A Smorgasbord of Applications of Fourier Analysis to Number Theory

Transcription

1 A Smorgasbord of Applications of Fourier Analysis to Number Theory by Daniel Baczkowski Uniform Distribution modulo Definition. Let {x} denote the fractional part of a real number x. A sequence (u n R is uniformly distributed modulo if # { n N : u n [a, b] } = b a N for every a < b. Let U N be a measure with unit masses at the points u n for n N. Then, Û N (k = N e( ku n where e(θ = e πiθ is the notation attributed to Vinogradov. Theorem. (Weyl s Criterion The following are equivalent:. {u n } is uniformly distributed. For every integer k, ÛN(k = o(n as N 3. If F is properly Riemann-integrable on [, ] then N N F (u n = F (xdx. Sometimes the proof of the first two equivalences is called Weyl s Criterion. The proof can be found in numerous books and, in fact, can be found in Michael Filaseta s class notes for Transcendental number theory which will be offered next semester, Spring 8. Now, the last equivalence follows via a standard measure theory type of argument. Note that second equivalence yields the third for F equal to a characteristic function on an interval. Thus, the same holds for a finite sum of characteristic functions, a simple function. Since the simple functions are dense in L [, ] (hence in C[, ], one can easily deduce that the second and third remarks are equivalent. A pertinent result of Weyl s Criterion is that if θ is irrational, then the sequence (nθ is uniformly distributed modulo. This can be easily seen via the summing of a geometric series and applying the above theorem. Indeed, for every k N e(knθ = e ( k(n + θ e ( kθ e ( kθ e(kθ = O k(. The result and notions above, and those listed below, have been extended to the case of sequences of vectors in R m for any m. In what follows, we will be concerned with just real or complex valued sequences. We now give another consequence of Weyl s Criterion which provides many more examples of sequences which are uniformly distribute mod.

2 Theorem. Let ( f(n be a sequence of real numbers such that f(n = f(n + f(n is monotone as n increases. Also, let f(n = and n f(n =. n n Then, the sequence ( f(n is uniformly distributed mod. For a proof of this result, see []. From this one can easily derive Corollary. (Fejér s Theorem Let f(x be a function defined for x that is differentiable for x x. If f (x tends monotonically to as x and if x x f (x =, then the sequence ( f(n is uniformly distributed mod. Indeed, the mean value theorem shows that for sufficiently large n, f(n satisfies the conditions of the previous theorem. From this particular result, one may show uniform distribution of numerous sequences, for example (i. (αn σ log τ n with α, < σ <, τ arbitrary; (ii. (α log τ n with α, τ > ; (iii. (αn log τ n with α, τ <. In the next section in Theorem 5, we show a generalization of the above corollary and dive deeper into the subject by examining some of the available tools. Exponential Sums To little surprise, the theory of exponential sums began with Gauss. Although, it was Weyl s contributions that later accelerated the development of the subject. These special sums are merely finite Fourier series (i.e. a trigonometric polynomial. Throughout we let P denote a polynomial with real coefficients. The following is a type of exponential sum commonly called a Weyl sum: N e ( P (n. In general, one would take any real valued function f in place of P. In this way, exponential sums form a bridge between complex Fourier Analysis and number theory. One exemplary problem is Waring s problem which asks whether for every natural number n there exists an associated positive integer s such that every natural number is the sum of at most s k th powers of natural numbers (for example, every number is the sum of at most 4 squares, or 9 cubes, or 9 fourth powers, etc.. The affirmative answer, known as the Hilbert-Waring theorem, was provided by Hilbert in 99. Perhaps, Weyl ignited the spark by utilizing exponential sums to answer related questions in number theory. Nevertheless, today it appears that an entire branch of number theory is devoted to this topic alone. Again, the usefulness of Weyl s Criterion continues as one can show that P (n is uniformly distributed modulo if P has at least one irrational coefficient. Before proceeding to some major results in the subject, we introduce the idea of what is sometimes called Weyl differencing. Notice that N e ( P (n = = N m= N h= N+ N e ( P (m P (n = n N h n N N n h= n e ( P (n + h P (n e ( P (n + h P (n = N + R N h= e ( P (n + h P (n where R denotes the real part of a complex number. Notice the reasoning behind the name differencing because if P (x has degree d then P (x + h P (x has degree d. Repeating this differencing procedure

3 d times leaves one with a geometric series which can be easily estimated. In this way, Weyl proved that {P (n} was uniformly distributed modulo. Later van der Corput found a way to improve, or perhaps further the study of, Weyl s differencing. The parameter h above going from to N was somewhat disadvantageous. This was alleviated and moreover generalized in what follows. Theorem 3. (van der Corput Let u,..., u N be complex numbers and H an integer such that H N. Then, N N H u n H(N + H u n + (N + H (H hr u n u n+h Proof. Define u n = for n and n > N. Note that So, the Cauchy-Schwarz inequality yields N H u n (N + H By the note, = (N + H N+ h= H N+ N+ u n h = H N u n = N+ h= u n h. u n h = (N + H h= h= h= N+ r= N+ u n h + (N + H R u n r s= s<r u n s u n r u n s N u n. Lastly, the latter double summation contains terms of the form u n u n+h where n N and h H. Since n is uniquely determined and there exists H h choices for each n, the sum equals N (H h u n u n+h = (H h h= h= u n u n+h. Theorem 4. (van der Corput Let {x n } be a sequence of real numbers. If for every positive integer h the sequence {x n+h x n } is uniformly distributed modulo, then {x n } is uniformly distributed modulo. Proof. For all h, we have that {x n+h x n } is uniformly distributed. Fix a nonzero integer m and let u n = e(mx n. By Theorem 3, for any H N we have that where C = N N e(mx n N + H HN + C h= N h e ( m(x n x n+h (N + H (H h(n h H N. By our hypothesis, we now have that sup Since this holds for every positive integer H, N N N e(mx n H. N e(mx n =. Thus, Weyl s Criterion yields that {x n } is uniformly distributed mod which concludes the theorem. 3

