STRINGS OF CONSECUTIVE PRIMES IN FUNCTION FIELDS NOAM TANNER

STRINGS OF CONSECUTIVE PRIMES IN FUNCTION FIELDS NOAM TANNER Abstract In a recent paper, Thorne [5] proved the existence of arbitrarily long strings of consecutive primes in arithmetic progressions in the polynomial ring F q [t] Here we extend this result to show that given any k there exists a string of k consecutive primes of degree D in arithmetic progression for all sufficiently large D 1 Introduction and Statement of Results Let F q be the finite field with q elements and let F q [t] denote the corresponding polynomial ring in one variable The ring F q is analogous to Z see [3]) Therefore, questions concerning the distribution of primes ie monic irreducible polynomials) are naturally of interest The Riemann Hypothesis can be proven in this setting and results such as the Prime Number Theorem for arithmetic progressions have been proven with the strongest possible error terms [3] This paper is concerned with the distribution of primes in arithmetic progressions In both F q [t] and Z, one can show the existence of odd behavior of primes in arithmetic progressions Let p 1 = 2, p 2 = 3, denote the sequence of primes in order ie p n is the nth prime) In 2000, Shiu [4] proved that given a and m Z with a, m) = 1 and any k, there exists a string of consecutive primes p r+1 p r+2 p r+k a mod m In 2007, Thorne [5] extended this result to F q [t] Thorne showed that for arbitrary monic polynomials a and m with a, m) = 1, given k, then there exists a constant D 0 depending on k, q, and m) such that for any D > D 0 there exists a string of consecutive primes p r+1 p r+2 p r+k a mod m, of degree at most D Consecutive here means with respect to lexicographical order As can be seen from the proofs, the choice of lexicographical order is arbitrary One could choose any ordering such that for two primes of different degrees, the prime of higher degree is greater 2000 Mathematics Subject Classification 11N05, 11T55 Key words and phrases Primes, function fields, Shiu s theorem, Maier matrix 1

2 NOAM TANNER Even though these results concerning strings of primes may be predicted from probablistic models see [1]), the fact that one can prove this odd behavior in both cases is quite surprising Although Thorne proves the existence of strings of consecutive primes for arbitrarily large degree, his construction guarantees the existence of these strings for scattered degrees In this paper we will show that these strings can be found among the primes of every sufficiently large degree: Theorem 11 If a and m are monic polynomials with a, m) = 1 and k is a positive integer, then there exists an integer D 0 depending on q, m, and k) such that for every D D 0 there is a string of k consecutive primes of degree D such that p r+1 p r+2 p r+k a mod m Moreover, we have that for sufficiently large k depending on q and m) ) 1/ 1 log D0 11) k log log D 0 ) 2 The implied constant depends only on q 2 Preliminary Results We fix a finite field F q throughout We are interested in the distribution of primes ie, irreducible monic polynomials) in the polynomial ring F q [t] Except as noted and always when referring to primes), we will assume all of our polynomials to be monic For a residue class a modulo m, let πa, m; n) denote the number of primes in F q [t] of degree n congruent to a modulo m By the Prime Number Theorem for arithmetic progressions [3], we have 21) πa, m; n) = 1 q n n + O q n/2 whenever a, m) = 1 Here the Euler φ-function is defined by = F q [t]/mf q [t]) As an important special case we have ) 22) πn) = qn q n/2 n + O, n where πn) denotes the number of primes of degree n Moreover, a simple exact formula for πn) is given in [3] Thorne s and Shiu s theorems make use of the Maier matrix method [2], which we describe as follows See also Granville s article [1] for an exposition about the method used for the integers and other related results) In the case of Z, let Q be a certain product of small primes and x 1, x 2, and y be integers such that x 1 < x 2 and n ),

