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Statistics of Primes (and Probably Twin Primes) Satisfy Taylor's Law from Ecology
Joel E. Cohen
To cite this article: Joel E. Cohen (2016) Statistics of Primes (and Probably Twin Primes) Satisfy Taylor's Law from Ecology, The American Statistician, 70:4, 399-404, DOI: 10.1080/00031305.2016.1173591
To link to this article: http://dx.doi.org/10.1080/00031305.2016.1173591
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INTERDISCIPLINARY Statistics of Primes (and Probably Twin Primes) Satisfy Taylor’s Law from Ecology
Joel E. Cohen The Rockefeller University and Columbia University, New York, NY, USA; Department of Statistics, Columbia University, New York, NY, USA; Department of Statistics, University of Chicago, Chicago, IL, USA
ASBTRACT ARTICLE HISTORY Taylor’s law, which originated in ecology, states that, in sets of measurements of population density, the Received November sample variance is approximately proportional to a power of the sample mean. Taylor’s law has been veri- Revised March V x fied for many species ranging from bacterial to human. Here, we show that the variance ( )andthemean KEYWORDS M x x x ( ) of the primes not exceeding a real number obey Taylor’s law asymptotically for large . Specifically, Generalized Taylor’s law; 2 V(x) ∼ (1/3)(M(x)) as x → . This apparently new fact about primes shows that Taylor’s law may arise in the Power law; Prime number absence of biological processes, and that patterns discovered in biological data can suggest novel questions theorem; Twin prime in number theory. If the Hardy-Littlewood twin primes conjecture is true, then the identical Taylor’s law holds conjecture; Uniform also for twin primes. Taylor’s law holds in both instances because the primes (and the twin primes, given the distribution; Variance conjecture) not exceeding x are asymptotically uniformly distributed on the integers in [2, x]. Hence, asymp- function totically M(x) ∼ x/2, V(x) ∼ x2/12. Higher-order moments of the primes (twin primes) not exceeding x satisfy a generalized Taylor’s law. The 11,078,937 primes and 813,371 twin primes not exceeding 2 × 108 illustrate these results.
1. Introduction Taking the square root of both sides, the coefficient of varia- 1 tion of the primes not exceeding x, cv(x) ≡ V (x) 2 /M(x), there- The empirical pattern that has come to be called Taylor’s law fore satisfies (henceforth TL) originated from ecological studies of the abun- √ √ danceofinsectandplantpopulations(Bliss1941;Frackerand v( ) = = / ≈ . . lim→∞ c x a 3 3 0 5774 (2) Brischle 1944;HaymanandLowe1961; Taylor 1961, 1986). TL x states that, in sets of measurements of population density, the Whymightthismatter?Tobiologists,thisapparentlynew sample variance is usually nearly proportional to a power of the finding confirms again, as the two previous citations have shown samplemean.TLhasbeenverifiedempiricallyandmodeledthe- inothercases,thatTLmayariseforreasonsthathavenothing oretically for population densities of many species ranging from to do with ecological processes such as birth, death, migration, bacterialtohuman.TLalsohasbeenconfirmedandmodeled or competition. To mathematicians, this finding confirms again, for many other nonnegative quantities in fields