Riemann Zeta Function

Transcription

1 Riemann Zeta Function Bent E. Petersen January 3, 996 Contents Introduction The Zeta Function The Functional Equation The Zeros of the Zeta Function Stieltjes and Hadamard Odds and Ends. Euler Relation Bibliography Introduction The notes on the Riemann zeta function reproduced belo are informal lecture notes from to lectures from the graduate complex variable course that I taught years ago. Much of the material is cribbed from the books of Edards and Conay (see bibliography) and, of course, from Riemann s 859 paper on the distribution of primes. The only original mathematics that I can claim is any errors that I may have added. I have not updated the notes except to correct errors. These notes ere prepared using L A TEXε. The original notes, distributed in February of 976, ere duplicated using hand ritten ditto masters. We ve come a long ay in desktop mathematical document preparation! The purpose of these lectures on the zeta function as to illustrate some interesting contour integral arguments in a nontrivial context and to make sure that the students learned about the Riemann hypothesis an important part of our mathematical heritage and culture. Thanks to Mary Flahive for pointing out numerous errors.

2 Riemann Zeta Function The Zeta Function If Re z +ɛ here ɛ>then n k=m k z = n k=m k Re z n k=m k ɛ () implies n z converges uniformly on { z C Re z +ɛ }. Thusthe series ζ(z) = n z () converges normally in the half plane H = { z C Re z> } and so defines an analytic function ζ in H. The function ζ is called the Riemann zeta function. Note substituting nt for t yields Γ(z) = Therefore for Re z>. No e t t z dt = n z e nt t z dt. (3) ζ(z)γ(z) = e nt t z dt (4) e nt = e t e t = ( e t ) (5) if t>. If z = x + iy then e nt t z dt = ( e t ) t x dt. (6) For large t e have ( e t ) e t and for small t e have ( e t ) t. It follos the integral in equation (6) converges if x>. Then by the Fubini Tonelli theorem e may interchange the order of integration in equation (4) (here e think of the summation as an integral relative to the appropriate measure). Thus ( ζ(z)γ(z) = e t ) t z dt (7)

3 Class Notes February for Re z>. Equation (7) is the very first result derived in Riemann s famous 8 page paper Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse, 859. The integral in equation (7) is badly behaved for Re z near since then the integrand behaves roughly as t for small t. Riemann therefore considers a related contour integral here e avoid the origin I(z) = γ (e ) ( ) z d. (8) Figure : The contour for equation (8). Here γ is the contour along the real axis from to δ>, counterclockise around the circle of radius δ ith center at the origin, and then along the real axis from δ to. Wetake to have argument π hen e are going toards the origin and argument π hen e are going toards. (Strictly speaking e should open this contour up a little and then pass to a limit, or else vie it as lying in the appropriate Riemann surface.) The integral (8) converges for all z and defines an entire function. Moreover, by Cauchy s theorem it is independent of the choice of δ>. Note moreover that (e ) has a removeable singularity at the origin and so by Cauchy s theorem I(k) = for k =, 3, 4, (9) (since hen z is an integer, the integrals along the real axis in (8) cancel and so e may regard γ as just the circle of radius δ).

4 4 Riemann Zeta Function No I(z) = + + δ =δ δ ( e t ) z(log(t) iπ) dt e t (e ) ( ) z d ( e t ) z(log(t)+iπ) dt e t. () We cannot use the Cauchy formula to evaluate the middle integral in (), but ith = δe iθ e have d = i dθ and so since (e ) has a removeable singularity at the origin e see the integral is bounded by Cδ z Re.Inparticular the integral goes to as δ provided that Re z>. Thus letting δ e obtain I(z) = ( e πiz e πiz) ( e t ) t z dt () =i sin(πz) ζ(z)γ(z) if Re z>. Recalling the functional equation for the gamma function Γ( z)γ(z) = π sin(πz), () e obtain ζ(z) = Γ( z) I(z). (3) πi No equation (3) has been proved for Re z>, but the right side is analytic in the hole plane, except that Γ( z) has simple poles at z =,, 3,.On the other hand I(z) has zeros at z =, 3,.Thusζ(z) is actually analytic in C {}. Atz = there is at orst a simple pole. We see the pole is actually there by computing the residue lim z (z )Γ( z) I(z) = I() πi πi = πi =δ (e ) d = lim (e ) ( ) =. (4) We have shon Theorem. The zeta function ζ continues analytically to a meromorphic function in C ith a simple pole at z =. The residue at the pole is.