STRINGS OF CONSECUTIVE PRIMES IN FUNCTION FIELDS 3 y < Q We consider the following matrix of integers: Qx 1 + 1 Qx 1 + 2 Qx 1 + y Qx 1 + 1) + 1 Qx 1 + 1) + 2 Qx 1 + 1) + y Qx 2 + 1 Qx 2 + 2 Qx 2 + y In the case of F q [t], let Q be a product of small prime polynomials monic irreducible polynomials) Let each polynomial f i be of degree s < deg Q By choosing Q so that it is divisible by a large enough power of t, we ensure that the rows of the matrix are ordered lexicographically) Let each g i be a polynomial of degree r We consider the following matrix of polynomials where α and β are positive integers): Qf 1 + g 1 Qf 1 + g 2 Qf 1 + g α Qf 2 + g 1 Qf 2 + g 2 Qf 2 + g α Qf β + g 1 Qf β + g 2 Qf β + g α In either case, the columns form arithmetic progressions modulo Q In the case of Z, for those Q which meet appropriate conditions on the associated Dirichlet L-functions, each column will contain roughly the expected number of primes In the case of F q the analogous conditions are trivial, so for all sufficiently large Q all columns will have approximately the expected number of primes Thorne and Shiu proved their results using a clever choice of Q, the Prime Number Theorem, and sieve-theoretic lemmas Shiu s choice of Q ensured that the majority of i with 1 < i < y are congruent to a mod m Thorne s choice of Q ensured that the majority of g i in the matrix were congruent to a mod m In order to estimate the number of i and g i congruent to a mod m the number of columns containg primes congruent to a mod m), Thorne and Shiu used a combinatorial argument to find expressions for these quantities Once they found these expressions, they used sieve-theoretic lemmas to bound these values The Prime Number Theorem enabled them to estimate the number of primes per column and it followed that the majority of primes in the matrix were congruent to a mod m In order to prove the theorem, we introduce a parameter b 0 which will allow us to establish the existence of a string of consecutive primes for a range of degrees We will then prove that these ranges overlap and cover all sufficiently large D We conclude this section by presenting our choice of notation Throughout q will denote the cardinality of the base field, m and a will denote monic polynomials in F q [t], p will denote a monic) prime element of F q [t], and f and g will denote generic elements of F q [t] Q will denote a certain product of primes, as in the introduction In Section 3, c, d, u will denote positive integers, analogous to the quantities y, z, t appearing in [4] We will write fx) gx) to mean that fx) > Cgx) for some

4 NOAM TANNER positive constant C and large enough x The constant C will depend only on q, but the range of allowable x may depend on other variables as noted 3 Proof of Theorem 11 The proof of Theorem 11 adapts Thorne s proof [5] We as in [5]) introduce a variable c, as well as variables d and u which will be chosen as unbounded, nondecreasing functions of c satisfying d < u = oc) We assume that the quantity c is sufficiently large in relation to q and m; the same will then also be true of u and d With this restriction, constants implied by and will depend only on q For a 1 we define a set of primes P by 31) P := {p : deg p c, p 1, a mod m} {p : u deg p c, p 1 mod m} {p : deg p c + d u, p a mod m} For the case a = 1 we define instead: { {p : deg p c, p 1 mod m} 32) P := {p : u deg p c + d u, p 1 mod m} Although the latter definition does not give optimal bounds for a = 1, it simplifies our proof by allowing us to treat both cases together We introduce a parameter b 0 which will allow us to prove the existence of a string of consecutive primes for a range of degrees We will then prove that these ranges overlap and cover all sufficiently large D Let 33) Q := mt c+d+1 p We then define a Maier matrix M consisting of the following set of integers: 34) M := Qf + g deg f=2 deg Q+b deg g=c+d Here f and g range over monic polynomials of the indicated degrees The g are arranged in lexicographic order and since t c+d+1 Q each row in the matrix is arranged in lexicographic order Each column of M will be the arithmetic progression of all monic polynomials of degree 3 deg Q + b which are congruent to a fixed g modulo Q By 21), each column containing primes will contain asymptotically the same number of primes Moreover, whether or not a particular element Qf + g is congruent to a modulo m depends only on g, and our choice of Q will ensure that most g with g, Q) = 1 fall into the desired congruence class

We then define sets STRINGS OF CONSECUTIVE PRIMES IN FUNCTION FIELDS 5 35) S := {h : deg h = c + d, h, Q) = 1, h a mod m}, T := {h : deg h = c + d, h, Q) = 1, h a mod m} Thorne shows that S is much larger than T for appropriate choices of u, c, and d This implies see [5]) that either the matrix contains a string of consecutive primes in arithmetic progression of length 36) 1 u d or ) 1/ 37) π 2 deg Q+b /q where π denotes the total number of primes congruent to a mod m, in the matrix From the estimation of S, it follows see [5]) that 38) π C m q c+d q 3 deg Q+b c + d u1/ 3 deg Q + b)φq) Into this last equation we incorporate the estimate: 39) 3 deg Q + b deg m + c + d + 1 + b + q c + q c 1 + q c + b If we set b such that 0 b 9q c+1, we obtain: 310) 3 deg Q + b q c, where the constant depends on q Using Mertens estimate see [5]), it can be shown that the following holds for all a: q deg Q 311) φq) m ) 1/ c c1 2/ c + d u) 1/ cu 1/ u Thus in the case of 37), if b is an integer between 0 and 9q c+1, we conclude from these estimates and 38) that there exists a row that contains C m q d consecutive primes all of degree 3 deg Q + b From 36), 37), and the above remarks we conclude that the matrix will contain an arithmetic progression of primes of length 1 u ) ) 1/, min Cm q d d With the choices u = c/2 log c and d = 2 log c we obtain a progression of length 312) 1 ) 1 c log 2 c Thus, we have proven the following lemma:

6 NOAM TANNER Lemma 31 Using the notation of the previous theorem, if large enough c is given, then there exists a string of consecutive primes of degree D in arithmetic progression of length 1 c for every degree D such that 3 deg Q D 3 deg Q + 9q c+1 ) 1 log 2 c We now use this lemma to prove the theorem by creating a sequence of intervals of degrees D containing consecutive primes in arithmetic progression These intervals will overlap and will cover all sufficiently large D In order to prove that these intervals overlap, we must first obtain estimates for deg Q Note that deg Q = deg m + c + d + 1 + deg p For a 1, deg p = c iπj, m; i) + j 1,a c kπ1, m; k) + k=u c+d u l=u lπa, m; l), recalling the notation in 21) For a = 1, deg p = c iπj, m; i) + j 1 c+d u l=u lπ1, m; l) Using the Prime Number Theorem we will now provide an upper bound for deg p that holds for all a: deg p c iπi) = c q i + Oq i/2 )) = c q i + Ocq c/2 ) = qc+ q 1 + Ocqc/2 ) 2q c + Ocq c/2 ) Since u = oc), then for large c deg p 2q c + Ocq c/2 ) 2 + ɛ )q c, for some ɛ with 0 < ɛ < 1

STRINGS OF CONSECUTIVE PRIMES IN FUNCTION FIELDS 7 Now we will bound deg p from below If a 1, then: c c deg p iπj, m; i) + iπ1, m; i) = = j 1,a As u = oc), then for large c c 1 2 i=u ) q i + Oq i/2 ) ) q c+ q ) + ) q c+ q q u 1 q 1 + Ocqc/2 ) for some ɛ with 0 < ɛ < If a = 1, then: c deg p iπj, m; i) = = j 1 c c ) q i + Oqi/2 ) i=u q u 1 q 1 + Ocqc/2 ) ) q c+ q 1 + Ocqc/2 ) Once again, since u = oc), then for large c deg p ) q c+ q 1 + Ocqc/2 ) ) ɛ q c, ) ) q i + Oq i/2 ) ) ɛ q c, with 0 < ɛ < If = 1, then every prime is congruent to a mod m so that the theorem is trivial We can therefore assume that > 1 Since c was assumed to be large enough and u = oc), for any a relatively prime to m there exist ɛ and ɛ so that deg Q = m + c + d + 1 + [ deg p ] )q ɛ c, 2 + ɛ )q c, where 0 < ɛ < ) and 0 < ɛ < 1 Thus, we have the desired estimate for deg Q and we are ready to show that the intervals overlap Incorporating the result of Lemma 31, these estimates above show that for every large c one can find consecutive primes in arithmetic progression of degree D where 313) D [ 32 + ɛ )q c, 3 ɛ ] )q c + 9q c+1

8 NOAM TANNER Letting c be the lower bound of c for which 312) and the estimates presented above for D and deg Q hold, note that for any c c our argument produces strings of length 1 ) 1 c log 2 c Therefore we have a sequence of intervals I c = [32 + ɛ )q c, 3 ɛ)qc + 9q c+1 ], such that for every D I c, there exist strings of consecutive, congruent primes of degree D of length 1 ) 1 c log 2 c However, since 32 + ɛ )q c+1 < 9q c+1, every interval I c overlaps with I c+1 Thus, we see that I c = [32 + ɛ )q c, ) c c Let c 0 be the smallest integer c such that Lemma 31 guarantees that the Maier matrix formed by choosing c 0 contains k consecutive primes Since c 0 is sufficiently large, we have that for any degree D such that D D 0 := 9q c 0, there exist strings of degree D consecutive primes of length k 1 c0 log 2 c 0 ) 1 Finally, since D 0 = 9q c 0, we have that c 0 log D 0 Since k 1 ) 1 c 0 log 2 c 0, for sufficiently large k this previous bound implies 11) Sufficiently large in terms of how large c has to be for the conclusion of Lemma 31 to hold, and this depends in turn on q and m) 4 Acknowledgements I would like to thank Frank Thorne and Ken Ono for their invaluable assistance with every aspect of this project, and to the anonymous referee for many helpful suggestions I would also like to thank the other instructors and students at the 2007 UW-Madison REU for all of their helpful comments References [1] A Granville, Unexpected irregularities in the distribution of prime numbers, Proceedings of the International Congress of Mathematicians Zürich, 1994), 388-399, Birkhäuser, Basel, 1995 [2] H Maier, Primes in short intervals, Michigan Math J 32 1985), 221-225 [3] M Rosen, Number theory in function fields, GTM 210, Springer-Verlag, New York, 2002 [4] D K L Shiu, Strings of congruent primes, J London Math Soc 61 2000), 359-373

STRINGS OF CONSECUTIVE PRIMES IN FUNCTION FIELDS 9 [5] F Thorne, Irregularities in the distributions of primes in function fields, J Number Theory, to appear Department of Mathematics, Princeton University, Princeton, New Jersey 08544 E-mail address: ntanner@princetonedu