Equidistribution and the Riemann hypothesis

Publicacions Matemàtiques, Vol 38 (1994), 51-55 . EQUIDISTRIBUTIO N AND THE RIEMANN HYPOTHESI S H . L HUTSO N Dedicated to the memory of Beth Lynn Tatarsk y A bstract  In this paper we demonstrate the relationship between unifor m distribution modulo 1 of the sequence cp`' , p prime, and the zer o free regions of the Riemann zeta function . A classical theorem of Franel [8], [5] says that each digit in th e j -th place after the decimalpoint in a table of natural numbers raise d to the a -th power appears equally often on the average if a E (0,1 ) . This follows immediately from the uniform distribution modulo 1 of th e sequence c n' , n = 1, 2, 3, . . . for each positive c [8], [3] . We revive thi s result by demonstrating the relationship between uniform distribution modulo 1 of the sequence cp a , p prime, and the zero free regions of th e Riemann zeta function . To acquire insight into the sequence cp a we consider T(x) which i s defined by E 1n(p)  x not a prime powe r Ilf(x) =  p v Çx + 0}+T(x – 0 ) 2  otherwise . W e use von Mangoldt ' s importantformula [3], [6] : x p (l)  T( x) = x - ( ' ( 0 )/(( 0 ) — 1/21n(1 - x2) - ~ - ; x > 1 P where the sum is over thecomplex zeros (p) of the zeta function (C( .$ ) ) and is taken on the order of increasing ~ im(p)I . Our first result is from the folklore on the analytic theory of primes . I t provides the first hint that the problem of equidistribution of a sequenc e of the form {f(p) — [f( p )]} ([x] denotes the integer part of x) is relate d to the zero free regions of ((s) . In the following e(x) = exp (27r2x) . W e also let 7r(x) denotethe number of primes Ç x . p 52  H . L . HUTSO N Theorem 1 . If f (x) is a real function with continuous derivative the n p > e (f( p )) (f(t))dtx J t P—l e(f (t)) dt+0 (In(x) ) (The sum on the right extenás over the complex zeros of ((s) and th e terms are added in the order of increasing ~ Im(p) 1) . Proof : Le t and observe tha t x g(x)  e(f(x) ) ln(x ) (3)  9 ' ( t )T( t ) dt = — E 9( pv ) ln ( p ) + 9 ( x ) T ( x ) p 1 < x (to see Chis use integration by parts and Stieltjes integration) . One thus ha s (4)  g ' (t)T(t) dt  E e ( f  + T(x)e(f(x) ) pu<x  ln(x ) Define ES n (x) _) (i(P " )) pC x and note that the sum on the sum on the right of (4) can be written a s E L E Sn(x l/n ) n rt> 1 The prime number theorem shows tha t – E ES n (x l/n ) m,>1  –ES l (x ) 7r(x)  R-(x ) Thus we hav e (5)  ES1(x)  x '(x)  1 2 g'(t)T(t)dt+P(x)T(x ) 7r(x) +O EQUIDISTRIBUTIDN  5 3 One now substitutes the right hand side of von Mangoldt's formula fo r *(t) (1) in the integral and then integrates term wise (this needs justifica - tion : see [3, page 162] } to get upon integration by parts and replacemen t of Ir(x) by 7r/ 1n(x) (De la Vallee Poussin's error estimate in the prim e number theorem [4] shows that this does no damage to the o term ) ES 1 (x)  e( f (x))(*(x) - x)  x p - 1  + e(f(x))  + 7r(x)  x  P P 1n(x)  e f (t))  ln(x) i x t P-l e(f (t))  1 +  dt —  ix ln ( t )  E  x 2  ln ( t )  dt + 0 (ln(x) ) p (several quantities have been absorbed finto the 4 term) . Again th e aboye stated error term in the prime number theorem shows that th e first two expressions do not change the asymptotic estimate . Finally on e checks that all occurrences of In in the integrals can be eliminated wit h out effecting the error . To see this observe tha t (6) f2 x In(x ) ln(t) ) dt = 1n(x)Li(x) — x + 0(1) = o (1nx) ) The result follows upon using two integrations by parts . The alteratio n of the lower limit of integration is just for convenience and causes n o difficulty . ■ Notethat if the limit as x increases of the first term of (2) is zero and if the limit of the sum can be evaluated term wise then the sequenc e { f (p) - - [f(p)]} is equidistributed . This follows from Weyl's criterion fo r uniform distribution modulo 1 ([2, page 108] } and from the fact tha t Re (p) < 1 . Unfortunately the evaluation of the limit of the sum ter m wise is a delicate matter . Nonetheless we have : Theorem 2 . lf ((s) is non zero for Re(s) > 1 - 2a, o < cx < 1/4 then the sequenc e {cp' - - [cp]} is equidistributed for all positive c and thus, under the state d condition, each digit in the j-th place after the decimal point in a table o f the primes raised to the a-th power appears equally often in the average . Proof : We show that the limit of the right' hand side of formula (2 ) with f(x) = cx' is zero . The first term yields to [7, page 961 . We hav e 1 fe(ct a )dt k C ~ x 54  H . L . HUTSO N far an appropriate constant k . For the second term integrate by parts t o obtai n —e(f ~ x )) E x P 1 e( .Ï~l)) E ~ + E ~ J f x tP~-a–le(f (t)) d t pp where q = a 2 ' n ric . The limit of the first term is zeroby the prime numbe r theorem . The second expression disappears as x increases because th e sum converges . (Recall the ordering of the terms) . Now integrate th e quantities in the remaining expression by parts to obtai n xp+c,—le(f(x))  e (f( 1 )) p(p + a)  x p p(p + a ) E  h PAP + a ) where h is a constant . Under the stated condition on ((s) the absolut e values of the terms in each sum are dominated by quantities which de - crease to zero as x increases and which add to a finite value . (In th e integrals use the trivial estimate) . The final statement follows as in th e proof of Franel ' s classical result . ■ Reference s 1. H . BOHR AND E . LANDAU, Beitráge zur theorie der Riemannsche n Zetafunktion, Math . Ann . 74 (1913), 3-30 . 2. J . DIEUDONNÉ, "Treatise on Ana l ysis, " vol . 2, Academic Press, Ne w York, 1970 . 3. W . J . ELLISON AND M . M . FRANCE, " Prime Numbers, " John Wi - .ley and Sons, New York, 1985 . 4. H . M . EDWARDS, " Riemann ' s Zeta Function," Academic Press , New York, 1975 . 5. J . FRANEL, A propos des tables de logarithmes, Vjschr . Naturf . Ges . Zurich 62 (1917), 286-295 . G . A . E . INGHAM, " The Distribution ofPrime Numbers, " Cambridg e Tracts in Math . 30, CambridgeUniversity Press, London, 1932 . 7 . E . C . TITCHMARSH, " The Theory of the Riemann Zeta -Punction, " Second Edition, Oxford University Press, Oxford, 1988 . (7) p tp + 2a–le(f (t))Ci í P EQUIDISTRIBUTION  5 5 8 . G . PóLYA AND G . SZEGO, "Problems and Theorems in Analysi s 1," Springer - Verlag, Berlin, 1978 . Department of Mathematic s The U .S . Merchant Marine Academ y Kings Point, N . Y . 1102 4 U .S .A . Primera versin rebuda el 3 de Maig de 1993 , darrera versid rebuda el 4 de Febrer de 1994