Short and fuzzy derivations of five remarkable formulas for primes

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1 SHORT AND FUZZY DERIVATIONS Short ad fuzzy derivatios of five remarkable formulas for primes THOMAS J. OSLER. Itroductio The prime umbers have fasciated us for over 600 years. Their mysterious behaviour has yielded a few ew secrets i the last 50 years, but these secrets have come at a high price. To uderstad fully the mathematics behid these moder results requires that the reader be a epert i fuctios of a comple variable ad the may kow features of the zeta fuctio. Regardless of the difficulties ivolved i precise proofs, there are relatively simple argumets that (vaguely) suggest their truth without eedig much effort. I this paper we provide simple motivatio for several remarkable formulas ivolvig prime umbers. Our reasoig is far from precise. We use a combiatio of eperimetal computer results ad some reasoable ad fuzzy mathematical aalysis. The reader is o doubt used to precise mathematical argumets. Ufortuately, such argumets are ot possible if a very short attempt is made to see a shadow of truth i these deep results. There might be times whe the reader will bear up ad thik that's ot eactly true. Please forgive us, we are oly lookig for some ghost of evidece to suggest the truth of our theorems. To get started, we eed a little otatio. Let p deote the th prime umber. We say that f () is asymptotic to g (N), ad write f (N) g (N), if f (N) lim. For eample, we ca write. A importat N g(n) = N3 + N + 5 N 3 eample is Stirlig's formula for large factorials N! πnn N e N. I our fuzzy thikig, we will at times fid it coveiet to thik of the relatio f (N) g(n) as meaig that the two fuctios are approimately equal for large N. We assume that the reader is familiar with the asymptotic formula for the harmoic umbers N where = logn + γ γ = is kow as Euler's gamma. Usig relatively elemetary mathematics we will fid five remarkable asymptotic formulas. First we will argue the truth of Mertes' theorem for primes p ( p p) e γ log. Secodly we will argue ituitively that the fiite sum of the reciprocals of the first N primes is give by the asymptotic formula N log (log p N ) + M,
2 THE MATHEMATICAL GAZETTE where M (Mertes' costat). Net we will argue that the desity of the primes o the ais ca be estimated by σ () log. For our fourth result we suggest that the umber of primes less tha or equal to, commoly deoted by π (), is give by π () log. This is called the prime umber theorem. Fially, we reaso that the th prime umber ca be approimated by log.. The fiite Euler product (Mertes' theorem) We ow compare the fiite Euler product ad the fiite harmoic series ( p N th harmoic umber) pn. Usig the geometric series we have = ( + + p + p 3 + ). To uderstad the coectio betwee this product ad the harmoic series, we eamie the case where N = 3. We have 3 = = ( )( )( ) = Notice that this last sum is of the form m where m = a 3 b 5 c ad the powers are oegative itegers. Clearly if we have N factors i the Euler product, we get
3 SHORT AND FUZZY DERIVATIONS 3 = m where m = p a p b p c 3 p h N. I other words, m is a atural umber whose prime factorisatio ivolves oly the first N primes. By the uiqueess of prime factorisatio, o term is repeated i this sum. Thus as N approaches ifiity, the Euler product approaches the full harmoic sum. We ow cocetrate o the fiite Euler product with N factors, ad compare it to the fiite harmoic series p N, which from the last aalysis is oly a portio of the series geerated. Sice log p N + γ it seems p N p reasoable to study R(N) =, i the hope that it approaches a log p N costat C as N grows large. Usig Mathematica we geerate the table: So we have N R(N) (covergig to a costat C) , , , ,000, where it appears from our table that C Sice p N C log p N, () log p N + γ, we ca also write this as C pn We have just cojectured Mertes' theorem []. This theorem ca be stated ( p p) as e γ log, where γ = is Euler's costat. I other words the costat C emergig from our table is C = e γ = Of course, our table suggested that R(N).
4 4 THE MATHEMATICAL GAZETTE approaches a costat, but it gave o hit that the costat could be epressed as e γ. We have ot foud ay easy way to suggest this. 3. The sum of the reciprocals of primes Startig with the fiite Euler product umerator ad deomiator by e p ( ) e p e p = to get = ( N ( p p ) e p e ( p ) e p) ( e N p)., we multiply The first product o the right i this last epressio coverges as approaches ifiity. Usig Mathematica we get We ow have ( ) e p = β βe p N p. Takig logarithms we get log ( p ) N From () we have C log p, so we get p N + log β. N + log β log (log p N ) + log C. Thus we ca fially write N log (log p N ) + M, () as N grows large. (Here M = logc logβ log.7807 log ). This shows us that the sum of reciprocals of the primes diverges. The accepted value of M is kow as the Mertes' costat ad is M = The desity of the primes Let σ () deote the desity of the prime umbers o the ais. This N
5 SHORT AND FUZZY DERIVATIONS 5 meas that the umber of primes i the iterval a < < b is give by b a σ () d. If δ () deotes the delta fuctio, the σ () = δ ( ). (Recall that δ () =, ad that.) for = 0 δ () d = 0 otherwise But Now, usig this desity fuctio ad () we ca write N = ( log ) σ () d log (log p N). (3) d log (log p N ). (4) Comparig (3) ad (4) we have some justificatio (however weak) for estimatig the desity fuctio as σ () log. (5) 5. The prime umber theorem Now that we have (5), we ca estimate π (), the umber of primes less tha or equal to. We have from (5) Itegratig by parts we get π () = σ (t) dt π () t log t + (log t) dt = Itegratig by parts agai we get π () log + So for large we have the asymptotic formula π () log t dt. log + (log ) +. (log t) dt log. log. (6) The followig table was costructed with the help of Mathematica.
6 6 THE MATHEMATICAL GAZETTE p N N = π (p N ) π (p N ) log p N π (p N ) , ,000 9, ,88.5,000, ,394 99,895 p N log p N + p N (log p N ) + 6. Estimatig the th prime umber If we set = i (6) we have Therefore ad iteratig we get π ( ) = log, log. log (log ) = log + log log. Sice log log is very small we have our desired formula log. (7) Cotiuig the iteratio we get log + log (log ). Formula (7) is 8 per cet too small for the,000,000,000,000th prime, while the above formula is 3 per cet too large. This completes our heuristic study of the five prime umber asymptotic formulas. I [] Polya gives a heuristic study of formulas related to the twi prime cojecture. Refereces. G. H. Hardy ad E. M. Wright, A itroductio to the theory of umbers (3rd ed.), Claredo Press, Oford (954) p G. Polya, Heuristic reasoig i the theory of umbers, The America Mathematical Mothly, 66 (5) (May 959) pp THOMAS J. OSLER Mathematics Departmet, Rowa Uiversity, Glassboro, NJ 0808 USA