4 Theorem 5. Let k be a positive integer and f(x be a function defined for x which is k times differentiable for x x. If f (k (x tends monotonically to as x and if x x f (k (x =, then the sequence ( f(n is uniformly distributed mod. Proof. We use induction on k. For k =, we use the above theorem of Fejér s. Let f be a function satisfying the conditions of the theorem for k replaced by k +. Assume that statement is true for k. For a positive integer h, set g h (x = f(x + h f(x. Then, g (k h (x = f (k (x + h f (k (x for x x. So, the induction hypothesis yields that ( g h (n is uniformly distributed mod. Applying van der Corput s difference theorem above gives the desired result. In the next section, we will adumbrate one of the key role that exponential sums play in analytic number theory. 3 An Application of Exponential Sum Techniques Conjecture. (the Goldbach conjecture Every even integer 4 is the sum of two primes. Conjecture. (the odd Goldbach conjecture Every odd integer 7 is the sum of three primes. Theorem 6. (Vinogradov Every sufficiently large odd integer is the sum of three primes. Notice that the Goldbach Conjecture states that every even integer 4 can be written as the sum of two primes. This conjecture is stronger than the above theorem. Simply note that if the conjecture were true, then adding three to every even integer represented by two primes yields the theorem. Nevertheless, Goldbach s Conjecture is an arduous, mind-boggling task which still remains open today. But, it was Vinogradov who made significant progress on this open problem by proving the aforementioned theorem. Define r(n to be the number of ways to write n as the sum of three primes. Also, let S n (θ = p n e(pθ where the summation runs over all primes p n. Observe that ( Sn (θ 3 = e ( (p + p + p 3 θ p n p n p 3 n which in turn, as e(nxdx is if n = and for other integers n, gives e( nθ ( S n (θ 3 dθ = p n p n p 3 n e ( (p + p + p 3 nθ dθ = r(n. Thus, we have a nice formula for r(n. We do not wish to rigorously prove the remarkable theorem above, but let us merely shed some light on how the above ideas apply to solving such a problem. The idea of Hardy and Littlewood was to analyze the above integral via cleverly dissecting the interval [, ] into say U n, the union of some very small disjoint intervals, and the rest we call V n. In particular, for every θ V n, and moreover for some integer parameter u Letting θ = a q + ɛ q with gcd(a, q = and ɛ (log n u q n (log n u. I = e( nθ ( S n (θ 3 dθ and I = e( nθ ( S n (θ 3 dθ U n V n clearly r(n = I + I. It can be shown that I n /(log n 3 and I = o ( n /(log n 3. In fact, this establishes Theorem 6. The important tool to bound I is a result from analytic number theory: 4

5 Theorem 7. Let π(n; q, a = #{n x : n is prime, n a (mod q}. If q (log n u and gcd(a, q =, then π(n; q, a = n dt φ(q log t + O( ne c log n ( where c > and φ(q is the number of positive integer q that are relatively prime to q (Euler s phi function. (Note Vinogradov s original proof was more complicated because the uniformity of q was not as large as in the above theorem. Utilizing this fact, one may show that for a q (log n u with gcd(a, q = S n (a/q = µ(q φ(q n dt log t + O( ne c log n where µ( denotes the Möbius function. This follows from breaking the sum S n (a/q into two sums, the first over all p n with p q and the second over all p n with p q. The latter is O(log q which easily gets absorbed into the claimed error term. The first turns out to be = e(ma/q π(n; q, m m q gcd(m,q= in which, upon substitution of the fact in (, the result in ( follows. Next, one notes that S n (a/q + β = n e(jβ [ S n (a/q S n (a/q ]. j= Thus, under the above constraints and for β / one sums by parts, utilizes (, and after one more summation by parts obtains that S n (a/q + β = µ(q φ(q j m e(jβ log j + O( n e c log n. Now, the remainder of the estimation for I is straightforward substitution with careful analysis of the error terms. Much more work is required to prove the desired estimation for I. For an excellent expository of the proof, see [4]. Considerable progress on the odd Goldbach conjecture, the easier case of Goldbach s conjecture, has been made throughout the years. First in 93, assuming the Riemann Hypothesis, Hardy and Littlewood showed that it follows for all sufficiently large integers. Then in 937, as discussed above, Vinogradov removed the dependence on the Riemann Hypothesis, and proved that it is true for all sufficiently large odd integers n (but did not quantify sufficiently large. In 956, Borodzkin found such a bound that worked, namely n Further improvements occurred in 989 when Chen and Wang reduced this bound to 43. Nevertheless, the exponent needs to be reduced dramatically before computers are able to aid in the finitely many cases leftover. References [] C. Caldwell, The Prime Glossary, [] L. Kuipers and H. Niederreiter, Uniform Distribution of Sequencesn, John Wiley and Sons, 974. [3] H. L. Montgomery, Harmonic Analysis as found in Analytic Number Theory, Twentieth century harmonic analysis a celebration (Il Ciocco,, p. 7 93, NATO Sci. Ser. II Math. Phys. Chem., 33, Kluwer Acad. Publ., Dordrecht,. [4] S. Wainger On Vinogradov s estimate of Trigonometric sums and the Goldbach-Vinogradov theorem, Mémoires de la S. M. F., tome 5 (97, p ( 5