“On the Number of Primes Less Than a Given Magnitude”

1. “On the Number of Primes Less Than a Given Magnitude” David Sklar San Francisco State University dsklar@sfsu.edu Eric Barkan ebarkan@jps.net Bruce Cohen Lowell High School, SFUSD bic@cgl.ucsf.edu http://www.cgl.ucsf.edu/home/bic Asilomar - December 2009 Ver. 20.00

2. (d) Worksheet

3. -1 1 1 1 1 0 -1 0 -1 0 -1 Worksheet

4. Hence Worksheet

5. Riemann and Prime Numbers Georg Friedrich Bernhard Riemann (1826-1866) was one of the greatest mathematicians of the 19th century. Most of his work concerns analysis and its developments, but in 1859 he published his only paper on number theory. It was entitled “Über die Anzahl der Primzahlen unter einer gegebenen Grösse” (On the Number of Primes less than a given Magnitude). In this eight page paper, he obtained a formula for the function p (x) (the number of primes less than or equal to x). The methods he introduced became the foundation of modern analytic number theory. This presentation is an attempt to provide a comprehensible overview of the essential content of Riemann’s paper.

6. Example: An important function describing the distribution of primes among the natural numbers is What Are Prime Numbers and Why Do We Care About Them? Answer: A Prime number is an integer greater than 1 whose only divisors are 1 and itself. (Note: 1 is not prime.) The first few primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ... , 91, 97, 101 Answer: Primes are important because they may be viewed as the multiplicative “building blocks” of the natural numbers. This idea is embodied in The Fundamental Theorem of Arithmetic: Every integer greater than one can be written as a product of prime numbers in one and only one way (aside from a reordering of the factors). Note: this is a step function whose value is zero for real x less than 2 and jumps up by one at each prime.

9. Notation: means . If we say that f (x) is asymptotically equal to g(x) or simply that f (x) is asymptotic to g(x). Analytic Number Theory Analysis: The theory of functions of a real or complex variable, especially using methods of the calculus and its generalizations Analytic Number Theory: The study of the integers using the methods of analysis A central result of analytic number theory concerns the distribution of primes. Specifically it concerns smooth approximations to p(x) for large x. The Prime Number Theorem: (Note: The theorem was conjectured by Gauss in 1796 and proved in 1896 using methods that were introduced by Riemann 1859.)

11. For , This is in contrast to the situation in Taylor series, in which the independent variable is the base in each term Prime Numbers and the Zeta Function One of the most important objects in analytic number theory is the zeta function, usually denoted byz (s). This notation is due to Riemann, but the function was first studied by Euler in the 1730’s. The most basic definition for this function, and the one first considered by Euler, is in terms of an infinite series: Example: Note that here the independent variable s appears as the exponent of each term. The importance of the zeta function in the theory of prime numbers is due to a remarkable alternative representation of z (s) discovered by Euler in 1737. This new version, known today as Euler’s product, takes the form of an infinite product over the prime numbers p:

12. He begins with Euler's product representation for z(s) (the zeta function). He points out that the sum and product define the same function of a complex variable s where they converge, but “They converge only when the real part of s is greater than 1”. I. II. Then, with the phrase “however, it is easy to find an expression of the function which is always valid” he sketches a theory of the zeta function in which he: Extends the domain of z to include all complex numbers s except A. Shows that the extended z (s) has zeros ( all with ) B. C. Proves, states and conjectures certain results about these zeros (including the Riemann Hypothesis) Obtains a product representation of z in terms of its zeros D. He obtains log z(s) as an integral involving a step function J(x) that jumps at prime powers III. Uses Fourier inversion to express J(x) as an integral involving log z(s) IV. Evaluates the integral to obtain a formula for J(x) V. Uses Möbius inversion to obtain a formula for p(x) in terms of J(x) VI. Proposes an improved approximation to p(x) in terms of Li(x) VII. A Brief Outline of Riemann’s Paper

13. ( ) V VI III IV I II V VI VII ? , a sum over the “mysterious” zeros of zeta is a sum of “oscillating” functions (“the music of the primes”), provides the terms that converge to the jumps. A Ramble Through of Riemann’s Paper Riemann’s main formula is an expression for J (x) a step function that jumps only at prime powers. The height of the jump at pn is 1/n . ( first movie ) Assuming the Riemann Hypothesis this sum can expressed in terms of the imaginary parts of the zeros of zeta.

