Krzysztof Maślanka LI S CRITERION FOR THE RIEMANN HYPOTHESIS NUMERICAL APPROACH

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1 Opuscula Mathematica Vol. 4/ 004 Krzysztof Maślanka LI S CRITERION FOR THE RIEMANN HYPOTHESIS NUMERICAL APPROACH Abstract. There has been some interest in a criterion for the Riemann hypothesis proved recently by Xian-Jin Li[9]. The present paper reports on a numerical computation of the first 3300 of Li s coefficients which appear in this criterion. The main empirical observation isthatthesecoefficientscanbeseparatedintwoparts.oneofthesegrowssmoothlywhile the other is very small and oscillatory. This apparent smallness is quite unexpected. If it persisted till infinity then the Riemann hypothesis would be true. Keywords: Riemann zeta function, Riemann hypothesis, Li s criterion, numerical methods in analytic number theory. Mathematics Subject Classification: M6, Y60.. INTRODUCTION Thedistributionofprimenumbersisanoldprobleminnumbertheory.Itisveryeasy tostatebutextremelyhardtoresolve,seefigure.inhisfamouspaperwrittenin 859 Bernhard Riemann connected this problem with a function investigated earlier by Leonhard Euler. He also formulated certain hypothesis concerning the distribution of complex zeros of this function. At first this hypothesis appeared as a relatively simple analytical conjecture to be proved sooner rather than later. However, future development of the theory proved otherwise: since then the Riemann hypothesis (hereafter called RH) is commonly regarded as both the most challenging and the mostdifficulttaskinnumbertheory[].itstatesthatallcomplexzerosofthezeta function,definedbythefollowingseriesif Re s > ζ(s) = n= n s () 03

2 04 Krzysztof Maślanka and by analytic continuation to the whole plane, are located right on the critical line Re s =.RH,iftrue,wouldshedmorelightonourknowledgeofthedistribution of prime numbers. More precisely, the absence of zeroes of ζ(s) in the half-plane Re s > θimpliesthat(see[5],theorem30) π(x) = li(x) + O(x θ log x) () where π(x) is the number of primes not exceeding x and li(x) denotes logarithmic integral.therefore,thevalue θ = (asriemannconjectured)makesthetheorem usefulsincetheerrortermin()isthesmallestpossible. Fig.. The distribution of prime numbers can be most naturally described using function π(x)whichgivesthenumberofprimeslessthanorequalto x.theargument xcanbe any positive real number and π() gives 0. On small scales π(x) has apparently randomlike behavior. On the basis of extensive empirical material two asymptotes of π(x) were independentlyandalmostsimultaneouslydiscovered: log(x) (A.-M. Legendre, lower smooth x curve)andlogarithmicintegral: li(x) := x dt, x > (C.F.Gauss,uppersmoothcurve) 0 log t. LI S CRITERION In 997 Xian-Jin Li[9] presented an interesting criterion equivalent to the Riemann hypothesis(see Fig. ).

3 05 Li s Criterion for the Riemann Hypothesis... Theorem. RH is true if and only if all coefficients λn := dn n s ln ξ(s) n Γ (n) ds (3) s= are non-negative, where s ζ(s). ξ(s) = (s )π s/ Γ + ΖHzL (4) ä 0 0 ä 0 ä 30 ä 40 ä ReHzL 50 ä ImHzL Fig.. Real and imaginary parts of the zeta-function of Riemann. Vertical lines denote 0 first complex zeros on the critical line Re z = An equivalent definition of λn is (see [9], formula.4): λn = n X ρ ρ (5) where the sum runs over all (paired) complex zeros of the Riemann zeta-function. However, the above definitions of λn are not suitable for numerical calculations. It should be mentioned that we know only one exact value, namely λ which may be expressed in closed form using mathematical constants: λ = X ρ ρ = + γ log 4π = The idea of relating Riemann Hypothesis to the rate of growth of certain sequences of real numbers is an old one. For example, in 96 Marcel Riesz proved that RH is equivalent to the assertion that X ( )k+ xk x/+ǫ, (k )!ζ(k) k= ǫ>0

