Riema s Hypothesis ad Stieltjes Cojecture Riema s Hypothesis ad Stieltjes Cojecture Dr. Clemes Kroll Abstract It is show that Riema s hypothesis is true by showig that a equivalet statemet is true. Eve more, it is show that Stieltjes cojecture is true. Key words Riema hypothesis, Stieltjes cojecture, Möbius fuctio, Mertes fuctio 1) Itroductio Riema stated his hypothesis i 1859 [1]: the o-trivial zeroes of his zeta-fuctio i the complex plae are all o the lie with real part 1/2. Closely related is the Möbius fuctio µ() [2], [9, page 234] which idicates if there are eve or odd umbers of distict primes ad which ca be used for a equivalet formulatio of the Riema hypothesis (5), precisely: - µ() = 0, if has oe or more repeated prime factors (is ot square-free). - µ() = (-1) k, if is a product of k distict primes. This is: µ() = 1 if there is a eve umber (icludig zero) of distict primes ad µ() = -1 if there is a odd umber of distict primes. Furthermore, the Mertes fuctio M(x) [5, page 370] is summig up the Möbius fuctio: x (1) M(x) = µ(k). Here is a sequece of equatios, aalysis ad theorems aroud the Möbius fuctio that will be used i this paper: µ(k) ( ) - (2) Equatio: lim = 6/π 2 ; asymptotic desity of square-free umbers q() = µ(), [9, page 270]. - (3) Equatios: Similarly [3], [10, page 606], ad usig Iverso s otatio, the asymptotic desities of µ(k)=1 or µ(k)=-1 are: lim ( [µ(k)=1 ]) = 3/π 2 ( ; ad lim µ(k) [µ(k)= 1]) = 3/π 2. ( ) - (4) Equatio: lim = 0; average order of µ, which is equivalet to the prime umber theorem [11, page 64]. - (5) Theorem: M(x) = O(x 0.5+ε ), ε>0, is equivalet to Riema s hypothesis, usig big O otatio [4], [5, page 370], [6, page 47], [7, page 251]. We will make use of this theorem to verify the Riema hypothesis. - (6) Theorem: M(x) = Ω(x 0.5 ); this shows a lower boud, usig Ω otatio [5, page 371]. Cotributed by Dr. Clemes Kroll 1/6
Riema s Hypothesis ad Stieltjes Cojecture - (7) Cojecture: M(x) = O(x 0.5 ); Stieltjes cojecture, implyig Riema s hypothesis [4], [7, page 250]. Dejoy s probabilistic iterpretatio of Riema s hypothesis [8, pages 268f] is ot used for the proof i chapter 3). 2) Outlie of the Proof Step 1: It follows from (2) ad (9) a liear term ad a remaider term O( x) for the summatory fuctio of square-free umbers (big O otatio). Step 2: The same remaider term for the summatory fuctio of the set of umbers with µ(k)=1 ad the set of umbers with µ(k)=-1 ca be cocluded. Step 3: Fially, the equatio for M(x) is created, usig the result of step 2, which shows Riema s hypothesis. Supportig evidece is preseted i chapter 4). 3) Riema s Hypothesis Lookig at the square-free umbers (2), there is a remaider term i [9, page 270]: (8) µ(k) = (6/π 2 ) + O( ), where the sum - usig Titchmarsh s otatio [5, page 370] - is oted as: x (9) Q(x) = µ(k) = (6/π 2 ) x + O( x). The remaider term (big O otatio) takes care of details ot described by the liear term. This does ot imply radomess, it shows that additioal terms are of order O( x). Let us defie with Iverso s otatio: x (10) Q+1(x) = [µ(k) x = 1], the summatory fuctio over all umbers with µ(k)=1. (11) Q-1(x) = [µ(k) = 1], same for all umbers with µ(k)=-1. From (10) ad (11) together with (9) ad (1) there is [10, page 606]: (12) Q(x) = Q+1(x) + Q-1(x), ad: (13) M(x) = Q+1(x) - Q-1(x). From (3) it is kow that Q+1(x) ad Q-1(x) both have liear terms, but I am ot aware of ay publicatio regardig the remaider terms. So there is: (14) Q+1(x) = (3/π 2 ) x + O(f+1(x)), ad: (15) Q-1(x) = (3/π 2 ) x + O(f-1(x)), where both f(x) limit the order of additioal terms. From (9) it is kow that the remaider term of (12) is O( ). Hece O(f+1(x)) ad O(f-1(x)) both are of maximal order O( ), otherwise there would be a cotradictio. From this, (3), (13) ad the calculatio rules of big O we coclude: (16) M(x) = O( x), complyig with (6). This is Stieltjes cojecture ad (16) implies Riema s hypothesis by (5). Hece Riema s hypothesis is true. Cotributed by Dr. Clemes Kroll 2/6
