Abstract. Keywords: conjecture; divisor function; divisor summatory function; prime numbers; Dirichlet's divisor problem

Transcription

1 A ew cojecture o the divisor summatory fuctio offerig a much higher predictio accuracy tha Dirichlet's divisor problem approach * Wiki-like trasdiscipliary article (Ope developmet iterval: 28 -?) - workig paper preprit [] - Versio. (7.6.28) * Adrei-Lucia Drăgoi [2] (versio. published o: ; last update o: ) Pediatricia specialist with both iterdiscipliary ad trasdiscipliary idepedet research i physical scieces (Bucharest, Romaia), dr.dragoi@yahoo.com * Importat ote: The latest (free) versio of this article ca be dowloaded from this URL. * Abstract This paper presets a ew cojecture o the divisor summatory fuctio (also i relatio with prime umbers), offerig a much higher predictio accuracy tha Dirichlet's divisor problem approach. Keywords: cojecture; divisor fuctio; divisor summatory fuctio; prime umbers; Dirichlet's divisor problem Importat ote (). This atypical URL-rich paper (which maximally exploits the layer of hyperliks i this documet), chooses to use Wikipedia liks for all the importat terms used. The mai motivatio for this approach was that each Wikipedia web-article cotais all the mai referece (icluded as edotes) o the most importat terms used i this paper: it s simply the most practical way to cite etire collectios of importat articles/books without usig a overwhelmig list of footote/edote refereces. The secodary motivatio (for usig Wikipedia hyperliks directly icluded i keywords) was to assure a click-away distace to short ecyclopedic moographs o all the (importat) terms used i this paper, so that the flow of readig to be miimally iterrupted. Importat ote (2). This paper also exploits the advatages of the hierarchic tree-like model of presetig iformatioal cotet which is very easy to be kept updated ad well orgaized. [] Olie paper preprit DOI:.3/RG [2] Adrei-Lucia Dragoi research pages o: ResearchGate, Academia.edu, ViXra, GJoural;

2 2 I. Itroductio ) Itroductio to the divisor fuctio. From the umber theory literature ad give a matrix M = which couts (o its 2d colum) the umber of (trivial plus o-trivial) positive 3 d divisors ( d ) for each (o-zero) atural umber (A5 OIE sequece), d (i simplified otatio ad usually oted d ( ) or τ ( ) ; aka the divisor fuctio d ( ) ( ν )( ν ) ( ν ) = k + with ν, ν 2... ν k beig the expoets of the prime factorizatio [URL2] ν ν2 νk j of = p p2... pk = p ν ; also kow i sigma otatio as the special case σ ( ) = d = d ( ) of σ x ( ) d k j= j x = d [URL] with x beig a real d or complex umber) appears i a umber of remarkable idetities (icludig relatioships o the Riema zeta fuctio ad the Eisestei series of modular forms) ad has some well-kow properties like [URL, URL2, URL3, URL] : a) d < 2, for ay (o-zero) atural umber ; b) I 838, P. G. L Dirichlet showed that the (arithmetic) average umber of divisors d ( av) d k / has the property d d = ( ) + ( γ ) = ( ) ( )( ) l 2 [URL-MathWorld] ad that divisor summatory av av D fuctio (DF) = dk ( = d( av) ) otatio of DF (with the simplified otatio replacig the stadard θ = ) has the property that D = ( ) + ( γ ) + O( ) D( ) d( k) ( ) l 2 for ay atural umber (this predicted by Dirichlet was abbreviated as ( D) or D so that to be distiguished from ad to be compared with the other predicted ( pr) proposed by the cojecture preseted i this paper) [URL, URL2, URL3], with the followig explaatios, defiitios ad otatios: i) Euler Mascheroi (gamma) costat γ (the limitig differece betwee the harmoic series ad the atural logarithm) is predefied as γ k ( ) = lim l = / x.5772 floor( x) (with floor fuctio floor( x ) predefied as the greatest iteger i x ad x beig a real umber)

3 3 ii) the big O (Bachma Ladau or asymptotic) otatio is predefied such as: f ( x) O( g( x)) or f ( x) = O( g( x)) if ad oly if there exists both a real costat c > ad a fiite real umber x, such that f ( x) c g( x) for ay real umber x x O θ of D is the target of Dirichlet's divisor iii) the theta expoet ( ) θ of o-leadig term ( ) problem (DDP) which is: to fid the smallest value of θ (oted θ mi ) for which θmi + t ( ) ( γ ) = ( ) l + 2 O for ay t >. Util preset, it is widely cojectured that coj. θ mi = / ad θ? mi was already formally demostrated to be i double closed iterval [ /, 3/ 6(.39) ] by M.N. Huxley i 23. DDP is oe of the major arithmetical problems still usolved up to preset, but the ew cojecture preseted i this paper offers a practical alterative method to approximate (also i relatio with prime umbers), idepedetly to D ad its predicted ( D). iv) Oe cosequece of ( D) defiitio is that a radomly chose (o-zero) atural umber has a expected umber of divisors d d( av) l ( ) + ( 2γ ) l ( ) ( < 2 ), which implies that ( D) l( k ) l(!). The graph of the ratio d / l( ) (with red liear tred lie added) ad the graph of its absolute error (i base- logarithmic scale) log d / l( ) (which idicates a iterestig oscillatig accuracy of the approximatio d / l( ) ) are preseted ext d / l( ) Figure Itro-a. The values of the ratio d / l( ) for the atural umber,.