beyond ecology as physics has long done, the possible payoff of looking in math- (review by Eisler, Bartos, and Kertész 2008), such as agricul- ematics for novel patterns suggested by biology. To statisticians, ture, developmental biology, meteorology, genetics, cancer epi- this finding confirms again the power of probabilistic thinking demiology, HIV/AIDS epidemiology, demography, computer and statistical analysis in bridging diverse disciplines and pro- science, stock market analysis, currency trading, and physics viding essential tools of discovery. (where TL is called variously fluctuation scaling or “big” or Taylor’s law holds because, as we shall show, as x → ,the “giant” or “large” number fluctuations). TL has been observed primes not exceeding a real number x are asymptotically uni- numerically in mathematical structures such as integer parti- formly distributed on the integers in [2, x]. Hence asymptotically tions and integer compositions (Xiao, Locey, and White 2015) as x → , and the absolute value of the Mertens function (Kendal and x x2 Jørgensen 2011). To our knowledge, TL has not previously been M(x) ∼ , V (x) ∼ . (3) demonstrated, numerically or theoretically, for the mean and 2 12 variance of the primes or twin primes. Higher-order moments of the primes not exceeding x satisfy Here, we show that the variance V(x)andthemeanM(x) a generalization (7) of Taylor’s law to higher-order moments. = = of the primes not exceeding a real number x obey Taylor’s law To illustrate, the primes not exceeding x 10 are p1 , = , = , = = π asymptotically for large x. Specifically, 2 p2 3 p3 5 p4 7. For x 10, the number (x) of primes not exceeding x is π(x) = 4, the mean M(x)ofthose b V (x) ∼ a(M(x)) or equivalently 4primesisM(x) = 4.25 = (2 + 3 + 5 + 7)/4, and the variance − = = 2 + 2 + 2 + 2 lim V (x)(M(x)) b = a V(x)ofthose4primesisV(x) 3.6875 (2 3 5 7 )/4– x→∞ 4.252.ThisV(x) is the exact variance (the average squared devi- with a = 1/3, b = 2. (1) ation of each prime from the mean), not the unbiased estimate
CONTACT Joel E. Cohen [email protected] The Rockefeller University and Columbia University, New York, NY . © American Statistical Association 400 J. E. COHEN of√ sample variance. The coefficient of variation of the 4 primes The prime number theorem (e.g., Crandall and Pomerance is 59/17 (ࣈ 0.4518). 2005; Montgomery and Vaughan 2007) states that as nonnega- Anaturalnumberp isdefinedtobeatwinprimeifandonly tive real x →∞, if p is a prime and p+2isaprime.Forexample,3and5areallthe π( ) ∼ ( ). twin primes not exceeding 10. (By this definition, 7 is not a twin x li x (5) + = prime because 7 2 9 is not a prime.) The Hardy-Littlewood The Hardy-Littlewood twin prime conjecture (e.g., Crandall twin prime conjecture (see Equation (6)) specifies a counting and Pomerance 2005, p. 15, their (1.7); Sebah and Gourdon function for the twin primes not exceeding x. Conditional on 2002,p.5,their(2)) states that, for some constant C2 > 0, as the Hardy-Littlewood twin prime conjecture, the twin primes nonnegative real x →∞, not exceeding x are asymptotically uniformly distributed on the π ( ) ∼ ( ). integers in [2, x]and,ifM2(x), V2(x), cv2(x) are the mean, vari- 2 x 2C2li2 x (6) ance, and coefficient of variation of the twin primes not exceed- Approximately C ≈ 0.6601618158 ...