“On the Number of Primes Less Than a Given Magnitude” Asilomar - December 2009 Bruce Cohen Lowell High School, SFUSD David Sklar San Francisco State University Ver Eric Barkan

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The methods he introduced became the foundation of modern analytic number theory. Georg Friedrich Bernhard Riemann ( ) 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  (x) (the number of primes less than or equal to x). This presentation is an attempt to provide a comprehensible overview of the essential content of Riemann’s paper. Riemann and Prime Numbers

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.) Answer: Primes are important because they may be viewed as the multiplicative “building blocks” of the natural numbers. This idea is embodied in The first few primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29,..., 91, 97, 101 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). Example: The Fundamental Theorem of Arithmetic: Note: this is a step function whose value is zero for real x less than 2 and jumps up by one at each prime. An important function describing the distribution of primes among the natural numbers is

Analytic Number Theory Analytic Number Theory: The study of the integers using the methods of analysis 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.) A central result of analytic number theory concerns the distribution of primes. Specifically it concerns smooth approximations to  (x) for large x. Analysis: The theory of functions of a real or complex variable, especially using methods of the calculus and its generalizations If we say that f (x) is asymptotically equal to g(x) or simply that f (x) is asymptotic to g(x). Notation: means.

Prime Numbers and the Zeta Function One of the most important objects in analytic number theory is the zeta function, usually denoted by  (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: The importance of the zeta function in the theory of prime numbers is due to a remarkable alternative representation of  (s) discovered by Euler in This new version, known today as Euler’s product, takes the form of an infinite product over the prime numbers p: This is in contrast to the situation in Taylor series, in which the independent variable is the base in each term For, Example: Note that here the independent variable s appears as the exponent of each term.

A Brief Outline of Riemann’s Paper He begins with Euler's product representation for  (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. 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: II. Extends the domain of  to include all complex numbers s except A. Shows that the extended  (s) has zeros ( all with ) B. Proves, states and conjectures certain results about these zeros (including the Riemann Hypothesis) C. Obtains a product representation of  in terms of its zeros D. He obtains log  (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  (s) IV. Evaluates the integral to obtain a formula for J(x) V. Uses Möbius inversion to obtain a formula for  (x) in terms of J(x) VI. Proposes an improved approximation to  (x) in terms of Li(x) VII.

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 p n is 1/n. VVIIIIIVIIIVVIVII? (), 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. ( first movie ) Assuming the Riemann Hypothesis this sum can expressed in terms of the imaginary parts of the zeros of zeta.

Euler’s Product for  (s) Euler noted that this result implies that there are infinitely many primes. 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). In 1737 Euler discovered a remarkable product representation for the zeta function. Specifically, he found (stated in modern terms) for Consider what happens as s approaches 1 (from the right). This would give 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.

and existence implies that every term (except 1) is eventually eliminated. A Derivation of Euler’s Product for  (s) At the k th step in this process all of the terms with denominators n s, 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,

A Derivation of Euler’s Product for  (s) So and

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

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

Riemann’s formula for log  (s) in terms of J(x) and therefore Now, if we define u(x), the unit step function, by then we may write x u(x)u(x) so 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

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

Riemann’s formula for log  (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  (s) in terms of J(x) Using

Riemann’s Formula for J(x) in terms of log  (s) Riemann had now obtained a formula for log  (s) in terms of the prime-power step function J(x): 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  (s) that he discovered. Now, if he could just solve this equation to get J(x) in terms of log  (s) he would have obtained complete information about prime powers (and thus about primes) from an expression involving only the smooth function  (s) and some elementary functions.

Integration Contour for Riemann’s Formula Re Im 1 a

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  (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  s) which has meaning throughout the complex plane, except at s = 1. To actually evaluate the integral, his idea was to write  (s) as a product of simpler functions. Then log  (s), and the integral, would break up into a sum of terms simple enough that the individual integrals could be evaluated more easily.

Before he could write  (s) as a product, he needed to clean it up a bit. Recall that  (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  (s) that is well behaved throughout the complex plane: where  refers 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  (s) that don’t lie on the real line. These are known as the non-trivial zeros of  (s). Riemann’s Plan to Evaluate His Integral

The Zeros of the Zeta Function 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. 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. 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. 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. For,

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). Re Im -2 Trivial Zeros Non-Trivial Zeros Critical Line 1 0 Region Represented by the Infinite Series Extended Region

Riemann’s Plan to Evaluate His Integral This is directly analogous to factorization of a polynomial p  s) in terms of its roots 1, 2, …, n in the form Since is defined and well behaved throughout the complex plane, Riemann could factor it in terms of its zeros  1,  2 … (the nontrivial zeros of zeta):

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

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

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:

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

Riemann’s Formula’s for J and  Riemann pointed out that and used M ö bius inversion to obtain Riemann then suggested an improved version of the Prime Number Theorem

Bibliography [2] T. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, New York, 1976 [8] [1] M. Abramowitz, I. Stegun, Handbook of Mathematical Functions, Dover, New York, 1965 [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 [9] [6] Bernhard Riemann, Gesammelte Werke, Teubner, Leipzig, (Reprinted by Dover Books, New York, 1953.) [7]

END OF MAIN TALK

Oscillations and Li(x r ) (assuming RH)

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

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

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

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

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 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, Note: the sine function plays two roles in this example; it provides the periodic oscillations, and its zeros provide the frequencies. expand in a Fourier-like series

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

Definitions,