14. In 1737 Euler discovered a remarkable product representation for the zeta function. Specifically, he found (stated in modern terms) for Euler’s Product for z(s) Euler noted that this result implies that there are infinitely many primes. Consider what happens as s approaches 1 (from the right). This would give Now, the sum on the left is just the harmonic series, which diverges. Hence, the product on the right must have infinitely many factors (or it would be finite). But there is exactly one prime for each factor in the product, so there must be infinitely many primes. This was the first new proof of this fact since Euclid’s proof, 2000 years earlier.

15. A Derivation of Euler’s Product for z(s) At the k th step in this process all of the terms with denominators ns,where the smallest prime factor of n is the k th prime, are subtracted from the sum. The Fundamental Theorem of Arithmetic implies that every n has a unique smallest prime factor. Uniqueness implies that no term is subtracted twice, and existence implies that every term (except 1) is eventually eliminated.

16. A Derivation of Euler’s Product for z(s) So and

17. Now, recall the Taylor series for , Riemann’s formula for log z(s) in terms of J(x)

18. At this stage, Riemann has Riemann’s formula for log z(s) in terms of J(x)

19. Riemann now converts his single sum to an integral essentially as follows. (Actually, we have made the argument slightly more modern, but equivalent, by use of the unit step function). First, note that for so and therefore Now, if we define u(x), the unit step function, by u(x) x then we may write Riemann’s formula for log z(s) in terms of J(x)

20. Using Riemann’s formula for log z(s) in terms of J(x) IV ??

22. Using Riemann’s formula for log z(s) in terms of J(x) IV Since the sum in the integrand is the step function J(x) we have arrived at Riemann’s formula for log z (s) in terms of J(x)

23. Riemann’s Formula for J(x) in terms of log z(s) Riemann had now obtained a formula for log z(s) in terms of the prime-power step function J(x): Now, if he could just solve this equation to get J(x) in terms of log z (s) he would have obtained complete information about prime powers (and thus about primes) from an expression involving only the smooth function z (s) and some elementary functions. He was able to do this by the method of Fourier inversion, yielding The integral on the right is a contour integral along a path in the complex plane. We can’t develop the theories of contour integration and Fourier inversion in this talk, but the details of these theories will not be needed to understand the essentials of what follows. What matters is that Riemann was able to evaluate this integral by using certain special properties of z (s) that he discovered.

24. Im Re 1 a Integration Contour for Riemann’s Formula

25. Riemann’s Plan to Evaluate His Integral Riemann now had his integral formula for J(x): First, note that the path of integration is not the real line. Thus the integrand must be evaluated for complex values of s. The original definition of z (s) as an infinite series would allow a direct evaluation, but is not sufficient to support Riemann’s attack on the integral. He was able to construct an analytic continuation of z (s) which has meaning throughout the complex plane, except at s = 1. To actually evaluate the integral, his idea was to write z (s) as a product of simpler functions. Then log z(s), and the integral, would break up into a sum of terms simple enough that the individual integrals could be evaluated more easily.

26. Riemann’s Plan to Evaluate His Integral Before he could write z (s) as a product, he needed to clean it up a bit. Recall that z (s) blows up at s = 1. This behavior is a problem if one is to factor a function in the way Riemann had in mind. In short, he ended up defining a new function x (s) that is well behaved throughout the complex plane: where Grefers to the Gamma function. The details of this choice were driven by a certain symmetry (the “functional equation”) that Riemann had discovered, as well as by the need for a function well behaved throughout the complex plane. This function has the further property that its only zeros are the zeros of z (s) that don’t lie on the real line. These are known as the non-trivial zeros of z (s).

27. The two specific forms for the zeta function (infinite series and Euler product) are intrinsically incomplete due to the fact that they are only defined for s > 1. In fact, both the series and product blow up for . This is a problem, because a central property of the zeta function is its set of roots or “zeros”, ie. values of s where the function value is zero. However, a glance at either of the two representations reveals that the zeta function is never zero for s > 1 (every term of the sum and every factor of the product is > 0). Thus the zeta function has no zeros for . For , Now, Riemann was able to find an analytic continuation of the zeta function to the entire real line and, in fact, to the entire complex plane, except for the point . The Zeros of the Zeta Function He found that this extended function does have zeros on the negative real axis (the “trivial” zeros) and in the strip of the complex plane with real part between 0 and 1. It is these latter “non-trivial” zeros which play a major part in the theory.