5 Class Notes February Riemann no goes on to deduce the functional equation for the zeta function, but first he remarks that ζ vanishes at the negative even integers. This fact may be seen as follos: if n is an integer then ζ( n) = n! I( n) (5) πi here (by the parenthetical remark folloing equation (9)) I( n) = (e ) ( ) n d γ =( ) n =δ e (6) d n+ No (e ) is analytic in a neighborhood of the origin. Thus e = n= n! B n n (7) for < π. ThenumbersB n defined by equation (7) are called the Bernoulli numbers (though there is some disagreement on ho these numbers should be defined). Since e + (8) is an even function e see that B = / and the other odd Bernoulli numbers all vanish. We can easily compute the even ones: for example B = 6 B 4 = 3 B 6 = 4 B 8 = 3. No by Cauchy s integral formula e have and therefore (n +)! πi =δ e ζ( n) =( ) n B n+ n + for each integer n. It follos ζ( ) = ζ( 4) = ζ( 6) = =. (9) d n+ = B n+ () () These roots are called the trivial zeros of the zeta function. The remaining roots are called the nontrivial zeros or critical roots of the zeta function.

6 6 Riemann Zeta Function The Functional Equation Figure : The contour for equation (). Let γ n be the contour consisting of to circles centered at the origin and a radius segment along the positive reals. The outer circle has radius (n + )π and the innner circle has radius δ<π. The outer circle is traversed clockise and the inner one counterclockise. The radial segment is traversed in both directions. We make the same conventions concerning the argument of as above. If e open the contour a little bit along the real axis e can employ the residue theorem, and then pass to a limit, to obtain (e πi γn ) ( ) z d ( = Res (e ) ( ) z = k= n n,k n ( (πik) z +( πik) z ). k= ),=πik ()

7 Class Notes February Since ( e z log(i) e z log( i)) e obtain πi i z +( i) z = i = ( ) zπi e e zπi i ( πz ) =sin, (e γn ) ( ) z d =(π)z sin (3) ( πz ) n k z. (4) k= No on the circle =(n+)π e have (e z ) is bounded independently of n and e have ( ) z / Re z. Thus if Re z< the integral over the large circle tends to as n. It follos that πi γ (e ) ( ) z ( πz ) d =(π)z sin n z. (5) By equation (3) it no follos that ζ(z) =(π) z sin ( πz ) Γ( z) ζ( z) (6) for Re z<. By uniqueness of analytic continuation equation (6) is valid for all z. Noteζ( z) has a simple pole at z = and roots at the positive odd integers greater than, Γ( z) has simple poles at the positive integers, and sin(πz/) has roots at the even integers. Thus e see explicitly that all the singularities on the right side of equation (6), except z =, are removeable. Equation (6) is Riemann s functional equation for the zeta function. Riemann gives a second proof of the functional equation. Since he is otherise economical in the extreme, this duplication is something of a mystery. Edards comments on this mystery in a footnote in section.6 in his book (see bibliography). The Zeros of the Zeta Function Riemann s paper starts by quoting the folloing result of Euler: ζ(z) = ( ) p z n (7)

8 8 Riemann Zeta Function if Re z>. Here p,p,p 3, is the sequence of prime numbers. To prove this result note that (geometric series) and therefore ( ) p z n = p mz n (8) m= N k= ( ) p z k = n z N,j (9) j= here the integers n N,,n N,,n N,3, are all the integers that can be factored as a product of poers of the primes p,p,,p N. Letting N e obtain equation (7). Since the product (7) contains no zero factors e see ζ(z) if Re z>. (3) Suppose no that ζ(z) =andre z<. Since ζ( z) the functional equation (6) implies ( πz ) Γ( z) sin =. (3) Since Γ has no roots e have z =k here k<is an integer, that is, z is a trivial zero. The strip { z C Rez } is called the critical strip. We have seen that all the nontrivial roots of ζ lie in the critical strip. Suppose no z is in the critical strip and ζ(z) =. Since sin(πz/) esee that ζ( z) =. That is, the nontrivial roots (or critical roots) are symmetric ith respect to the critical line Re z =. Note also, since ζ(z) is real for real z, it is trivial that the roots of the zeta function are symmetric ith respect to the real axis. For his study of the distribution of primes Riemann needs to estimate the numbers of roots of the zeta function in a box in the critical strip symmetric about the critical line. He obtains such an estimate (finally proved in 95 by van Mangoldt) and then remarks that the number of zeros on the critical line in the box is about the same (no one has ever proved this statement). He then goes on to say that it is very likely that all the roots in the critical strip lie on the critical line. This statement is the Riemann hypothesis. Riemann goes on to say:

9 Class Notes February One ould of course like to have a rigorous proof of this, but I have put aside the search for a such a proof after some fleeting vain attempts because it is not necessary for the immediate objective of my investigation. (This translation is from Edard s book.) No, 7 years later, after a great deal of effort by many mathematicians, there is still no compelling reason to believe or to disbelieve the Riemann hypothesis. Hardy in 94 proved that ζ has an infinite number of roots on the critical line. Also in 94 Bohr and Landau proved that in a certain sense most of the critical roots lie on the critical line. That is, if δ>then lim T ( number of roots z ith Im z T, δ +/ Re z number of roots z ith Im z T, Re z ) =. (3) In 9 Hardy and Littleood shoed that the number of roots on the imaginary segment [ /, /+it ]isatleastkt for all large T,forsomeconstantK. In 94 Selberg shoed the number of roots on this segment is a least KT log(t ) for all large T. Selberg s ork implies that a positive fraction (in an appropriate sense) of the critical roots lie on the critical line. The MIT Mathematics Department Alumni Nesletter in February 975 announced that N. Levinson had proved at least /3 of of the critical roots lie on the critical line. During 975 I heard a rumor that Levinson had improved his estimate to at least 98 per cent. On October, 975, Levinson died. Rosser, Yohe and Schoenfeld in 968 shoed that for a certain T the zeta function has 3,5, roots in the box { z C Im z T, Re z } and that all these roots are simple and lie on the critical line. Lehman in 966 had obtained the same result, but just for the first 5, roots. It turns out that certain behavior observed during the calculations may imply that eventually a fe critical roots, not on the critical line, ill be found. No one really knos. Stieltjes and Hadamard From the Euler product formula e have ζ(z) = ( ) p z n (33)

