorials), the digamma function (the extended harmonic numbers), and trigonometry. We derive Euler's Sine Product Formula, which we then use to prove the gamma and digamma functions' reflection formulas. Finally, we derive a related formula for calculating cotangent.Chapters:00:00 Intro0:43 Background and Notation3:24 The Digamma-Cotangent Connection5:09 The Gamma-Sine Connection6:04 The Sine Product Formula9:59 Proving the Gamma-Sine Connection12:22 The value of (1/2)!13:07 Proving the Digamma-Cotangent Connection14:21 The True Logarithmic Derivative15:52 An Infinite Sum for Cotangent17:46 Final Thoughts
When Riemann zeta function produces 0 result, ζ(s) = 0, at least one of these terms is 01) 2s = 0 → s = -∞2) πs-1 = 0 → s = -∞3) sin(πs/2).Γ(1-s) = 0 → s ∈ {negative even numbers}4) ζ(1-s) = 0 = ζ(s)Point #3 gives trivial zeros, while point #4 gives non-trivial zeros.when 1-s=s → 1=2s → s=0.5But ζ(0.5) <> 0
ζ(s) + ζ(1-s*)=0ζ(s) - ζ(1-s*)=0
What if you could group the trivial zeros of the Riemann zeta function with any hypothetical non-trivial zero off the critical line. And what if this group could not exist within the critical strip? Wouldn't this proof that the Riemann hypothesis is correct?
But...is there a Pattern to predict Primes without Calculations?
When a small imaginary number is added to the plot, we get different peculiarity.https://www.wolframalpha.com/input?i=plot%28im%28zeta+%28x%2Bi%2F10e13%29%29%29+from+-42+to+-40 It is not clear yet if it's just an artefact caused by rounding error in the algorithm, or it's an inherent feature of Zeta function. Although, the first case seems to be more likely.
Someone thinks that he might have solved the problem//www.youtube.com/watch?v=PvUrbpsXZLU
Riemann's hypothesis can be proven by showing that ζ(s) - ζ(1-s*) = 0 if and only if s=1-s*, which implies that Re(s)=1/2
The statement "The functional equation also implies that the zeta function has no zeros with negative real part other than the trivial zeros" is true, but understanding why requires some familiarity with complex analysis and the properties of the Riemann zeta function. Here's a breakdown:1. What is the Riemann zeta function?The Riemann zeta function, denoted by ζ(s), is a function defined for complex numbers 's' (except for 's = 1'). It plays a crucial role in understanding prime numbers and other important areas of mathematics.2. What are the trivial zeros?The zeta function has some "obvious" zeros, called trivial zeros, at negative even integers (-2, -4, -6, etc.). These occur because the series definition of the zeta function diverges at these points.3. What is the functional equation?The functional equation is a powerful relationship between the zeta function at 's' and its value at 1 - s. It essentially "mirrors" the behavior of the function across the line Re(s) = 1/2 (where Re(s) denotes the real part of 's').4. How does the functional equation imply no zeros with negative real parts?Suppose ζ(s) had a non-trivial zero with a negative real part, say at s = -a (where a is positive). By the functional equation, this would imply another zero at s = 1 + a, which has a positive real part.However, we know that the zeta function only diverges at negative even integers, and it is non-zero for all other positive real values. Therefore, a zero at 1 + a would contradict this property.This contradiction arises because having a zero with a negative real part essentially "unpairs" the relationship between s and 1 - s through the functional equation. Since such "unpairing" leads to inconsistencies, it follows that such zeros cannot exist.5. Important note:This explanation necessarily simplifies a complex mathematical argument. For a rigorous understanding, deeper knowledge of complex analysis and the properties of the zeta function is required.
No, there are no zeros of the Riemann zeta function with real part greater than 1/2. This is a known result in mathematics, but with two levels of "known":Level 1: Proven results:For Re(s) > 1: We can prove using the Euler product formula that the zeta function is never zero in this region.For Re(s) = 1: While not as simple as the first point, it has also been proven that the zeta function does not have any zeros on the line Re(s) = 1.Level 2: Unsolved but highly likely:The Riemann Hypothesis: This famous conjecture posits that all non-trivial zeros of the zeta function lie on the "critical line" where Re(s) = 1/2. Although unproven, it is widely believed to be true based on extensive numerical calculations and theoretical evidence.In conclusion, with high confidence, we can say that there are no zeros of the Riemann zeta function with real part greater than 1/2. If you're interested in learning more about the zeta function and its zeros, I can provide further information or point you towards relevant resources.
Riemann knew that the non-trivial zeros of the zeta function were symmetrically distributed about the line s = 1/2 + it, and he knew that all of its non-trivial zeros must lie in the range 0 ≤ Re(s) ≤ 1. He checked that a few of the zeros lay on the critical line with real part 1/2 and suggested that they all do; this is the Riemann hypothesis.https://en.wikipedia.org/wiki/Riemann_hypothesis