4 06 Krzysztof Maślanka The same idea, in its formulation very similar to Li s approach, appeared also in 99inalittle-knownpaperbyJerryKeiperinwhichhepresentedcertainnewand efficient method of calculating Stieltjes constants[6]. This method has been later implemented in Wolfram s package Mathematica. Xian-Jin Li was clearly unaware of Keiper s result.(keiper tragically died in a bicycle accident in 995.) In the present paper I shall put forward an effective method for calculating coefficients introduced by Li. The gathered data investigated numerically up to n = 3300 reveals unexpected properties: it contains a strictly growing trend plus extremely small oscillations superimposed on this trend. Thefollowingdecompositionof λ n isalreadyimplicitlygiveninarecentpaper by Bombieri and Lagarias([], Theorem ): λ n = (log(4π) + γ) n n + ( n ( ( ) j) j j ) ζ(j) j= }{{} n ( ) n η j j j= }{{} oscillations trend (6) λ n + λ n Using the language of signal theory(perhaps not very common but sometimes appropriate in number theory) one can say that the decomposition(6) uniquely splits thebehaviorofthesequenceof {λ n }intoastrictlygrowingtrend λ n andcertain tinyoscillations λ n superimposedonit.itmaybeprovedthatthetrendcanbe expressed as λn = d n [ ( ( Γ(n) ds n s n ln π s/ Γ + ))] s (7) s= Itmayalsobeprovedthatthetrendisindeedstrictlygrowingas ntendstoinfinity. Itisevidentthat(7)differsfromthemaindefinition(3)simplybyreplacing ξ(s)by muchsimplerfunction: π s/ Γ( + s/).ontheotherhand,theoscillatorybehavior of λ n isnotsoevident,neverthelessitmaybeinvestigatednumerically.ishallreturn to this decomposition later. The starting point of Li s approach to RH is certain transformation of the complexplaneintoitselfusingthemap s z = /s(whichisaspecialcase ofmöbiustransformation).underthistransformationthehalf-plane Re s > is mappedintotheunitdisk(withthecriticalline Re s = becomingtheunitcircle, seefigs.3and4).thiswasli soriginalidea.however,hewasinspiredbystudying A.Weil sproofofrhforfunctionfieldsoverfinitefieldswherethecriticallineis also transformed into a unit circle[0].

5 07 Li s Criterion for the Riemann Hypothesis Fig. 3. Mo bius transformation of the complex plane used by Li. The lower part is an image of the upper part under s 7 z = /s in which the critical line is mapped into unit circle centered at the origin. (See text for details) ) on a small part of the transformed complex plane containing Fig. 4. Plot of / ζ( z all nontrivial zeroes. Nontrivial zeros are visible as sharp pins. The apparent lack of peaks in the center is an artifact. All complex zeroes are very crowded near z = and the corresponding peaks are increasingly thinner

6 08 Krzysztof Maślanka 3. THE MAIN DERIVATION Ithasbeenknownsincemedievaltimesthattheharmonicseries k= /kdiverges. ThiswasprovedlongagowiththeuseofelementarymethodsbyNicoled Oresme in the 4th century(in his work Questiones super Geometriam Euclidis), and, much later, independently, by Pietro Mengoli(in his book on arithmetic series Novae quadraturae arithmeticae, 650) as well as, using yet another method, by the Bernoulli brothers. A natural question emerges: how fast does this series diverge? It turns out that its divergence is weak, more precisely: logarithmic. The quantitative answer to this question implies the definition of the following famous number called the Euler Mascheroni constant: γ := lim x k log x = 0, (8) k x Itsnaturalgeneralizationisthesequence γ n definedby γ n = ( )n lim n! x k (log (log k)n x)n+ (9) n + k x where γ 0 = γ.thesearetheso-calledstieltjesconstants ).Another similar very usefulsequencedenotedby η n isdefinedby η n := ( )n lim Λ(k) n! x k (log (log k)n x)n+, (0) n + k x where Λ(k) is the so-called von Mangoldt function defined for any positive integer k as: { log pif kisaprime poranypowerofaprime p n Λ(k) = () 0otherwise The von Mangoldt function is related to the Riemann zeta function ζ(s) by ζ (s) ζ(s) n= Λ(n) n s Theabovesequences(9)and(0)areimportantontheirownrightsincethey appear in the Laurent expansions for ζ(s) and its logarithmic derivative around ) It should be noted that the function StieltjesGamma[n] implemented in Wolfram s Mathematica,which employs Keiper s algorithm[6], uses a different convention. It is related to our γ nvia γ n = ( )n StieltjesGamma[n] n!