(Sebah and Gourdon ing x,then(1), (2), and (3)remainvalidwithM replaced by M2 2 2002, p. 5). Although li2(x) →∞as x →∞,ithasnotyetbeen and V replaced by V2, and a generalized TL also holds asymp- π ( ) →∞ →∞ totically. proved that 2 x as x . We review some elementary information about the uni- form distribution. By definition, a real-valued scalar random variable X has the uniform distribution on [0, x], where 2. Definitions and Prior Results x > 0, if and only if, for every r ∈ [0, 1], Pr{X ≤ rx}= r. If f(x)andg(x) are real-valued functions of a real x, then by defi- If X has the uniform distribution on [0, x], then X has nition f(x)isasymptotictog(x)andwewrite f (x) ∼ g(x) if and mean E(X) = x/2, second raw moment E(X2) = x2/3, vari- only if, as x → ,wehavef(x) → and g(x) → and f(x)/g(x) ance var(X√) = E(X2) − (E(X))2 = x2/12, and coefficient of →1. The relation “∼” is transitive. We write f (x) = O(g(x)) variation 3/3. For m ≥ 0, the uniform distribution on [0, x] if and only if, for some real C > 0andallrealx (possibly in a has mth raw moment E(Xm) = xm/(m + 1) and mth central specified subset), | f (x)|≤C|g(x)|.Thesymbol“ࣕ”means“is moment E((X − x/2)m) = 0ifm is an odd integer (since defined as.” the uniform distribution is symmetric about its mean x/2) and (( − / )m) = m/ = We now define TL formally, in three variants: exact (=), E X x 2 x cm if m is an even integer, where cm approximate (ࣈ), and asymptotic (∼). Suppose a nonnegative 2m(m + 1) (which is sequence A058962 in the Online Encyclo- real-valued random variable X(x) whose distribution depends pedia of Integer Sequences). on a parameter x has mean M(x)andvarianceV(x). Let X be Given these moments, it follows that if X is uniform on [0, x], > ( ) = ( 1 )( ( ))2 the set (finite or infinite) of possible values of x. Then, by defi- then for all x 0, TL holds exactly, that is, V x 3 M x , nition, the family {X(x)}xࢠX satisfies TL if and only if there exist and if j > 0, k ࣙ 0, then the generalized TL holds exactly, that two constants independent of x,namely,a > 0andrealb,such is, b b that V(x) = a(M(x)) exactly for all x,orV (x) ≈ a(M(x)) k ( ) ∼ ( j + 1) j k approximately (with error term unstated but small), or V x E(Xk) = (E(X j )) j . (7) a(M(x))b asymptotically as x → . k + 1 By a natural extension (Giometto et al. 2015), a family Suppose Y has the uniform distribution on [2, x], that is, for = ࣈ {X(x)}xࢠX satisfies a generalized TL in three variants ( , , fixed x > 2andeveryr ∈ [0, 1], Pr{Y − 2 ≤ r(x − 2)}= r.If ∼ ) if and only if there exists a set of real pairs (j, k)including X has the uniform distribution on [0, x], then, as x →∞,every = = j 1, k 2, and for each such pair (j, k) there exist real con- moment of X asymptotically equals the corresponding moment > ( ( ))k stants a jk 0andb jk independent of x such that E[ X x ] of Y because Pr{X ∈ (0, 2)}→0. For example, E(Y ) = 2 + ( ( ( ))j )b jk = and a jk E[ X x ] are equal exactly ( )orapproximately (x − 2)/2 = 1 + x/2 ∼ E(X) = x/2. (ࣈ)orasymptotically(∼). TL is the special case j = 1, k = 2of the generalized TL. Giometto et al. (2015)gavetheoreticaland ecological examples of a generalized TL. 3. Analytical Results for Primes and Twin Primes N = Now we turn to primes and