The Prime Unsolved Problem in Mathematics

Transcription

1 The Prime Unsolved Problem in Mathematics March, 0 Augustana Math Club

2 859: Bernard Riemann proposes the Riemann Hypothesis.

3 859: Bernard Riemann proposes the Riemann Hypothesis. 866: Riemann dies at age 39, having published only 9 papers.

4 859: Bernard Riemann proposes the Riemann Hypothesis. 866: Riemann dies at age 39, having published only 9 papers. 900: The RH makes Hilbert s list of 3 unsolved problems for the 0 th century (only 3 remain).

5 859: Bernard Riemann proposes the Riemann Hypothesis. 866: Riemann dies at age 39, having published only 9 papers. 900: The RH makes Hilbert s list of 3 unsolved problems for the 0 th century (only 3 remain). 000: The RH is chosen as one of 7 Millenium Problems and assigned a $,000,000 bounty.

6 OUTLINE

7 OUTLINE () Where are the PRIME NUMBERS?

8 OUTLINE () Where are the PRIME NUMBERS? () What is the ZETA FUNCTION?

9 OUTLINE () Where are the PRIME NUMBERS? () What is the ZETA FUNCTION? (3) Where are the ZETA ZEROS?

10 OUTLINE () Where are the PRIME NUMBERS? () What is the ZETA FUNCTION? (3) Where are the ZETA ZEROS? the RIEMANN HYPOTHESIS

11 OUTLINE () Where are the PRIME NUMBERS? () What is the ZETA FUNCTION? (3) Where are the ZETA ZEROS? the RIEMANN HYPOTHESIS (4) How does this CONNECT to other areas of math and physics?

12 PART Where are the PRIME NUMBERS?

13 The PRIME COUNTING FUNCTION π()

14 The PRIME COUNTING FUNCTION π()

15 The PRIME COUNTING FUNCTION π()

16 The PRIME COUNTING FUNCTION π()

17 What are the chances that a number of a certain size is prime?

18 What are the chances that a number of a certain size is prime? P( ) ~

19 What are the chances that a number of a certain size is prime? P ~ ) ( P ~ 5 3 ) (

20 What are the chances that a number of a certain size is prime? P ~ ) ( P ~ 5 3 ) ( P ~ 5 3 ) (

21 What are the chances that a number of a certain size is prime? P ~ ) ( P ~ 5 3 ) ( P ~ 5 3 ) ( 3 series: Geometric

22 3 3 ) ( P P ~ 5 3 ) (

23 3 3 ) ( P P 4 3 ~ ) ( P ~ 5 3 ) (

24 3 3 ) ( P P 4 3 ~ ) ( P ~ ln ) ( P ~ 5 3 ) (

25 3 3 ) ( P P 4 3 ~ ) ( P ~ ln ) ( P ln ~ ) ( P ~ 5 3 ) (

26 PRIME NUMBER THEOREM P( ) ~ ln ( ) ~ lnt dt ~ Li( ) ~ ln

27 Testing the Prime Number Theorem Li() π()

28 Testing the Prime Number Theorem Li() π()

29 Testing the Prime Number Theorem Li() π()

30 Testing the Prime Number Theorem Li() π()

31 PART What is the ZETA FUNCTION?

32 P ~ 5 3 ) ( P ~ ln 4 3 ~ ) ( Recall

33 P ~ 5 3 ) ( P ~ ln 4 3 ~ ) ( Recall n p n p numbers whole primes /

34 P ~ 5 3 ) ( P ~ ln 4 3 ~ ) ( Recall n p n p numbers whole primes / n s p s n p /

35 P ~ 5 3 ) ( P ~ ln 4 3 ~ ) ( Recall n p n p numbers whole primes / n s p s n p / (s) ZETA FUNCTION

36 THE ZETA FUNCTION ζ(s) () () 6.6

37 EXTENDING THE ZETA FUNCTION While ζ(s) only converges for s>, there are ways to rewrite it so that: () we still get the same behavior when s> () the function is defined for s< as well