7 Li s Criterion for the Riemann Hypothesis s =. There are different conventions in the literature when defining these numbers, here I have adopted those of Bombieri and Lagarias[]: ζ(s + ) = s + γ n s n () ζ ζ (s + ) = s + η n s n (3) Integrating the second equation(3) with respect to s, inserting the result into the first one and equating coefficients in the appropriate powers of the variable s, onecanfindexplicitrelationsbetweenthe γ n andthe η n : ( ) s n+ η n n + = log + γ n s n+ (4) η n s n+ n + = k= ( ( ) k ) k s k γ n s n k Nowintroducethecoefficients c (k) n defined by c (k) n s n = ( ) k γ n s n Employing a certain formula from[4](formula 0.34, i.e., raising a power series to an arbitrary integral exponent) one can express the c coefficients by the following recurrence relations: c (k) 0 = γ k (5) c (k) m = m [km (k + )i]γ m i c (k) i mγ i=0 Thematrixofcoefficients cdependson {γ n }: m = 0 m = m = m = 3 m = 4 k = γ 0 γ γ γ 3 γ 4 k = γ0 γ 0 γ γ + γ 0 γ γ γ + γ 0 γ 3... k = 3 γ0 3 3γ0 γ 3γ 0 γ + 3γ 0 γ k = 4 γ0 4 4γ0γ k = 5 γ (6) (In what follows only the upper triangular part of this infinite matrix will be needed, seefig.5)

8 0 Krzysztof Maślanka Fig.5.Signsofthecoefficientsofmatrix c(5)for k = 50with rowsandcolumns labelled as in(6). Little white squares denote plus sign, black squares denote minus sign; grey squares mark unused entries of the matrix Withthehelpof(4)thecoefficients η n mayfurtherbeexpressedusingthe elementsofthematrix cas Fromthiswehave: η n = (n + ) n k=0 ( ) k+ k + c(k+) n k (7) η 0 = γ 0 (8) η = +γ 0 γ η = γ γ 0 γ 3γ η 3 = +γ 4 0 4γ 0 γ + γ + 4γ 0γ 4γ 3 η 4 = γ γ 3 0γ 5γ 0 γ 5γ 0γ + 5γ γ + 5γ 0 γ 3 5γ 4,... Finally,theoscillatingpartsof λ n areexpressibleaspolynomialsinthestieltjes constants: n ( ) n λn = η j. (9) j j=

9 Li s Criterion for the Riemann Hypothesis... Usingnow(8)and(9)wefinallyobtain: λ = γ 0 (0) λ = γ 0 γ 0 + γ λ3 = 3γ 0 3γ 0 + γ γ 3γ 0 γ + 3γ λ4 = 4γ 0 6γ 0 + 4γ 3 0 γ γ + γ 0 γ + 4γ 0γ γ + γ 4γ 0 γ + 4γ APPLICATIONS AND CONCLUSIONS The recurrence formulae(5) together with(7) and(9) allow in principle to computeboth η n and λ n witharbitraryaccuracyforanyvalueof n,butitisclear thatwithincreasing nthenumberoftermsincreasesveryrapidly ).Itwouldbe desirabletosimplifythepolynomialsin(8)and(0),oratleasttorevealsome hiddenregularitiesinthem, butidoubtwhetherthis is possible.the Table demonstrates that it would be even impractical to write down explicit expressions for,say, λ n for ngreaterthan5or0. Table n #oftermsin η n (eq.8) #oftermsin λ n (eq.0) ) Using the On-Line Encyclopedia of Integer Sequences ( njas /sequences/) one can see thatnumber oftermsin η n isrelated to the number ofpartitions of n(partition number, PartitionsP[n] in Mathematica notation), which grows like exp(const n),whereasthenumberoftermsin λ nisequaltothenumberofsums Sofpositive integers satisfying S n(sum[partitionsp[k], {k,,n}] in Mathematica notation). Using conventions of the above mentioned Encyclopedia the first is denoted by A00004 and the second by A06905.