twin primes. Let {1, 2, 3, …} We now interpret the primes and twin primes not exceed- O = N be the natural numbers, {0} be the nonnegative inte- ing a real number x ∈ R as families of random variables. R = R = , ∞) P = = , = 2 gers, + [0, )and 2 [2 .Let { p1 2 p2 For every x ∈ R and for every p ࢠ P(x), define X(x)to , = , = ,... P = 2 3 p3 5 p4 7 }betheprimesand 2 {3,5,11, be the random variable that takes the value p ࢠ P(x)with ={ ∈ P| + ∈ P} ∈ 17, …} p p 2 be the twin primes. For x probability 1/π(x). (Recall that π(x) is the number of primes R P( ) ≡{ ∈ P| ≤ }, P ( ) ≡{ ∈ P | ≤ } +, define x p p x 2 x p 2 p x . not exceeding x.) Then for any r ∈ [0, 1], the cumulative For any finite set S, define #S as the number of elements inS.By distribution function of X(x) is, by definition, F ( )(rx) = π( ) ≡ P( ) X x definition, x # x isthenumberofprimesnotexceeding Pr{X(x) ≤ rx} and the mean and variance of X(x)areM(x) ≡ x,andπ2(x) ≡ #P2(x) is the number of twin primes not exceed- 2 E(X(x)) = (p1 +···+ pπ(x))/π(x) and V (x) ≡ E(X(x) ) − ing x.If0ࣘ x < 2, then π(x) = π2(x) = 0. ( ( ( )))2 = ( 2 +···+ 2 )/π( ) − ( ( ))2 E X x p1 pπ(x) x M x . For x ∈ R2, the logarithmic integrals are defined by Similarly, for every x ∈ R2 and for every p ࢠ P2(x), define X2(x) to be the random variable that takes the value p ࢠ x x dt dt P2 π 2 ∈ , ( ) ≡ , ( ) ≡ . (x)withprobability1/ (x). Then for any r [0 1], the li x li2 x 2 (4) 2 log t 2 log t cumulative distribution function of X2(x) is defined to be THE AMERICAN STATISTICIAN: INTERDISCIPLINARY 401
( ) = { ( ) ≤ } ( ) FX2 (x) rx Pr X2 x rx and the mean M2 x and variance Similarly, for the twin primes, define V (x) of X2(x) are defined similarly. 2 x t · dt x dt μ (x) ≡ , (10) Theorem 1. The primes not exceeding x are asymptotically uni- 2 2 2 ∈ , →∞ ( ) ∼ 2 log t 2 log t form on [2, x], that is, for any r [0 1], as x , FX(x) rx x t2 · dt x dt r. σ 2( ) ≡ −(μ ( ))2. 2 x 2 2 2 x (11) Proof. For any x ∈ R2, the number of primes not exceeding x 2 log t 2 log t π( ) ∈ , is x , by definition. Therefore, for any r [0 1], the number → π( ) Corollary 2. As x , of primes not exceeding rx is rx . Therefore the fraction of primes not exceeding x that do not exceed rx is π x2 x M(x) ∼ μ(x) ∼ ∼ , (12) #P(rx) π(rx) π(x) 2 FX(x)(rx) = = . #P(x) π(x) V (x) ∼ σ 2(x) . (13) By the prime number theorem and Lemma A.1 (see →∞ π( ) ∼ ( ) ∼ / Moreover, if the Hardy-Littlewood twin prime conjecture (6) the Appendix), as x , x li x x log x. Therefore holds, then π(rx) ∼ (rx)/ log(rx) and (π( ))2 ( ) ∼ μ ( ) ∼ C2 x ∼ x, rx M2 x 2 x π ( ) (14) π(rx) log rx r log x 2 x 2 ∼ = → r as x →∞. π(x) x log x + log r ( ) ∼ σ 2( ) , V2 x 2 x (15) log x 3 C2(π(x)) Theorem 2. If the Hardy-Littlewood twin prime conjecture (6) ∼ 1. (16) π (x) π(x2) holds,thetwinprimesnotexceedingx are asymptotically uni- 2 form on [2, x]. Proof. As M(x) and μ(x) are both asymptotically x/2, they are asymptotic to one another, and similarly for the other moments. Proof. For any x ∈ R , thenumberoftwinprimesnotexceed- 2 From the definition (8)andLemmaA.1, ing x is π2(x), by definition. Therefore, for any r ∈ [0, 1], the x2 number of twin primes not exceeding rx is π2(rx). Therefore, ( 2) μ( ) ∼ li x ∼ 2logx = x ∼ ( ). the fraction of twin primes not exceeding x that do not exceed x x M x P ( ) π ( ) li(x) 2 # 2 rx = 2 rx →∞ π ( ) ∼ log x rx is .If(6)holds,then,asx , 2 x #P2 (x) π2 (x) ( ) ∼ x ( 2 ) π( 2 ) 2C2li2 x 2C2 2 by Lemma A.1 (the Appendix), and like- li x ∼ x (log x) The prime number theorem also gives ( ) π( ) .This π ( ) ∼ rx li x x wise rx 2C 2 . Therefore, 2 2 (log rx) proves (12). Similarly, from the definition (9) and Lemma A.1,