Riema s Hypothesis ad Stieltjes Cojecture (4) Supportig Evidece Part 1 I [7, page 323] while followig Dejoy s probabilistic iterpretatio of Riema s hypothesis it is argued that from a strict 1:1 correlatio betwee umbers with µ(k)=1 ad umbers with µ(k)=-1 the Riema hypothesis follows. Let us have a look o equatios (3) ad (4). They suggest that µ(k)=1 is as frequet as µ(k)=-1. But still there might be a huge deviatio from a strict 1:1 correlatio. (17) Theorem: There is a bijectio betwee the set of odd square-free umbers ad the set of eve square-free umbers. (18) Theorem: There is a bijectio betwee the set of umbers with µ(k)=1 ad the set of umbers with µ(k)=-1. Allocatio of square-free umbers multiply with 2 divide by 2 Set A: all odd square-free umbers with eve umber of prime factors Set B: all eve square-free umbers with odd umber of prime factors 1 µ=1 2 µ=-1 3*5, 3*7,, 5*7, 5*11, 3*5*7*11, 3*5*7*13, bijectio 2*3*5, 2*3*7,, 2*5*7, 2*5*11, 2*3*5*7*11, 2*3*5*7*13, Set C: all odd square-free umbers with odd umber of prime factors Set D: all eve square-free umbers with eve umber of prime factors 3, 5, 7, 3*5*7, 3*5*11, 3*5*7*11*13, 3*5*7*11*17, µ=-1 bijectio 2*3, 2*5, 2*7, µ=1 2*3*5*7, 2*3*5*11, 2*3*5*7*11*13, 2*3*5*7*11*17, (19) Picture Proof: I picture (19) all square-free umbers are allocated ito four sets A, B, C, D. By costructio these four sets do ot share ay commo umbers ad together these four sets cover all square-free umbers. There is a bijectio betwee sets A ad B, ad a bijectio betwee sets C ad D, both implemeted as a multiplicatio with 2 or divisio by 2 respectively. From this there is a bijectio betwee {A C} ad {B D} which delivers theorem (17). There also is a bijectio betwee {A D} ad {B C} which delivers theorem (18) ad is more strict tha (3) which states same asymptotic desity. (4) Supportig Evidece Part 2 (20) Theorem: There is a bijectio betwee the set of odd square-free umbers with a odd umber of prime factors ad the set of odd square-free umbers with a eve umber of prime factors (sets A ad C i picture (19)). (21) Theorem: There is a bijectio betwee the set of eve square-free umbers with a odd umber of prime factors ad the set of eve square-free umbers with a eve umber of prime factors (sets B ad D i picture (19)). Proof: picture (22) shows the square-free umbers geerated with a set of four odd prime umbers. The umber of elemets with a certai umber of prime factors are couted. Cotributed by Dr. Clemes Kroll 3/6
Riema s Hypothesis ad Stieltjes Cojecture For 0, 1, 2, 3, 4 prime factors there are 1, 4, 6, 4, 1 elemets respectively, which is determied by the biomial coefficiets. Numbers of elemets i Set 1 ad Set 2 are the same. Example with primes 3, 5, 7, 11 Set 1: all odd square-free umbers with eve umber of prime factors 0 prime factors: { 1 } = 1 2 prime factors: { 3*5, 3*7, 3*11, 5*7, 5*11, 7*11 } = 6 4 prime factors: { 3*5*7*11 } = 1 Set 2: all odd square-free umbers with odd umber of prime factors 1 prime factor: { 3, 5, 7, 11 } = 4 3 prime factors: { 3*5*7, 3*5*11, 3*7*11, 5*7*11 } = 4 If Set 1 ad Set 2 are created with more ad more odd prime factors, the result stays the same: Set 1 = Set 2. The reaso is the formula for the sum of alteratig biomial coefficiets: (22) Picture k=0 ( k )( 1)k = 0. This formula holds for arbitrary may prime factors (alteratively Set 1 = Set 2 could be show by iductio whe addig the ext odd prime factor) ad thus delivers theorem (20). Takig 2 as oe of the prime factors i the sets delivers theorem (21). From theorems (17), (18), (20) ad (21) it follows: A = B = C = D with the four sets defied i picture (19). (4) Supportig Evidece Part 3 Let S be a ifiite set of umbers: S={a, b, c, }; where a, b, c, are arbitrary umbers. For ease of otatio let us defie a fuctio r that returs k if t is k times icluded i S: (23) r(t, S)=1, if {t} S; r(t, S)=2, if {t, t} S; r(t, S)=k, if {t, t,, t} S, with {t, t,, t} =k ; otherwise r(t, S)=0. (Returs maximum possible k). If all elemets i S are differet, r is a idicator-fuctio, showig if a umber t is i S. Summatory fuctio of r (, ); S0={1, 2, 3,, 30, } = 40 35 30 25 20 15 (, ); = 40 35 30 25 20 15 S 10 5 10 5 S2 0 1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627282930 0 1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627282930 (24) Picture Cotributed by Dr. Clemes Kroll 4/6
Riema s Hypothesis ad Stieltjes Cojecture The summatory fuctio of r is show i picture (24), o the left had side there is a simple example with a set S0 cotaiig atural umbers startig with 1. O the right had side there is a more complex S resultig i r(k, S) = cost + O( ). The dotted lie represets the liear term, the rage betwee the cotiuous lies shows the effect of the big O (illustrative). Summatory r of aother set S2 is show additioally, described below. Now let us check what happes to r(k, S) = cost + O( ) whe all elemets of S are multiplied with some umber c, for example 2 like i picture (19). Let us call this set S2, with S2={a 2, b 2, c 2, }. The there is r(k, S2) = cost*/2 + O( 1/2 ). Recallig the ivariace of big O regardig costat factors results i: (25) r(k, S2) = cost /2 + O( ). Similarly, havig a set S with eve elemets oly like sets B ad D i picture (19) all elemets i S ca be divided by 2 ad S3={a/2, b/2, c/2, }. Like above there is: (26) r(k, S3) = cost 2 + O( ). From (25) ad (26) it is see that for liear scalig operatios o elemets of a set S with r(k, S) = cost + O( ), the remaider term stays i the same order O( ), ad the liear term is re-scaled iversely to the multiplicative factor but still is a liear term. This applies to sets A, B ad to sets C, D i picture (19). This is show for iteger multipliers or divisors. There are multiple more geeral settigs tha used here regardig the multipliers or the fuctios withi the big O. Let us come back to the summatory fuctio of r: r(k, S) ad have two ifiite sets S1 ad S2. Without loss of geerality, it is assumed that the elemets i the sets show up i ascedig order. From this orderig there directly is: (27) r(k, {S1 S2}) = r(k, S1) + r(k, S2). This supports the coclusio regardig the big O terms i (14) ad (15). Fially takig the four sets i picture (19) we have: r(k, A) = (1.5/π 2 ) + O( ); r(k, B) = (1.5/π 2 ) + O( ); r(k, C) = (1.5/π 2 ) + O( ); r(k, D) = (1.5/π 2 ) + O( ). Cotributed by Dr. Clemes Kroll 5/6
Riema s Hypothesis ad Stieltjes Cojecture (5) Refereces [1]: Berhard Riema: Über die Azahl der Primzahle uter eier gegebee Größe. (19. Oktober 1859). I: Moatsberichte der Köigliche Preußische Akademie der Wisseschafte zu Berli. 1860, S. 671 680. Versio i Eglish: [6, page 201] [2]: http://mathworld.wolfram.com/moebiusfuctio.html [3]: oeis.org/wiki/möbius_fuctio [4]: https://e.wikipedia.org/wiki/mertes_cojecture [5]: Edward Charles Titchmarsh: The Theory of the Riema Zeta-Fuctio. Oxford 1986 [6]: P. Borwei, S. Choi, B. Rooey, A. Weirathmueller: The Riema hypothesis. A resource for the afficioado ad virtuoso alike, Caad. Math. Soc., Spriger-Verlag, 2008 [7]: Joh Derbyshire: Prime Obsessio, Riema ad the greatest usolved problem i mathematics, Joseph Hery Press, 2003 [8]: H. M. Edwards: Riema s Zeta Fuctio, Dover Publicatios Ic, Mieola, New York, 1974 [9]: G. H. Hardy ad E. M. Wright: A itroductio to the theory of umbers, Oxford Uiversity Press, 1975 [10]: E. Ladau: Hadbuch der Lehre vo der Verteilug der Primzahle, Leipzig ud Berli, Verlag B. G. Teuber, 1909 [11]: T. M. Apostol: Itroductio to Aalytic Number Theory, New York, Spriger Verlag, 1976 Cotributed by Dr. Clemes Kroll 6/6