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https://en.wikipedia.org/wiki/Riemann_hypothesisIn mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2. Many consider it to be the most important unsolved problem in pure mathematics.[1] It is of great interest in number theory because it implies results about the distribution of prime numbers. It was proposed by Bernhard Riemann (1859), after whom it is named.The Riemann hypothesis and some of its generalizations, along with Goldbach's conjecture and the twin prime conjecture, make up Hilbert's eighth problem in David Hilbert's list of twenty-three unsolved problems; it is also one of the Clay Mathematics Institute's Millennium Prize Problems, which offers a million dollars to anyone who solves any of them. The name is also used for some closely related analogues, such as the Riemann hypothesis for curves over finite fields.The Riemann zeta function ζ(s) is a function whose argument s may be any complex number other than 1, and whose values are also complex. It has zeros at the negative even integers; that is, ζ(s) = 0 when s is one of −2, −4, −6, .... These are called its trivial zeros. The zeta function is also zero for other values of s, which are called nontrivial zeros. The Riemann hypothesis is concerned with the locations of these nontrivial zeros, and states that:The real part of every nontrivial zero of the Riemann zeta function is 1/2.Thus, if the hypothesis is correct, all the nontrivial zeros lie on the critical line consisting of the complex numbers 1/2 + i t, where t is a real number and i is the imaginary unit.

Visualizing the Riemann zeta function and analytic continuation

A Youtube channel seems to be dedicated to explain this problem, and I find it as one of the best explanation online.https://www.youtube.com/c/zetamath/videos

Factorials, prime numbers, and the Riemann HypothesisToday we introduce some of the ideas of analytic number theory, and employ them to help us understand the size of n!. We use that understanding to discover a surprisingly accurate picture of the distribution of the prime numbers, and explore how this fits into the broader context of one of the most important unsolved problems in mathematics, the Riemann Hypothesis.

The Basel Problem Part 1: Euler-Maclaurin ApproximationThis is the first video in a two part series explaining how Euler discovered that the sum of the reciprocals of the square numbers is π^2/6, leading him to define the zeta function, and how Riemann discovered the surprising connection between the zeroes of the zeta function and the distribution of the primes, leading ultimately to his statement of the Riemann Hypothesis. This video focuses on how Euler developed a method to approximate this sum to 17 decimal places, as well as how the Bernoulli numbers naturally appear as part of this problem.

The Basel Problem Part 2: Euler's Proof and the Riemann HypothesisIn this video, I present Euler's proof that the solution to the Basel problem is pi^2/6. I discuss a surprising connection Euler discovered between a generalization of the Basel problem and the Bernoulli numbers, as well as his invention of the zeta function. I explain Euler's discovery of the connection between the zeta function and the prime numbers, and I discuss how Riemann's continuation of Euler's work led him to state the Riemann hypothesis, one of the most important conjectures in the entire history of mathematics.Sections of this video:00:00 Intro01:24 Euler's Basel proof23:20 The zeta function and the Bernoulli numbers32:01 Zeta and the primes48:15 The Riemann hypothesis

Analytic Continuation and the Zeta FunctionWhere do complex functions come from? In this video we explore the idea of analytic continuation, a powerful technique which allows us to extend functions such as sin(x) from the real numbers into the complex plane. Using analytic continuation we can finally define the zeta function for complex inputs and make sense of what it is the Riemann Hypothesis is claiming.Chapters:00:00 zetamath does puzzles00:23 Recap02:40 Bombelli and the cubic formula08:45 Evaluating real functions at complex numbers12:33 Maclaurin series21:22 Taylor series27:19 Analytic continuation35:57 What goes wrong48:19 Next time

Complex Integration and Finding Zeros of the Zeta FunctionIn this video we examine the other half of complex calculus: integration. We explain how the idea of a complex line integral arises naturally from real definite integrals via Riemann sums, and we examine some of the properties of this new sort of integral. In particular, we consider some complications that arise when trying to apply the fundamental theorem of calculus to complex functions.We then bring these ideas to the central focus of this series: the zeta function and the Riemann hypothesis. By the end of the video, we will be able to use complex integrals to approximate the location of the zeroes of the zeta function (or those of any other complex function for that matter)!

https://en.wikipedia.org/wiki/Riemann_zeta_function#Riemann's_functional_equationThis zeta function satisfies the functional equationwhere Γ(s) is the gamma function. This is an equality of meromorphic functions valid on the whole complex plane. The equation relates values of the Riemann zeta function at the points s and 1 − s, in particular relating even positive integers with odd negative integers. Owing to the zeros of the sine function, the functional equation implies that ζ(s) has a simple zero at each even negative integer s = −2n, known as the trivial zeros of ζ(s). When s is an even positive integer, the product sin(πs/2)Γ(1 − s) on the right is non-zero because Γ(1 − s) has a simple pole, which cancels the simple zero of the sine factor.