10 Krzysztof Maślanka Usingtheaboveformulae(5),(7)and(0)Ihavecomputedquitealotofinitial valuesof η n and λ n.firstitwasnecessarytotabulatestieltjesconstants γ n with sufficient number of significant digits. In order to obtain these I used Mathematica 5 which can handle arbitrary precision numbers and performs automatically full control of accuracy in numerical calculations. This part of computations took over 60hoursonacomputerwithAMD667MHzprocessor. Themaincalculations(η n and λ n )werealsotimeconsuming(about0hours) and required considerable amount of computer memory(3 56 Mb). In particular, having 000 pre-computed Stieltjes constants, with 800 significant digits each, I calculated000 η n andover3300 λ n. The main conclusion which stems from the above calculations is contained in thefollowingplotsshowingthetrendof λ(fig.6a)andtheoscillatingpartof λ (Fig. 6b). Their sum gives the coefficients which appear in Li s criterion for RH. Note thatthescalesonbothplotsdifferbynearlytwoordersofmagnitude.asmentioned before,itiseasytoshowthatthetrend(7)isstrictlygrowing.therefore,ifthe oscillations were bounded or, at least, if their amplitude would grow with n slower thanthetrend,thenrhwouldbetrue.inotherwords,wehaveanewrhcriterion, which is simply a reformulation the original Li s result, but from the viewpoint of the present paper it has an obvious interpretation. It states that if for all positive integer n λ n λ n thenrhistrue. ThenumericaldatagatheredsofarandpresentedinFigures6aand6bisin itsfavor.ofcourse,oneshouldbearinmindthatinnumbertheorythenumerical evidence, no matter how convincing, may be just illusory. In fact, Oesterlé observed recently(inanunpublishednote,[5])thatifthefirst ncomplexzerosofzetaare located on the critical line, then the Li positivity criterion should hold for about thefirst n Licoefficients(see[],p.44).Therefore,directnumericalsearchfora possible counterexample to RH using Li s criterion is rather a hopeless task. FinallyIwouldliketostressoutthatsofartherearenopublishedextensive tablesofli scoefficients.severalnumericalvaluesof λ n withsmallaccuracyare given in[]. All the numerical data obtained during preparation of this paper as well as appropriate Mathematica notebook are available from the author. Note added in proof. The present paper, initially posted as preprint at e-print archive has since then motivated some further work, cf.[7] and [4]. In particular, Voros, using classic technique of saddle-point method, proved that the trend behaves asymptotically as λn ( + n lnn) + cn where c = (γ ln π).

11 Li s Criterion for the Riemann Hypothesis... 3 a) Λn b) Λ n n n 0 Fig.6.Thetrendof λ n(a)incomparisonwiththeoscillatingpartof λ n(b).notedifferent verticalscales.infact,thesumofthetrendandtheoscillatingpart,i.e.full λ n λ n + λ n, would look exactly like the upper plot since the amplitude of the oscillations is smaller thanthethicknessofthegraphline Acknowledgements IwouldliketoexpressmygratitudetoProf.JeffreyC.Lagarias,AT&TLabs,for confirming the effect of tiny oscillations, as well as for several remarks concerning furtherdevelopmentoftheideapresentedinthispaper.iwouldalsoliketothankdr. Mark W. Coffey, Department of Physics, Colorado School of Mines, for directing my attentiontosomemisprintsaswellasforsendingmeapreliminaryversionofhis paper prior to publication[3].

12 4 Krzysztof Maślanka REFERENCES []BianeP.,PitmanJ.,Yor,M.:ProbabilitylawsrelatedtotheJacobithetaand Riemann zeta functions, and Brownian excursions. Bull. Amer. Math. Soc. 38 (00), [] Bombieri E., Lagarias J. C.: Complements to Li s Criterion for the Riemann Hypothesis. J. Number Theory 77(999), [3] Coffey M.: Relations and positivity results for the derivatives of the Riemann ξ function. J. Comp. Applied Math. 66(004), [4] Gradshteyn I. S., Ryzhik I. M.: Tables of Integrals, Series, and Products. corr. enl.4thed.sandiego,ca:academicpress980. [5] Ingham A. E.: The distribution of prime numbers. Cambridge University Press, 93. [6] Keiper J. B.: Power Series Expansion of Riemann s ξ Function. Math. Comp. 58(99), [7] Lagarias J. C.: Li Coefficients for Automorphic L-Functions. posted at arxiv:math.nt/ , 8 May 004. [8] Lagarias J. C.: private communication, 003. [9] Li X.-J.: The Positivity of a Sequence of Numbers and the Riemann Hypothesis. J. Number Theory 65(997), [0] Li X.-J.: private communication, 003. [] Maślanka K.: February 004. [] Riemann G. B. F.: Ueber die Anzahl der Primzahlen unter eine gegebenen Grösse. Monatsberichte Königl. Preuss. Akad. Wiss. Berlin, 859, 67. [3] Riesz M.: Sur l hypothèse de Riemann. Acta Math. 40(96), [4] Voros A.: A Sharpening of Li s Criterion for the Riemann Hypothesis. submitted to C. R. Acad. Sci. Paris, Ser. I; 5 April 004. [5] Voros A.: private communication, 004. Krzysztof Maślanka maslanka@oa.uj.edu.pl Astronomical Observatory of Jagiellonian University ul. Orla 7, Cracow, Poland Received: May 8, 004.