On the Subsequence of Primes Having Prime Subscripts

Transcription

1 Journal of Integer Sequences, Vol. (009, Article On the Subsequence of Primes Having Prime Subscripts Kevin A. Broughan and A. Ross Barnett University of Waikato Hamilton, New Zealand kab@waikato.ac.nz Abstract We eplore the subsequence of primes with prime subscripts, (q n, and derive its density and estimates for its counting function. We obtain bounds for the weighted gaps between elements of the subsequence and show that for every positive integer m there is an integer arithmetic progression (an + b : n N with at least m of the (q n satisfying q n an + b. Introduction There are a number of subsets of primes with a conjectured density of a constant times / log. These include the primes separated by a fied even integer, Sophie Germain primes and the so-called thin primes, i.e., primes of the form e q where q is prime [3]. In order to gain some familiarity with sequences of this density we undertook an investigation of the set of primes of prime order, called here prime-primes, and report on the results of this investigation here. Some of the properties of this sequence are reasonably straight forward and the derivation follows that of the corresponding property for the primes themselves. Others appear to be quite deep and difficult. Of course the process of taking a prime indeed subsequence can be iterated, leading to sequences of primes of density / log 3, / log 4 etc, but these sequences are not considered here. In Section upper and lower bounds for the n th prime-prime are derived, in Section 3 the prime-prime number theorem is proved with error bounds equivalent to that of the prime number theorem, in Section 4 an initial study of gaps between prime-primes is begun, and in Section 5 it is shown that for each m there is an arithmetic progression containing m prime-primes.

2 Definition. A prime-indeed-prime or prime-prime is a rational prime q such that when the set of all primes is written in increasing order (p,p, (, 3,, we have q p n where n is prime also. Here is a list of the primes up to 09 with the prime-primes in bold type so q p 3 and q 0 p 9 09:, 3, 5, 7,, 3, 7, 9, 3, 9, 3, 37, 4, 43, 47, 53, 59, 6, 67, 7, 73, 79, 83, 89, 97, 0, 03, 07, 09. Definition. If > 0 the number of prime-primes q up to is given by π π ( : q. Then π π ( π(π( n χ P(n χ P (π(n, where χ P (n : π(n π(n, for n N, is the characteristic function of the primes. Bounds for the sequence of prime-primes: The following theorem and its corollary gives a set of useful inequalities for estimating the size of the n th prime-prime and for comparing it with the n th prime. Theorem 3. As n : q n q n < n(log n + loglog n(log n + loglog n n log n + O(n loglog n, > n(log n + loglog n(log n + loglog n 3n log n + O(n log log n. Proof. We use the inequalities of Rosser and Schoenfeld [4]. Namely p n < n(log n + loglog n / valid for n 0, p n > n(log n + loglog n 3/ valid for n, by first substituting p n for n, then using both the upper and lower bounds on each side, and finally simplifying. Corollary 4. The following inequalities are also satisfied by the (q n : as n : (a q n n log n + 3n log n loglog n + O(n log n, (b q n n log n, (c q n, q n+ (d q n log n p n, (e for all ǫ > 0 there is an n ǫ N such that for all n n ǫ, q n p +ǫ n, (f for n >, q n < p 3 n, (g For all m,n N, q n q m > q mn, (h For real a > and n sufficiently large, q an > aq n. Proof. The inequalities (a-(e follow from Theorem 3. Item (f follows by an eplicit computation up to n 000 and then by contradiction, assuming n(log n + loglog n < p n < n(log n + loglog n, for n > 000. Item (g follows from the corresponding theorem of Ishikawa [7] for primes and (h from that of Giordano [6].

3 3 The prime-prime number theorem and inequalities Here we obtain two forms for the asymptotic order of the counting function of the primeprimes: Theorem 5. As, π π ( log,and π π( Li(Li( + O ( ep( A log 3 5 loglog 5, for some absolute constant A > 0, where the implied big-o constant is also absolute. Proof. The first asymptotic relation follows from a substitution: As : π(π( π( ( π( log π( + O log π( + O( ( log log loglog ( + O log + O( log 4 log log / log ( loglog log(/ log + O(/ log + O log 4 ( loglog log [ + O( loglog ] + O log ( loglog log + O log 3. log 4 Now let ( : O ( ep( A log 3 5 loglog 5 and use the equation [8, 5] π( Li( + (. By the Mean Value Theorem there is a real number θ with θ < such that π(π( Li[Li( + (] + (π(, ( Li(Li( + log[li( + θ (] + (, Li(Li( + (.,, Proposition 6. The following inequalities are true for every integer k > and real,y sufficiently large: (a π π (k < kπ π (, (b π π ( + y π π ( + 4π π (y, (c π π ( + y π π ( y log y. 3