4 log d / l( ) Figure Itro-b. The values of log d / l( ) for the atural umber,. ***

5 5 II. ome ew cojectures o the divisor fuctio ) New cojectures o the divisor fuctio. The author of this paper discovered some ew cojectures o = d with relatively high accuracy i predictig ad which ca be used as alteratives to k Dirichlet s D: see ext. 2) Cojecture o. (C). e E = 3 / 2, a approximate equality which becomes progressively more exact with the growth of the atural umber to ifiity. C additioally states that E < for ay atural umber 82. ee the ext graph E = / e Figure C-a. The values of E for the atural umbers, a) Iterestigly, the values of the fuctio log E E to value : the absolute error measured logarithmically), teds to stabilize its values very close to -.5 so that E 3/2.968 for 82 : see the ext graph. (which measures the closeess of ( )

6 log -3 E Figure C-b. The values of log E for the atural umbers, 3) Actually, the value 3/2 x =.968 appears as the real target aroud which E teds to stabilize: see the ext graph of the fuctio log E x log E x Figure C-c. The values of log E x for the atural umbers,

7 7 a) Redefiitio (). Based o the progressive decrease of log E x, C ca be refied ad rewritte 3 / 2 e 3/2 as E = x( =.968) accuracy) the value of ( ) for ay. Note. C allows to rapidly predict (with relative high, such as l ( 3 x ). Defiig the predicted (pr) ( pr) = l 3 x, the ratio ( ) / is graphed below. The absolute error measured pr logarithmically as log ( pr) / is also graphed below (with a red liear tred lie added): from this (secod) graph, oe ca observe that log / log ( ), which is equivalet to ( pr) ( pr) / / ad ( pr) / / (as also see from the graph of ( pr) / ) ( pr) Figure C-d. The values of the ratio ( ) / for the atural umbers pr,

8 log ( pr) / Figure C-e. The values of log ( ) / pr for the atural umbers, b) Redefiitio (2). Based o the additioal property ( ) / / (which is equivalet to pr ( ) ), C ca be refied as l ( 3 x ) / ( / ) ( pr) / / ad atural umber, with. (Re)defiig ( x ) ( ) 3/2 x =.968 ( 2) = l 3 / /, the ratio ( pr2) / ad pr its associated log ( pr2) / is graphed ext.

9 9..5 ( pr2) / Figure C-2a. The values of the ratio ( 2) / for the atural umbers pr, log ( pr2) Figure C-2b. The values of log ( pr2) / for the atural umbers,. c) Redefiitio (3). The graph of log ( pr2) / oe ca also observe that log ( pr2) / log ( ), which is equivalet to ( pr 2) / / ad ( pr2) / / : this implies that l ( 3 ) / ( / ) ( / ) ( ) ( ) 2 x, which is equivalet to l 3 x / / so that a predicted ( pr3) ca be further refied as

10 ( ) ( ) 2 ( ) ( ) 2 ( pr3) = l 3 x / /. C additioally states that the fuctio (sup) = l 3 / / is a superior limit for for ay atural umber 82, so that ( ) = O : see the ext graph of the ratio (sup) > (sup) / ( 82) (sup) / Figure C-3a. The values of the ratio (sup) for the atural umbers /,. d) Redefiitio (). ( pr3) supports further refiig with eve higher accuracies, by usig a accessory fuctio f ( ) /l( ) =, so that ( ) ( ) 2 f ( ) ( pr) = l 3 x / /. e) The predicted (arithmetic) average umber of divisors d = / = l ( 3 x ) / ( / ) 2 ad d ( x ) ( ) 2 ( ) ( )( ) ( ) / l 3 / / f = = (with av pr pr D predictio d = ( ) ( γ ) ( av)( pr3) ( pr3) /l( ) f ( ) = ) ca be compared with ( )( ) l + 2. For example, d ( av )( pr 3) geerates much more accurate av D predictios for d( av) = dk / = / tha d ( av)( D) does: for compariso, the ratios d ( av)( pr3) / d ( av) ad ( av )( D ) / ( av ) d d are graphed ext i red ad blue respectively.

11 d / d ( av)( D) ( av) d / d ( av)( pr3) ( av) Figure C-. A compariso betwee the ratios d ( av)( pr3) / d ( av) (i red) ad d ( av )( D ) / d ( av ) (i blue), to emphasize the much higher accuracy of C whe compared to D i predictig d ( av) for the atural umbers,. f) Remark. C is also idirectly related to the prime-umber theorem because a importat elemet which slows dow the progressive growth of (ad the growth of the expoetial E implicitly, which is cojectured to remai subuitary for ay 82 ) is the frequecy of prime umbers (a frequecy maily defied by the prime umber theorem as / P / l( P ), also based o the primecoutig fuctio P, usually oted π ( ) ) which primes (p) all have d p = 2, a d p value which acts like a brake ad slowig the growth of ad E implicitly. ) Fial coclusio. Cojecture (C) has a major advatage of Dirichlet s estimatio of d ( av) (D) (ad = d implicitly), as C predicts d ( av) (ad implicitly) with much higher accuracy: ( av) ( av)( pr3) ( pr3) = / = l ( 3 ) / ( / ) 2 ad ( ) ( ) 2 ( ) ( )( ) ( ) / l 3 / / f d av pr pr x d x with f ( ) /l( ) =. = =,