rx 3 2 2 2 x · π (rx) (log rx) log x li x3 li x2 x 2 2 ∼ = r → r. σ 2( ) ∼ − ∼ 3logx − π ( ) + x ( ) ( ) 2 x x log x log r li x li x x 2 ( )2 log x log x x2 = ∼ V (x) . This proof remains valid regardless of the value of C2 > 0. 12 This proves (13). The proofs of (14)and(15)followthesame Corollary 1. For any x ∈ R,(1), (2), and (3)holdand(7)holds procedure. In the proof of (14), after proving that M (x) ∼ asymptotically (not exactly) for the number X(x)ofprimesnot 2 μ (x) because both are asymptotic to x/2, one notes that exceeding x. If the Hardy-Littlewood twin prime conjecture (6) 2 holds,thesameistrueforthenumberX2(x)oftwinprimesnot x2 ( )2 ( ( ))2 (π( ))2 exceeding x. x = 2 log x ∼ li x ∼ C2 x . x ( ) π ( ) 2 2 2li2 x 2 x (log x) 4. Logarithmic Integral Approximations Finally, (16) follows from dividing (14)by(12). Crandall and Pomerance (2005, pp. 10–11) “note that one useful, albeit heuristic, interpretation of [the logarithmic integral (4)in 5. Numerical Comparisons of Asymptotic Formulas the prime number theorem (5)] is that for random large integers with Exact Counts x the ‘probability’ that x is prime is about 1/ln x.” This perspec- × 8 tive suggests, and rigorous proof provided below will confirm, For 31 values of x selected from 10 through 2 10 ,Icom- an alternative asymptotic expression for the mean and the vari- puted (starting from the “primes” function of Matlab, Release ance of the primes (and twin primes) not exceeding x.Define 2015a) the number, mean, variance, and cv of the primes (twin primes) not exceeding x, the asymptotic approximations to these x · x μ( ) ≡ t dt dt , quantities based on the ratios of logarithmic integrals, and x/2 x ( ) ( ) (8) 2 2 log t 2 log t and x /12. The 31 values of x examined were integral rounded x 2 · x approximations to increments by one-quarter on a log10 scale σ 2( ) ≡ t dt dt −(μ( ))2. 8 8 x ( ) ( ) x (9) from 10 to 10 , plus the geometric mean between 10 and 2 log t 2 log t 402 J. E. COHEN
Table . For selected x, values of the number, mean, variance, and cv of primes not exceeding x, the asymptotic number, mean, variance, and cv based on the logarithmic integral in the prime number theorem, and the limiting expressions given by the right sides of (). The limiting cv, ., is the same for all x. The notation E+nn means multiply by +nn.
Column x π(x) M(x) V(x)cv(x)li(x) μ(x) σ (x) σ (x)/μ(x) x/ x/
. . . . . . . . , . . . . . , . .E+ . . . .E+ . , .E+ ,, , . .E+ . . . .E+ . , .E+ ,, , ,, .E+ . . ,, .E+ . ,, .E+ ,, ,, .E+ . ,, ,, .E+ . ,, .E+ × ,, ,, .E+ . ,, ,, .E+ . E+ .E+
Table . For selected x, values of the number, mean, variance, and cv of twin primes not exceeding x, the asymptotic number, mean, variance, and cv based on the loga- rithmic integral in the twin primes conjecture, and the limiting expressions given by the right sides of (). The limiting cv, ., is the same for all x.