10 Riemann Zeta Function for Re z>. It follos that here ζ(z) = µ(n) n z (34) + if n = + if n is the product of an even number of distinct primes µ(n) = if n is the product of an odd number of distinct primes otherise. (35) Since ζ has a pole at, /ζ has a root at. It turns out that the series (34) actually converges for z = (van Mangoldt 897, de la Vallée Poussin 899). Thus e obtain Euler s curious formula (748) = (36) 7 Let M be the step function defined by M() =, M pieceise constant, M has ajumpµ(n) atn, andm is equal to the average of its left and right limits at each jump. Then for Re z>. ζ(z) = x z dm(x) =z M(x) x z dx (37) If M gros less rapidly than x α for some α> then the second integral above ill converge absolutely for Re z>α. Then, by uniqueness of analytic continuation, e can conclude that ζ has no roots in { z C Re z>α}. In 9 Littleood proved the converse and therefore: Theorem. The Riemann hypothesis is true if and only if for each ɛ> e have lim M(x) x x / ɛ =. (38) In 885 Stieltjes rote to Hermite that he had proved that M(x)x / is bounded for large x and that therefore the Riemann hypothesis is true. Stieltjes, hoever, as unable to recall his proof in later years. Hadamard in 896 published a paper on the zeta function in hich he shos that ζ has no roots on the line Re z =. He says that he is publishing his proof only because Stieltjes proof that there are no zeros in Re z>/ has not yet been published.

11 Class Notes February 976 Stieltjes avait démontré, en conformément aux prévisions de Riemann, que ces zéros sont tous de la forme + ti (le nombre t étant réel); mais sa démonstration n a jamais été publiée, et il n a même pas été établi que la fonction ζ n ait pas de zéros sur la droite Re (s) =. C est cette dernière conclusion que je me propose de démontrer. It no seems likely that Stieltjes had in fact made an error. Odds and Ends. Euler Relation If e form the Dirichlet product e have ζ(z) = d(n) n z (39) here d(n) is the number of divisors of n. We have similar series involving sums of divisors of n or the number of positive integers less than n relatively prime to n. See for example Conay s book. Note and therefore (n) z = z ζ(z) (4) ( z ) ζ(z) = ( ) n+ n z. (4) Thus e have ζ(z) = ( z) ( ) n+ n z (4) here the series converges uniformly, but not absolutely, for Re z δ for any δ>. Equation (4) therefore yields a representation of ζ(z) valid for Re z>.

12 Riemann Zeta Function The equation (4) can be analytically continued explicitly to obtain ζ(z) = ( Γ(z) e t ) t z dt if < Re z< t = ( Γ(z) e t t + ) t z dt if < Re z< (43) These representations are useful for dealing ith ζ(z) andζ( z) forz in the critical strip. If n is an integer then ζ( (n )) = ( ) n B n n (44) by equation (5). By the functional equation (6) Thus e obtain the Euler relation ζ( (n )) = (π) n Γ(n)( ) n ζ(n). (45) Taking n =,, e obtain familiar expressions ζ(n) = (π)n ( ) n+ B n. (46) (n)! n = π 6 and so on. n 4 = π4 9 Bibliography Riemann, B., Ueber die Anzahl der Primzahlen unter einer gegebenen Grósse, 859. In Gesammelte Werke, Teubner, Leipzig, 89. Reprinted by Dover, Ne York, 953. Hadamard, Jacques, Sur la distribution des zéros de la fonction ζ(s) et ses conséquences arithmétiques, Bull. Soc. Math. France 4 (896) 99-.

13 Class Notes February Edards, H. M., Riemann s Zeta Function, Academic Press, Inc., San Diego, Ne York, 974. Titchmarsh,E.C.,The Theory of the Riemann Theta Function, Oxford Univ. Press, London, Ne York, 95. Conay, J. B., Functions of One Complex Variable, Springer Verlag, Ne York, Berlin, 973. Copyright c 996 Bent E. Petersen. Permission is granted to duplicate this document for non profit educational purposes provided that no alterations are made and provided that this copyright notice is preserved on all copies. Bent E. Petersen 4 hour phone numbers Department of Mathematics office (54) Oregon State University home (54) Corvallis, OR fax (54) petersen@math.orst.edu eb: peterseb