4 Proof. (a We apply the theorem of Panaitopol [], namely that for all k > and sufficiently large π(k < kπ( to derive π π (k π(π(k π(kπ( < kπ(π( kπ π (. (b Now apply the inequality of Montgomery and Vaughan [9] as well as the theorem of Panaitopol: π π ( + y π(π( + y π(π( + π(y (c This follows directly from (b and Theorem 5. π(π( + π(π(y < π π ( + 4π π (y. In part (c compare the epression when π π is replaced by π [0, p. 34]. The following integral epression shows that, approimately, the local density of the prime-primes is dt/ log t: Proposition 7. As Li(Li( Proof. Use the epression [5, p. 86] in the case n, so Then let ( log t + loglog t ((loglog dt + O log 3 t log 4 t dt log t + O( loglog log 3 Li( log( + (n! log + + ( log n ( + O(. log n+ ( Li( log( + O(. log ( Then F( : Li(Li( y dt, where log t y log u du. F ( /[log log(li(] /[log ( loglog log + O( log ] loglog log [ + log + O((loglog log log + loglog log 3 + O((loglog log 4 4

5 so therefore F( ( log t + log log t log 3 t log dt + O((log log 4. To derive the second epression, split the integral for the second term in the integrand of the first epression at. 4 Etreme values of gaps between prime-primes Note that for n > the number of primes between each pair of prime primes is always odd, so q n+ q n 6. It is natural to conjecture that this gap size of 6 is taken on an infinite number of times, as is every even gap size larger than 6. Since for n >, q n+ q n [4], we have q n+ q n q θ n for θ.0. The same best current value for primes, due to Baker, Harman and Pintz [, ], namely p n+ p n ǫ p θ+ǫ n, θ 0.55 works for prime-primes. To see this replace by π( in their equation [, p. 56] π( π( log to derive the formula, for every ǫ > 0 and sufficiently large π π ( ǫ π π ( ǫ 0 log. The first proposition below is modeled directly on the corresponding result for primes. The second is also closely related to the derivation for primes, [3, p. 55]. Proposition 8. For every integer m > there eists an even number δ such that more than m prime-primes are at distance δ. Proof. Let n N, S : {q,,q n+ } be a finite initial subset of the ordered sequence of prime-primes, and let D : {q j+ q j : j n} be the n differences of consecutive elements of S. If D n, then m q n+ q (q q + (q 3 q + + (q n+ q n n m n m + O(. By Theorem 3, we can choose n sufficiently large so q n+ < n log n and the inequality for q n+ q is not satisfied. Therefore we can assume n is sufficiently large so D < n. Then one of the differences m must appear more than m times. Call the size of this difference δ. 5

6 Proposition 9. lim inf n q n+ q n log q n. Proof. Let ǫ > 0 be given. Define two positive constants α,β with α β + β with β > 3 and so < α <. Let ǫ > 0 be another positive constant with ǫ < /α /. Let L : + ǫ and let {q m,,q m+k } be all of the prime-primes in the interval [, β]. Now suppose (to obtain a contradiction that for all n with m n m + k we have Then q n+ q n L log q n. (β q m+k q m m+k L log q n nm Lk log. so But by Theorem 5, for all sufficiently large ( ǫ log < π π( < ( + ǫ log k π π (β π π ( ( ǫβ log β ( + ǫ log (β ǫ(β + log for sufficiently large. Therefore (β L(β ǫ(β +, or in other words ( + ǫ( αǫ, which is impossible. Therefore there eists an n such that q n [,β] and q n+ q n log q n < + ǫ so lim inf n (q n+ q n / log q n. 6

7 q Using a similar approach one can show that lim sup n+ q n n log q n. It is epected however that the limit infinum of the ratio should be zero and the limit supremum infinity. Fig. is based on the normalized nearest neighbor gaps for the first two million prime-primes with a bin size of 0.05 and with the -ais 60 corresponding to a normalized gap value of Figure : Normalized gap frequencies. 5 Prime-primes in arithmetic progressions Using the prime number graph technique of Pomerance [], applied to the sequences (n,q n and (n, log q n, we are able to demonstrate the eistence of infinite subsets of the q n such as the following: Proposition 0. (a There eists an infinite set of n N with q n q n i + q n+i for all 0 < i < n. (b There eists an infinite set of n N with q n > q n i q n+i for all 0 < i < n. Now we show that there are arithmetic progressions (an + b : n N containing specified numbers of prime-primes, not necessarily consecutive, satisfying q n an + b. Although modelled on the technique of Pomerance, [, Theorem 4.], the key step is introducing functions, named f(u and g(u, but delaying their eplicit definitions until sufficient information becomes available. Theorem. For every integer m > there eists an arithmetic progression (an + b : n N, with a,b N, with at least m prime-primes satisfying q n an + b. Proof.. Definitions: Let u > 0 be a real variable and f(u > 0 and g(u > 0 two real decreasing 7

