Riemann Hypothesis Ellen, Megan, Dan
Riemann Hypothesis • The nontrivial Riemann zeta function zeros, that is, the values of s other than -2,-4,-6….. such that δ(s)=0 all lie on the critical line Θ = R[s] = ½ (with real part ½)
Terms • Riemann zeta function • Riemann zeta function zeros • Trivial zeros: negative even integers -2,-4,-6…….. • Nontrivial zeros • Occur at certain value of t such that s = ½ + it
Terms • Critical line: The line R[s] = 1/2 in the complex plane
History of The Riemann Hypothesis • Bernhard Riemann, 1859 • On the Number of Primes Less Than a Given Magnitude • Found explicit formula for the number of primes π(x) less than a given number x.
History of the Riemann Hypothesis • Stieltjes (1885) published a note claiming to have proved the Mertens conjecture with c=1 • would have led to the Riemann hypothesis, but was neither published nor found. • Later the Mertens conjecture was found to be false. • H. Rademacher proved the Riemann Hypothesis was false in 1940s • reported in Time magazine, even after a flaw was found in the proof by Siegel. • De Branges has written many papers discussing approachs to prooving the generalized Riemann Hypothesis • claimed to prove the generalized Riemann hypothesis but no proofs have been found. • Conrey and Li proved a counterexample to de Branges's approach, in 1998 this shows that de Branges was incorrect.
Bernhard Riemann • (Georg Friedrich Bernhard Riemann) • Born September 17, 1826 in Breselenz in the Kingdom of Hanover (modern-day Germany) • Riemann was the second of six children. His father, Friedrich Bernhard Riemann, was a poor Lutheran pastor who fought in Napoleonic Wars. His mother, Charlotte Ebell, died before her children had reached adulthood. • From an early age, Riemann demonstrated exceptional math skills and calculation abilities. He was very shy and timid and feared public speaking which in turn led to numerous nervous breakdowns.
Riemann was taught solely by his father until he was ten, when his father got assistance from a local school teacher named Schulz. In 1840, he began going to school in Hanover where he lived with his grandmother until her death in 1842. He then moved to Luneburg, where he attended high school. Although his father encouraged him to study theology, he took a particular interest in mathematics. His talent in mathematics was noticed when a director of mathematics lent him Legendre’s book on the theory of numbers and he finished ready its 900 pages in six days. • In 1846, Riemann enrolled at the University if Gottingen and began studying theology as his father wished. His interest in mathematics grew as he attended several mathematics lectures, so with his father’s approval, he began studying math instead. At the University of Gottingen, Riemann studied under Moritz Stern and Gauss. Although some believed that Gottingen was the perfect place to study mathematics under Gauss, he was unapproachable and did not recognize Riemann’s genius.
In 1847, Riemann transferred to the University of Berlin where he studied under Steiner, Jacobi, Dirichlet, and Eisenstein. During this time, he first came up with the ideas on the theory of functions of a complex variable which led to some of his most important work. Two years later, he returned to Gottingen and received his doctor’s degree in 1851. This time, Riemann caught Gauss’s attention, and under his supervision he presented several essays on topics such as complex analysis, real analysis, and the foundations of geometry. In 1854, he was appointed as an unpaid lecturer, which led to a time of poverty until Dirichlet’s death, when he took his place as a fulltime professor in 1859. This same year he was elected a member of the Berlin Academy of Sciences, in which he was required to send a report of his most recent work. His report, titled "On the number of primes less than a given magnitude", now known as the Riemann hypothesis, is considered by some mathematicians to be the most important unresolved problem in mathematics. • Riemann married Elise Koch in 1862 and had a daughter. Later that same year, he caught a series of colds which led to tuberculosis. He travelled to warmer weather in Italy several times in an attempt to recuperate, but never fully did. His poor heath led to his death in 1866.
What Are We Concerned With? • There are two different types of zeros for the Riemann Zeta Function: trivial zeros and non-trivial zeros. • We are concerned with the non-trivial zeros, which are imaginary.
Has It Been Solved? • No • There has been no proof that works for this conjecture, as well as no counterexamples. • Proof of this hypothesis is actually number 8 on Hilbert’s problems, and number 1 on Smale’s List. • Basically, this problem is very difficult.
What Has Been Done? • Mathematicians have worked hard on this problem, with some success. In fact, the Clay Mathematics Institute offered a $1 million dollar prize for a proof! • So far, this hypothesis has been proven true for the first 10^13 trillion zeros. • Many mathematicians have shown that 40% of the roots of the function must lie on the critical line, including Weil, Conrey, Levinson and Selberg.
Fun Facts! • In A Beautiful Mind, Russell Crowe’s character tries to solve this problem. • In the show Numb3rs, Charlie says that one of the victims of a kidnapping was taken because he was close to figuring out the proof to this problem.
Back To Actual Math Stuff • This hypothesis actually has some importance in Mathematics, unlike some unsolved problems. • This, if proven false, would “create havoc in the distribution of prime numbers” (Havil).
How Is This Related To Primes? • Euler studied the sum of the reciprocals of the integers raised to the s power. • He noticed that this relates to Bernoulli numbers, such as (π^2)/6. • He then came up with a product that holds true for the real part being greater than 1, and Riemann then derived the Zeta Function. • The imaginary zeros, or the non-trivial zeros, all lie in the critical strip, 0 < Re(s) < 1, and are symmetric about the critical line, which is at Re(s) = ½.
Things To Remember • The problem can be generalized by finding that all of the zeros in the critical region have a real part of ½. • That is, for all s, Re(s) = ½. • This has not been proven or disproven, and is extremely difficult. • If disproven, this would cause major problems with the distribution of the primes. • This function deals with imaginary and real numbers, and the trivial zeros are (-2, -4, -6, ….. -∞) • Prove it and win $1 million dollars.