38 EXTENDING THE ZETA FUNCTION While ζ(s) only converges for s>, there are ways to rewrite it so that: () we still get the same behavior when s> () the function is defined for s< as well ( s) n ( ) for s s s n n 0

39 EXTENDING THE ZETA FUNCTION Riemann found a formula for s<0 ( s) s s s sin ( s) ( s)

40 EXTENDING THE ZETA FUNCTION Riemann found a formula for s<0 ( s) s s s sin ( Then he allowed s to be a comple variable : s s) ( iy s)

41 EXTENDING THE ZETA FUNCTION Riemann found a formula for s<0 ( s) s s s sin ( Then he allowed s to be a comple variable : s s) ( iy s) We now call this etended function the RIEMANN ZETA FUNCTION

42 PART 3 Where are the ZETA ZEROS?

43 Graphing comple functions is tricky ( 4) 0

44 Trivial zeta zeros: ( n) 0

45 Cool things happening near the imaginary ais (y) lots of potential zeros =½ seems to be important

46 Nontrivial zeta zeros: ( it ) 0 n t t t t

47 ( s) s s s sin ( s) ( s) ZETA ZEROS

48 non-trivial zeros trivial zeros

49 RIEMANN HYPOTHESIS All nontrivial zeros of ζ(s) lie along the line s it.

50 So far: over 0 trillion nontrivial zeros found, all of which lie on the critical line. But all it takes is a single countereample

51 What did Riemann want with this?

52 What did Riemann want with this? ( ) ( ) Li( ) Prime Counting Function Logarithmic Integral

53 What did Riemann want with this? ( ) ( ) Li( ) Prime Counting Function Logarithmic Integral Prime Number Theorem: Δ() is small

54 What did Riemann want with this? ( ) ( ) Li( ) Prime Counting Function Logarithmic Integral Prime Number Theorem: Δ() is small Riemann constructed an eact formula for Δ() in terms of the zeta zeros!

55 Li() Δ() π()

56 PART 4 How does this CONNECT to other areas?

57 RSA PUBLIC-KEY CRYPTOGRAPHY

58 RSA PUBLIC-KEY CRYPTOGRAPHY () Locate large prime numbers p and q (say, 00 or so digits each).

59 RSA PUBLIC-KEY CRYPTOGRAPHY () Locate large prime numbers p and q (say, 00 or so digits each). () Compute n = p q and publish n as a public key for encrypting messages.

60 RSA PUBLIC-KEY CRYPTOGRAPHY () Locate large prime numbers p and q (say, 00 or so digits each). () Compute n = p q and publish n as a public key for encrypting messages. (3) Decryption requires p and q, a very time-consuming task if the RH is true.

61 QUANTUM MECHANICS

62 QUANTUM MECHANICS () Consider a giant atomic nucleus (say, with thousands of p and n)

63 QUANTUM MECHANICS () Consider a giant atomic nucleus (say, with thousands of p and n) () The interactions are complicated and unknown, but we have had success with random matri models

64 QUANTUM MECHANICS () Consider a giant atomic nucleus (say, with thousands of p and n) () The interactions are complicated and unknown, but we have had success with random matri models (3) The eigenvalues of these giant random matrices turn out to have the same statistical properties as the zeta zeros

65 QUANTUM MECHANICS (4)This suggests that there might be a physical system whose QM energy levels correspond eactly to the zeta zeros.

66 QUANTUM MECHANICS (4)This suggests that there might be a physical system whose QM energy levels correspond eactly to the zeta zeros. (5) If such a system can be constructed, the RH is then a direct consequence of the requirement that these energy levels are real eigenvalues.

67 CONNECTIONS IN MATH Mathematics is more than a collection of facts and calculational techniques. Mathematics, at its most beautiful, shows deep connections between objects, ideas, or systems that seem to be unrelated.

68 CONNECTIONS IN MATH The real reward for proving or disproving the RH is not the yes/no answer itself, but rather how the techniques used will influence many different areas of mathematics.

69

70

71