https://en.wikipedia.org/wiki/Riemann_zeta_function#Other_resultsThe fact thatfor all complex s ≠ 1 implies that the zeros of the Riemann zeta function are symmetric about the real axis. Combining this symmetry with the functional equation, furthermore, one sees that the non-trivial zeros are symmetric about the critical line Re(s) = ½It is also known that no zeros lie on a line with real part 1.

https://en.wikipedia.org/wiki/Riemann_hypothesis#Zeros_on_the_critical_lineHardy (1914) and Hardy & Littlewood (1921) showed there are infinitely many zeros on the critical line, by considering moments of certain functions related to the zeta function. Selberg (1942) proved that at least a (small) positive proportion of zeros lie on the line. Levinson (1974) improved this to one-third of the zeros by relating the zeros of the zeta function to those of its derivative, and Conrey (1989) improved this further to two-fifths. In 2020, this estimate was extended to five-twelfths by Pratt, Robles, Zaharescu and Zeindler[22] by considering extended mollifiers that can accommodate higher order derivatives of the zeta function and their associated Kloosterman sums.

https://mathworld.wolfram.com/RiemannHypothesis.html It is known that the zeros are symmetrically placed about the line I(s)=0. This follows from the fact that, for all complex numbers s,1. s and the complex conjugate s* are symmetrically placed about this line.2. From the definition (1), the Riemann zeta function satisfies zeta(s*)=zeta(s)*, so that if s is a zero, so is s*, since then zeta(s*)=zeta(s)*=0*=0.It is also known that the nontrivial zeros are symmetrically placed about the critical line R(s)=1/2, a result which follows from the functional equation and the symmetry about the line I(s)=0. For if s is a nontrivial zero, then 1-s is also a zero (by the functional equation), and then 1-s* is another zero. But s and 1-s* are symmetrically placed about the line R(s)=1/2, since 1-(x+iy)*=(1-x)+iy, and if x=1/2+x', then 1-x=1/2-x'.

There must be some missing key ingredients not yet thought to be related to the problem.

Quote from: hamdani yusuf on 01/07/2022 05:59:51There must be some missing key ingredients not yet thought to be related to the problem.NoIt may not be provable.

Quote from: Bored chemist on 01/07/2022 13:32:43Quote from: hamdani yusuf on 01/07/2022 05:59:51There must be some missing key ingredients not yet thought to be related to the problem.NoIt may not be provable.Why? Do you have any reason to think that way? Or is it just your intuition?

some problems are impossible to solve- That's the incompleteness theorem

At this point, it should be obvious that direct attack on the problem is impossible.

Here are some interesting results which might be useful in solving the problem.Quotehttps://en.wikipedia.org/wiki/Riemann_zeta_function#Riemann's_functional_equationThis zeta function satisfies the functional equationwhere Γ(s) is the gamma function. This is an equality of meromorphic functions valid on the whole complex plane. The equation relates values of the Riemann zeta function at the points s and 1 − s, in particular relating even positive integers with odd negative integers. Owing to the zeros of the sine function, the functional equation implies that ζ(s) has a simple zero at each even negative integer s = −2n, known as the trivial zeros of ζ(s). When s is an even positive integer, the product sin(πs/2)Γ(1 − s) on the right is non-zero because Γ(1 − s) has a simple pole, which cancels the simple zero of the sine factor.

There may be additional true statements within the new, extended set of axioms that are unproveable within that extended set of axioms.

From above results, it can be inferred that for any point in the critical strip,ζ(s*)=ζ(1-s) if and only if s is in critical line,where s*=complex conjugate of s.

4) ζ(1-s) = 0 = ζ(s)

The fact thatζ(s) = (ζ(s*))*

Quote from: evan_au on 03/07/2022 10:45:45There may be additional true statements within the new, extended set of axioms that are unproveable within that extended set of axioms.And the RH might be one of them.Is there proof that there is a proof, or is it possible that the OP is wasting his time?

When Riemann zeta function produces 0 result, ζ(s) = 0, at least one of these terms is 01) 2^{s} = 0 → s = -∞2) π^{s-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=½ But ζ(½) <> 0

https://en.wikipedia.org/wiki/Riemann_zeta_function#Other_resultsThe fact thatOr if written in alternative notation, ζ(s)=(ζ(s*))*for all complex s ≠ 1 implies that the zeros of the Riemann zeta function are symmetric about the real axis. Combining this symmetry with the functional equation, furthermore, one sees that the non-trivial zeros are symmetric about the critical line Re(s) = ½It is also known that no zeros lie on a line with real part 1.