28. Non-Trivial Zeros Im 32.9 30.4 25.0 21.0 14.1 Trivial Zeros Re -2 1 -1 0 Region Represented by the Infinite Series Extended Region Critical Line The Zeta Function in the Complex Plane and The Riemann Hypothesis The Riemann Hypothesis All of the non-trivial zeros of the zeta function have real part 1/2 (ie., they lie on the critical line).

29. Since is defined and well behaved throughout the complex plane, Riemann could factor it in terms of its zeros r1 , r2 … (the nontrivial zeros of zeta): Riemann’s Plan to Evaluate His Integral This is directly analogous to factorization of a polynomial p(s) in terms of its roots l1 , l2, … , ln in the form

30. Riemann’s Plan to Evaluate His Integral At this point Riemann had written his new function x (s) in two ways: He then used these results to eliminate x (s) and to solve for z (s): This is the desired factorization of z (s). It yields immediately

31. Evaluation of Riemann’s Integral So, to evaluate his formula Riemann could substitute and evaluate term by term to get

32. Evaluation of Riemann’s Integral Riemann was a master of complex analysis. He was able to evaluate each of these complex contour integrals with the following results:

33. Riemann’s formula for J(x) Putting all this together he had Or, rearranging slightly

34. Riemann pointed out that and used Möbius inversion to obtain Riemann then suggested an improved version of the Prime Number Theorem Riemann’s Formula’s for J andp

35. Bibliography [1] M. Abramowitz, I. Stegun, Handbook of Mathematical Functions, Dover, New York, 1965 [2] T. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, New York, 1976 [3] Brian Conrey, The Riemann Hypothesis, Notices of the AMS, March 2003 [4] H. M. Edwards, Riemann’s Zeta Function, Dover, New York, 2001 (Republication of the 1974 edition from Academic Press) [5] Andrew Granville, Greg Martin, Prime Number Races, American Mathematical Monthly, Volume 113, January 2006 [6] Bernhard Riemann, Gesammelte Werke, Teubner, Leipzig, 1892. (Reprinted by Dover Books, New York, 1953.) [7] http://www.claymath.org/millennium/ [8] http://en.wikipedia.org/wiki/Prime_number_theorem [9] http://en.wikipedia.org/wiki/Riemann_zeta_function

36. END OF MAIN TALK

37. Oscillations and Li(xr) (assuming RH)

38. J (x) the number of primes x 1/2 the number of prime squares x 1/3 the number of prime cubes x primes less than or equal to are Hence the number of prime squares More generally, the number of prime squares The number of prime nth powers Using p(x) to Count Prime Powers prime squares less than or equal to 225 are Example The same reasoning leads us to the general result, We can use this result to write J(x) in terms of p(x).

39. We have where the Möbius function is defined as Note: These sums contain only finitely many terms, they terminate when since for . Thus the number of terms is . Relating p(x) and J(x) Möbius inversion, a technique from elementary number theory, can be used to write p(x) in terms of J(x).

40. Zeta Between 0 and 1 The sum on the right converges for and provides an analytic continuation of z into the interval between 0 and 1.

41. Speculations About Riemann’s Motivation “Music of the Primes” An example from classical Fourier (harmonic) analysis find a nice smooth approximation, Start with a simple staircase function, subtract to get a periodic “sawtooth function”, expand in a Fourier series

42. Speculations About Riemann’s Motivation “Music of the Primes” In our example from classical Fourier (harmonic) analysis We started with a simple staircase function, found a nice smooth approximation, subtracted to get a periodic “sawtooth function”, expanded in a Fourier series Note: the sine function plays two roles in this example; it provides the periodic oscillations, and its zeros provide the frequencies. By analogy Start with a step function that describes the distribution of primes, find a nice smooth approximation, subtract to get an oscillatory “sawtooth function”, find appropriate oscillatory functions, and a function whose zeros provide the appropriate frequencies, expand in a Fourier-like series

45. ( gamma interpolates the factorial function ) , for all z except 0, -1, -2, -3, … The Gamma Function G , for Re(z) > 0 ( by integration by parts ) ( the functional equation for the gamma function ) The functional equation provides an analytic continuation for the gamma function

46. Definitions ,