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Some number-theoretical constantsSome number-theoretical constantsarising as products of rational functions of p overprimesIn September 1999, Pieter Moreeasked me to help with high-precision calculations of someconstants arising in various contexts in elementary and analyticnumber theory. PARI/GP running on a few 333 and 360MHz UltraSPARC-IIi(tm) CPUssoon made short work of them. We pushed the calculations to justbeyond 1000 decimal digits.The basic reference for our method isP. Moree, Approximation of singular series and automata, to appear in Manuscripta Math. (1999); preprint available (DVI).Many of these constants appear with explanations and referenceson SteveFinch's FavoriteMathematical Constants site (abbreviated FMC in what follows),and are cross-referenced to the corresponding pages there.We regard the constants as given in the form of an Euler-typeproduct over rational terms1-f/g with rational coefficients,where the degree of the polynomial g is at least 2 plusthat of f,evaluated at all primes p, or sometimesat almost all primes (e.g. when one factor would vanish forp=2).Products of terms of the shape1+f/g are readily accommodated bymoving the sign into f. What really countsfor the computation, however, is the behavior ofg-f and of g.In its original form, the Euler product converges abysmallyslow. It has been folklore knowledge for some time that it canbe transformed into a product ofpowers of values of the Riemann zeta function prodk>1 zeta(k)-e[k],however, although the convergence tends to be a lot betterdue to the exponential convergence of zeta(k)to 1 as k increases, it is still unsatisfactory,and this product does not converge at all when the exponentse[k] grow too fast.The trick which makes these computations feasible isto compute the contributionsfrom the small and larger primes separately. By choosing appropriate cutoff points,we could obtain the desired 1000-digit accuracy using notmuch more than (typically) 20 or 30 minutes CPU time on a1999-vintage Sun workstation. If several computations of thistype with similar accuracy requirements are to be executed,one can save time by pre-computing the zeta(k)just once to the maximum required precision (and maximum requiredk). For a target precision of 1000 digits,this step takes 15 to 20 minutes; so the gain is considerable.Therefore, we usually ran batches of three or four or five computationssharing an array of pre-computed zeta values.There are some hidden duplications among our constants, in thefollowing sense: One can introduce or remove factors ofp2/(p2-1) in(g-f)/g, and compensate forthis by writing a suitable power of 6/pi2(inverse of zeta(2)) in front of the product.It is not obvious which rational function within such a familyshould be preferred. One can use this transformation to ensurethat the degree of g exceeds that of f by 3 or more,but this does not always lead to the simplest or most naturalshape for the numerator and denominator, and it does not helpto improve convergence or to make the computation faster.The inverse of any constant of this kind is another constantof this kind, with g and g-fswapping rôles.The TablesThis table provides an overview.Table entries are structured as follows: entry number;numerator f, denominator gand starting prime p0 in theoriginal formula prodp (1 - f(p)/g(p)), p >= p0;approximate value hyperlinked to the corresponding entry in thefull-precision table;and a reference hint about the context in which the constant arises.Entries in which f is `positive'(in an obvious sense) come first, followed by entries with `negative'f, and in each group we begin with entries wheref and g are of simple shape:often the numerator is 1, and g is a product offactors p and (p±1), beforeproceeding to other candidates. The exception to this rule is theinfinite sequence of Hardy-Littlewood constants of which the Twin Prime Constant is the first member, and whichtherefore appear immediately after it. The numbering is otherwisearbitrary. # f g p0 valueReference 01 1 p(p-1) 2 0.37395581361920228805 Artin's Constant 02 1 p2(p-1) 2 0.69750135849636590328 Rank 2 Artin Constant 03 1 p3(p-1) 2 0.85654044485354217443 04 1 p4(p-1) 2 0.93126518416000433439 05 1 (p-1)2 3 0.66016181584686957393 Twin Prime Constant 06 3p-1 (p-1)3 5 0.63516635460427120721,2.85824859571922043243 Hardy-Littlewood Constants C3 and D 07 6p2-4p+1 (p-1)4 5 0.30749487875832709312,4.15118086323741575717 Hardy-Littlewood Constants C4 and E 08 10p3-10p2+5p-1 (p-1)5 7 0.40987488508823647448 Hardy-Littlewood Constant C5 10 12p-1 p(p+1)p3 2 0.70444220099916559274,0.42824950567709444022 "Carefree" Constant 11 1 p2(p+1) 2 0.88151383972517077693 Quad. class numbers 12 1 p3(p+1) 2 0.94773326214367537594 13 1 p4(p+1) 2 0.97582415304766824168 14 1 p5(p+1) 2 0.98850439774124690875 15 1 p2-1 2 0.53071182047204479497 see 25 below 16 1 p(p2-1) 2 0.78816250003022070058 17 1 p2(p2-1) 2 0.90192603958708217138 20 13p-2 (p+1)2p3 2 0.77588351000389549962,0.28674742843447873411 "Strongly Carefree" Constant 25 2 p2 2 0.32263409893924467058,0.66131704946962233529 Feller-Tornier Constant 30 p+2 p3 3 0.72364840229820000941,0.74397119335037474469 Sarnak's Constant 32 p p3-1 2 0.57595996889294543964 Stephens' Constant 40 1 p2-p-1 3 0.71546823598995584509 41 1 p2+p-1 2 0.66958029053906236764 42 1 p2-2 2 0.38894518997956192931 45 p9 - (p-1)7(p2+7p+1) p9 2 0.00131764115485317811 Heath-Brown and Moroz's Constant 52 -1 p2(p-1) 2 1.33978415357434724660,2.20385659643785978787 Totient Constant 55 -1 (p-1)2 2 2.82641999706759157555 Murata's Constant 56 -1 (p+1)2 2 1.26655850147152857161 57 -1 p(p2-1) 2 1.23129114888860356277 60 -1 p2-p-1 2 2.67411272557002150896 61 -1 p2+p-1 2 1.41956288050548591932 Inverse of 10 above.ReferencesArtin's Constant 0.37396Density of the set of primes q, relative to the set ofall primes, such that a given positive integer (not a proper powerand with squarefree part incongruent 1 mod 4) is a primitive rootmodulo q.