RELATING THE RIEMANN HYPOTHESIS AND THE PRIMES BETWEEN TWO CUBES

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1 RELATING THE RIEMANN HYPOTHESIS AND THE PRIMES BETWEEN TWO CUBES COPIL, Vlad Faculty of Mathematics-Iformatics, Spiru Haret Uiversity, Abstract I this paper we make a evaluatio for the umber of primes betwee two cosecutive cubes, if we assume the Riema hypothesis There eists at least a prime betwee two cosecutive cubes More precisely, if we deote by N() the umber of primes betwee (-) ad, the N() / log Key-words: distributio of primes, the Riema hypothesis AMS classificatio: A4, M6, 6D07 Itoductio I 85 J Bertrad postulated that for every iteger there is a prime betwee ad Bertrad could ot prove his postulate, but he verified it for all < Chebyshev was the first to prove the cojecture i 85 This result was improved afterwards For eample, i 989, N Costa Pereira proved that if , the iterval, cotais a prime 57 The problem we study here comes from the itet to prove that betwee two cosecutive squares there is at least a prime This cojecture has ot bee either proved or ifirmated util ow L Skula made the hypothesis that for >, each of the sequeces +, +,, + ad + +, + +,, + cotais at least a prime A Schizel made a eve stroger assumptio, statig that for every real umber 8 betwee ad + (log ) there is at least a prime Sierpiski observed that from a result of A E Igham it results that for m fied ad large eough (depedig of m), betwee ad (+) there eists at least m primes, that represets obviously a weaker result tha the oe preseted here I this paper we evaluate the umber of primes betwee (-) ad 5

2 We give ow a list of the otatios used: π ( ) is the umber of primes less tha or equal to ; f ( ) ( ( )) = o g, if lim = 0 (where g( ) 0 for large values of ); f ( ) g( ) f ( ) g( ), if lim = (where g( ) 0 for large values of ); f ( ) g( ) s Γ () s = e d, where s > The relatio betwee the Riema ζ fuctio ad the distributio of primes The Riema ζ fuctio is defied as ζ () s = s =, where s is the comple variable s= σ + it, σ,t, σ > The ζ fuctio is aalytic ad has a aalytic cotiuatio for σ > 0, eceptig the poit s =, give by the formula s {} ζ () s = s d s+ s I s = the ζ fuctio has a simple pole with the residue Whe σ < 0 the aalytic cotiuatio of ζ is give by the formula s s π s ζ() s = π si Γ( s) ζ( s) It results that all o-positive eve umbers are roots for ζ, roots that are called trivial zeros It was proved that the o-trivial zeros of ζ are i the so called critic strip 0< Re(s) < Riema made the hypothesis that all the zeros of ζ are o the lie Re(s ) =, fact that is kow as the Riema Hypothesis ad that has ot bee proved util ow O the set of the zeros, a order has bee itroduced, as follows: Re(s ) < Re( s ), ad if the real parts are equal, the s s < s if < s if Im( s ) < Im( s ) Va der Lue has proved that the first zeros of ζ have the real part The prime umber theorem states that π ( ) This results takig ito log accout the fact that ζ () s 0, if Re( s) ad s 6

3 As ew zeros of ζ are computed, we obtai more precise iformatio for the prime distributio fuctio π by gettig fier iequalities For istace, after the first 5,000 zeros of ζ have bee computed, i 98, J B Rosser has proved that for 55 we have < π ( ) < log + log 4 I 96, after computig the first,50,500 zeros Rosser ad Schoefeld have proved that π ( ) <, for > e ad log π ( ) >, for 67 log The mai result Assumig the Riema hypothesis, L Schoefeld proves that π ( ) li( ) < log 8 for 657 () π ε dt dt where li( ) = lim ε 0 + log t log t 0 + ε ε > 0 We deote N ( ) the umber of primes betwee ( ) ad We prove the followig: Theorem If Hypothesis is a positive iteger, the N ( ), assumig the Riema log π Proof: From () we have li( ) log ( ) li( ) log, for π π < < + 8π It results that π ( ) > li log li ( ) ( ) log( ) 8π 8π = ( ) ( ) ( ) ( ) dt = log ( ) log( ) > log t 8π 8π ( ) 7

4 dt ( ) > log > log = log t 4π log 4π ( ) + = log = log, log 4π log log 4π for ( ) 657, therefore 5 Thus N ( ) > log () log log 4π We evaluate the last two terms i () We have = o ad log log log = o log Therefore, it results that N ( ) > + o () log log From () it also results that π ( ) π (( ) ) < li ( ) + log li (( ) ) ( ) log( ) 8π + 8π = dt = + log + ( ) log( ) < log t 8π 8π ( ) dt ( ) < + log < + log logt 4π log( ) 4π ( ) So we obtaied that N ( ) < + log (4) log( ) log( ) 4π Evaluatig the terms i (4), we get, log( ) log = o log( ) log şi 8

5 log = o log Therefore, we have N ( ) < + o log log From () ad (5) it results that N ( ) log (5) Cosequece Assumig the Riema Hypothesis, there eists at least a prime betwee two cosecutive cubes Proof: If we assume the Riema hypothesis, we have the relatio () N ( ) > log log log 4π I order to prove that betwee two cubes there is at least a prime, it is eough to prove that It is easy to see that for log > 0 (6) log log 4π, we have + 0 log log Therefore, it is eough to prove that 9 > (7) log 0 log 4π The previous relatio is equivalet to > (8) Deote > klog 4 = ad 4 5 log 6π > (9) 5 k = The relatio (9) is equivalet to 6π 9

6 k Let f :[, ), f ( ) = klog We have f '( ) = > 0 for > k, so f is icreasig for > k ; as f( k ) > 0 it results that f( ) > 0 for > k 4 ad, therefore, (8) is true for > k 80 Direct verificatios for lead us to state that betwee two cosecutive cubes there eists at least a prime 4 Remarks Eve assumig the Riema hypothesis, from Schoefeld s iequality () it does ot result that betwee ( ) ad there eists at least a prime because we obtai that π( ) π(( ) ) > log, log π ad the lower boud is o-positive for large eough REFERENCES Bertrad, J, Mémoire sur le ombre de valeurs que peut predre ue foctio quad o y permute les lettres qu elle referme, J L Ecole Royale Polytech, 8, (845), -40 Chebyshev, P, Mémoire sur les ombres premiers, J Math Pures Appl, 7, (85), Copil, V, ad Paaitopol, L, Primes Betwee Two k-th Power of Cosecutive Itegers, To appear Costa, Pereira N, Elemetary Estimate for the Chebyshev Fuctio ad the Möbius Fuctio, Acta Arith, 5, (989), 07-7 Sádor, J, Mitriović, DS, ad Crstici B, Hadbook of Number Theory I, Spriger, (006) Schoefeld, L, Sharper Bouds for the Chebyshev Fuctios θ ( ) ad ψ ( ) II, Mathematics of Computatio, 0, 4 (976), 7-60 Titchmarsh, E C, The Theory of the Riema Zeta-Fuctio, Claredo Press, Oford, (986) 0