Column x π x M x V x x x μ x σ x σ x μ x x x ( ) ( ) ( )cv( )li( ) ( ) ( ) ( )/ ( ) / / . . . . . . . . , . ,, . . . ,, . ,, , . .E+ . . . .E+ . , .E+ ,, . .E+ . . . .E+ . , .E+ ,, , ,, .E+ . . ,, .E+ . ,, .E+ , ,, .E+ . . ,, .E+ . ,, .E+ × , ,, .E+ . . ,, .E+ . .E+
2 × 108, namely, 10, 18, 32, 56, 100, 178, 316, 562, 1000, 1778, characterize all but the first five of the 31 values of x examined, 3162, 5623, 10,000, 17,783, 31,623, 56,234, 100,000, 177,828, that is, x ࣙ 178, but all these inequalities fail to hold for some 316,228, 562,341, 1,000,000, 1,778,279, 3,162,278, 5,623,413, smaller values of x.ThemeanM(x)oftheprimesnotexceeding 10,000,000, 17,782,794, 31,622,777, 56,234,133, 100,000,000, x satisfies μ(x) Figure . As functions of selected values of x, the (a) number, (b) mean, and (c) vari- Figure . As functions of selected values of x, the (a) number, (b) mean, and (c) vari- ance of primes not exceeding x, according to exact enumeration (marker ×), the ance of twin primes not exceeding x, according to exact enumeration (marker ×), formulas based on the logarithmic integral (marker ), and (for (b) and (c)) the the formulas based on the logarithmic integral (marker ), and (for (b) and (c)) the asymptotic expressions given by () (marker ). Panel (d) tests the power law () asymptotic expressions given by () (marker ). Panel (d) tests Taylor’s law ()by variance of primes mean of primes variance of twin primes mean of twin primes by plotting log( ) as a function of log( ), using plotting log( ) as a function of log( ), the same markers. In (d), the markers are perfectly linear with slope on log-log using the same markers. In (d), the markers are perfectly linear with slope on coordinates. The superposition of all three markers in (b), (c), and (d) confirms the log-log coordinates. The superposition of all three markers in (b), (c), and (d) con- accuracy of the asymptotic expressions. firms the accuracy of the asymptotic expressions. THE AMERICAN STATISTICIAN: INTERDISCIPLINARY 403 Figure . As functions of selected values of x, the coefficient of variation (cv) of the (a) primes and (b) twin primes not exceeding x, according to exact enumeration (marker ×), the formulas based on the logarithmic integral (marker ), and the limiting value . on the right side of () (marker ). that is, the ratio of logarithmic integrals approximates the actual one part in 104 for primes and agrees to better than one part in variance V(x) more closely than the variance of the asymptotic 103 for twin primes. uniform distribution approximates V(x). It is tempting to con- Comparing the 31 values for primes and twin primes shows ࣙ jecture that there exists an integer x0 such that for all x x0,the that, for all selected x ࣙ 10,μ(x)>μ2(x) and σ(x)/μ(x)< preceding inequalities hold. σ2(x)/μ2(x).Foreachselectedx ࣙ 32 (but not for x = Figure 1 plots the number, mean, and variance of the primes 18), M(x)>M2(x), cv(x) Table . For selected x, values of the number of primes and number of twin primes not exceeding x, their asymptotic mean x/, and the asymptotic estimates of their mean π(x2 )/π(x) C (π(x))2/π (x) from ()and 2 2 from (). The last column is asymptotic to from (). x π x π x x π(x2 )/π(x) C (π(x))2/π (x) C (π(x))3/(π (x)π(x2 )) ( ) ( ) / 2 2 2 2 .E+ .E+ . .E+ .E+ . .E+ .E+ . .E+ , .E+ , , . .E+ , , .E+ , , . .E+ , , .E+ , , . .E+ , , .E+ ,, ,, . .E+ ,, , .E+ ,, ,, . .E+ ,, ,, .E+ ,, .E+ ,, ,, .E+ ,,, .E+ ,,, ,, .E+ ,,, .E+ ,,, ,,, .E+ ,,, .E+ ,,, ,,, .E+ ,,,, .E+ ,,,, ,,, .E+ ,,,, .E+ ,,,, ,,,, .E+ ,,,, .E+ ,,,, ,,,, .E+ ,,,,, 404 J. E. COHEN 6. Conclusions and Open Questions Proof. Integration by parts gives By viewing a classic area of number theory in the light of an eco- x tm · dt xm+1 n x tm+1 · dt = + − C, logical