(more on FMC)Rank 2 Artin Constant 0.69750This, as well as the next two 0.85654and 0.93127, are higher analogues ofArtin's Constant, and are related to(among other things) the generation of prime residue class groupsmodulo a prime p by multiplicatively independentsets of r positive integers, a problem firststudied by Keith Matthews. The case r=1 correspondsto Artin's Constant.(The `other things' include generalizations of Artin's Conjectureto number fields of degree r, this being thecontext in which these constants recently starred inHans Roskam'sthesis.)More on FMC and in the papers listed below.K. R. Matthews, A generalisation of Artin's conjecture for primitive roots, Acta Arith. 29 (1976) 113-146; MR 53 #313.F. Pappalardi, On the r-rank Artin conjecture, Math. Comp. 66 (1997), 853-868; MR 97f:11082.L. Cangelmi and F. Pappalardi, On the r-rank Artin conjecture II, J. Number Theory 75 (1999) 120-132.H. Roskam, Artin's primitive root conjecture for quadratic fields, preprint available in PostScript. (1999).Twin Prime Constant 0.66016This is part of a conjectural density formula for the number oftwin primes not exceeding a given bound.(more on FMC)Hardy-Littlewood Constants C3=0.63517, C4=0.30749, C5=0.40987Part of an infinite family of which the TwinPrime Constant is the first member; the general formula forCn has g-f=pn-1(p-n)and g=(p-1)nand p0 the first prime larger thann.For these and the derived constantsD=(9/2)C3 andE=(27/2)C4 see againFMC."Carefree" Constant 0.42825 and "Strongly Carefree" Constant 0.28675Let us call a pair (a,b) of naturalnumbers "carefree" if a is squarefreeand coprime to b, "strongly carefree"if, in addition, b is also squarefree. The sets ofsuch pairs have natural densities 0.42825 and 0.28675, respectively,relative to all pairs of positive integers.Moreover, 0.28675 is also the natural density of pairwise coprimetriples of positive integers (relative to all such triples).See sections 2.7, 4.4, 4.7 in Schroeder's book.(more on FMC)M. R. Schroeder, Number Theory in Science and Communication: With Applications in Cryptography, Physics, Digital Information, Computing and Self-Similarity, 2nd ed., Springer-Verlag 1986. (There is a 1997 edition, too)P. Moree, Counting carefree couples, unpublished manuscript (1999); available from FMC as DVI file.Quadratic class number Constant 0.88151This is related to the average behavior of class numbers of real quadratic fields. See Cohen's book, section 5.10.1, page 290 or 291 (depending on the edition).H. Cohen, A Course in Computational Algebraic Number Theory, Springer-Verlag, 1993; MR 94i:11105.Feller-Tornier Constant 0.32263The density of natural numbers whose prime factorization containsan even number of distinct primes to powers larger than the first(ignoring any prime factors which appear only to the first power)equals (1 + 0.32263...)/2=0.661317....Note that up to a factor 1/zeta(2), this can equally wellbe computed from the entry number 15.W. Feller and E. Tornier, Mengentheoretische Untersuchungen von Eigenschaften der Zahlenreihe, Math. Ann. 107 (1933), 188-232Sarnak's Constant 0.72365 or 0.74397The second value, which is 5pi2/48 timesthe original product, is the one of interest in Sarnak's work aboutclass numbers. (more on FMC)P. C. Sarnak, Class numbers of indefinite binary quadratic forms II, J. Number Theory 21 (1985) 333-346; MR 87h:11027.Stephens' Constant 0.57596Density (up to a rational factor) of the setT(a,b) of primes qsuch that q divides ak-bfor some k, given multiplicatively independent integersa and b.(more on FMC)P.J. Stephens, Prime divisors of second-order linear recurrences, I, J. Number Theory 8 (1976) 313-332; MR 54 #5142.P. Moree and P. Stevenhagen, A two variable Artin conjecture, submitted (1999); preprint available (DVI).Heath-Brown and Moroz's Constant 0.0013176This is one factor in a formula for the number of primitive pointsof height not exceeding a given value on a cubic surface.Think of 1-f/g as coming from prod (1 - 1/p)7 (1 + (7p+1)/p2).(more on FMC)D. R. Heath-Brown and B. Z. Moroz, The density of rational points on the cubic surface X03=X1 X2 X3, Math. Proc. Cambridge Philos. Soc. 125 (1999) 385-395.Totient Constant 2.20386Slightly easier to compute as zeta(2) times thevalue 1.33978 obtained as shown above, this equals the sum sumn>0 1 / (n phi(n))where phi denotes the Euler totient function.It appears in the paper by Stephensmentioned above. Pieter Moree has recentlyimproved some of the estimates from that paper.A related constant, which should be computable in a similarfashion from logarithms of zeta values, is the sum over the1/phi(n) themselves, which was studiedby Landau.E. Landau, Über die zahlentheoretische Funktion phi(n) und ihre Beziehung zum Goldbachschen Satz. In: Collected works Vol.1, 106-115.Murata's Constant 2.82642This belongs into a context closely related to Artin's Constant.(more on FMC)L. Murata, On the magnitude of the least prime primitive root, J. Number Theory 37 (1991) 47-66; MR 91j:11082.G. Niklasch /Fri, 23 Aug 2002 14:15:53 MEST |
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