8 functions both tending to zero as u, to be chosen later. Let v u + u f(u so log v log u( + O ( f(u, and log u ( log Li(v log Li(u( + O ( f(u. log u ( Let T be a parallelogram in the first quadrant of the y plane bounded by the lines u, v, y Li(Li(u + ug(u + u, and k y Li(Li(u ug(u + u, k where k log Li(u log u. Claim: If y Li(Li( < ug(u then (, y T. This is demonstrated in. 4. below.. The upper bound: since y < Li(Li( + ug(u we have, for some u < ξ <, u y < Li(Li(u + log Li(ξ log ξ + ug(u, Li(Li(u + u log Li(u log u + ug(u. 3. The lower bound: We have Li(Li( ug(u < y and need Li(Li(u + u log Li(ulog u ug(u < y so it is sufficient to have u log Li(u log u u ug(u or log Li(ξ log ξ v u log Li(u log u v u ug(u, which is equivalent to log Li(v log v f(u[ log Li(u log u ] g(u, or by ( and ( log Li(v log v f(u log Li(u log u [ + O(f(u/ log u ] O( f(u f(u M log 3 u log 3 g(u (3 u for some M > Counting points and lines: by. and 3. each point (q n,n with u q n v is in T, so the number of such points is, by Theorem 5, bounded below by the numerator in the epression below. The number of lines with slope /k passing through the integer lattice points of T is bounded above by the denominator of this epression. Therefore for u sufficiently large: #points # lines uf(u log u (log Li(u log u4ug(u f(u/8 (4 (log u 4 g(u 8

9 Now let f(u : log α u and g(u : log β u For eample α, β 6.5. Then in (4 and in (3 where α,β > 0 are chosen so that α + 4 < β < α + 3. f(u (log u 4 g(u logβ α 4 u f(u (log u 3 g(u logβ α 3 u 0+, so choose u sufficiently large that f(u /(log 3 ug(u /M, thus ensuring the validity of the lower bound. With these choices, the number of points divided by the number of lines tends to positive infinity, so for every natural number m there is at least one line on the graph of (q n,n with m of these points. Finally we note that since k varies continuously with u, we can choose k N and so q n an + b with a,b N since a k. 6 Epilog Leading on from Section 5, a natural aim is to show that there are an infinite number of prime-primes congruent, say, to modulo 4, or some other eplicit arithmetic progression. Then show that every arithmetic progression (an + b : n N, with (a, b, contains an infinite number of prime-primes. This has been demonstrated numerically, with the number of prime-primes falling approimately evenly between the equivalence classes modulo a. This problem appears to have considerable more depth than the results given here. For eample it is, on the face of it, more difficult than the corresponding theorem of Dirichlet for primes, because the primes in residue classes given by that theorem do not appear in any particular order. 7 Acknowledgements We acknowledge the assistance given by a helpful referee, especially in making us aware of the result in the paper of Baker, Harman and Pintz. The figure was produced using Mathematica. References [] R. C. Baker and G. Harman, The difference between consecutive primes, Proc. London Math. Soc. (3 7 (996, [] R. C. Baker, G. Harman and J. Pintz, The difference between consecutive primes, II, Proc. Lond. Math. Soc. (3 83 (00,

10 [3] K. A. Broughan and Q. Zhou, Flat primes and thin primes, (preprint. [4] R. E. Dressler and S. T. Parker, Primes with a prime subscript, J. ACM (975, [5] H. M. Edwards,Riemann s Zeta Function, Dover, 00. [6] G. Giordano, The eistence of primes in small intervals, Indian J. Math. 3 (989, [7] H. Ishikawa, Über die Verteilung der Primzahlen, Sci. Rep. Tokyo Univ. Lit. Sci. Sect. A (934, [8] N. M. Korobov, Estimates of trigonometric sums and their applications Uspehi Mat. Nauk 3 (958, [9] H. L. Montgomery and R. C. Vaughan, The large sieve, Mathematika 0 (973, [0] H. L. Montgomery, Topics in Multiplicative Number Theory, Springer Lecture Notes in Mathematics, Vol. 7, Springer-Verlag, 97. [] L. Panaitopol, Eine Eigenschaft der Funktion über die Verteilung der Primzahlen, Bull. Math. Soc. Sci. Math. R. S. Roumanie, 3 (979, [] C. Pomerance, The prime number graph, Math. Comp. 33 (979, [3] K. Prachar, Primzahlverteilung, Springer-Verlag, 957. [4] J. B. Rosser and L. Schoenfeld, Approimate formulas for some functions of prime numbers, Illinois J. Math. 6 (96, [5] I. M. Vinogradov, A new estimate of the function ζ(+it. Izv. Akad. Nauk SSSR. Ser. Mat. (958, Mathematics Subject Classification: Primary A4; Secondary B05, B5, B83. Keywords: prime-prime, prime-prime number theorem, prime-prime gaps, prime-primes in progressions. (Concerned with sequence A Received December 5 008; revised version received January Published in Journal of Integer Sequences, January Return to Journal of Integer Sequences home page. 0