pattern, Taylor’s law, we have been led to apparently new n n + n+1 2 log(t) (m + 1) log x m 1 2 t log(t) facts about primes and twin primes. This stimulation of mathe- matics by biology is not a fluke, but a productive strategy (Cohen where C = 2m+1/[(m + 1)(log 2)n]. As x → , the ratio of the second term 2004). This discovery demonstrates, once again, that the bread- on the right to the first term on the right approaches 0. and-butter tools of probability and statistics (e.g., random vari- ables, the cumulative distribution function, expectation, vari- ance) matter in seemingly unrelated areas of pure mathematics. Acknowledgment It also gives one more confirmation that Taylor’s law can show The author thanks Lek-Heng Lim, Carl Pomerance, the editor Nicole Lazar, up where it is least expected. and the associate editor for helpful exchanges and suggestions, and Priscilla Many further mathematical questions arise from these ele- K. Rogerson for help. mentary results, and the tools of probability and statistics may prove useful in addressing these questions as well. How fast Funding does the mean of primes (twin primes) not exceeding x con- verge to x/2? How fast does the variance of primes (twin This work was partially supported by U.S. National Science Foundation primes) not exceeding x converge to x2/12? How fast do the grant DMS-1225529. mean and variance of primes and twin primes converge to the asymptotic formulas based on ratios of logarithmic inte- References gralsforthemeanandvarianceofprimesandtwinprimes? Answers to these questions may follow from a more pre- Bliss, C. I. (1941), “Statistical Problems in Estimating Populations of cise version of the prime number theorem, which bounds the Japanese Beetle Larvae,” Journal of Economic Entomology, 34, 221–232. [399] error of the logarithmic integral approximation (5): for some > ∈ R π( ) = ( ) + ( ( )) ( ) = Bui, H. M., and Keating, J. P. (2010), “On Twin Primes Associ- C 0, for all x 2, x li x O f x ,where f x ated with the Hawkins Random Sieve,” available at http://arxiv.org/ / − / x exp(−C(log x)3 5(log log x) 1 5) (Montgomery and Vaughan abs/math/0607196v3 [404] 2007, p. 194, their eq. (6.28)). Cohen, J. E. (2004), “Mathematics Is Biology’s Next Microscope, Only Bet- Of the inequalities observed numerically in the comparisons ter; Biology Is Mathematics’ Next Physics, Only Better,” Public Library of Science Biology, 12, 2017–2023. [404] between exact counts and asymptotic formulas in the previous Crandall, R., and Pomerance, C. (2005), Prime Numbers: A Computational section, which are true for all sufficiently large x? Perspective (2nd ed.), New York: Springer Science+Business Media. Can any or all of (1), (3), (2)and(7) for primes and their [400,401] analogs for twin primes be derived from Hawkins’srandom sieve Eisler, Z., Bartos, I., and Kertész, J. (2008), “Fluctuation Scaling in Com- or other random models of the primes (Lorch and Ökten 2007; plex Systems: Taylor’s Law and Beyond,” Advances in Physics, 57, 89–142. [399] Bui and Keating 2010)? Fracker, S. B., and Brischle, H. A. (1944), “Measuring the Local Distribution Which of these results can be extended to prime pairs with of Ribes,“ Ecology, 25, 283–303. [399] agappn+1 − pn other than 2 or to other sequences of primes Giometto, A., Formentin, M., Rinaldo, A., Cohen, J. E., and Maritan, A. (Lorch and Ökten 2007;BuiandKeating2010)? (2015), “Sample and Population Exponents of Generalized Taylor’s In addition to these new questions for mathematics is a ques- Law,” Proceedings of the National Academy of Sciences, 112, 7755–7760. [400] tion for further biological research. Do any processes involved in Hayman, B. I., and Lowe, A. D. 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