The zeta function, L-functions, and irreducible polynomials

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1 The zeta function, L-functions, and irreducible polynomials Topics in Finite Fields Fall 203) Rutgers University Swastik Kopparty Last modified: Sunday 3 th October, 203 The zeta function and irreducible polynomials For a nonzero polynomial F T ) F q [T ], we define its size F by: F q degf ) Note that F G F G We can now define the zeta function: ζs) F T ) F q[t ], monic F s Here in the land of polynomials, we are fortunate that ζ has a nice closed form expression: ζs) d 0 d 0 degf )d q d q ds q s, where we have convergence for all s with Rs) > This automatically gives us analytic continuation of ζ to all of C We note the following simple observations: The Riemann hypothesis: All zeroes of ζ lie on the line Rs) /2, simply because there are no zeroes! 2 ζ has poles at s + 2πin log q 3 Locally around s, ζ has the Laurent series expansion: q ds s ) log q + a 0 + a s ) + a 2 s ) 2 +, where Hs) a 0 + a s ) + is analytic and bounded in an open neighborhood of The unique factorization of polynomials into irreducibles gives us Euler s factorization of ζs): ζs) + + ) P 2s + P T ) F q[t ] monic, irreducible ) P T ) F q[t ] monic, irreducible We will exploit this formula to give us some understanding of the distribution of irreducible polynomials

2 Rough estimates Restrict s to lie in, ) Taking log of both sides: log ζs) log ) P T ) F q[t ] monic, irreducible m P ms P T ) F q[t ] monic, irreducible m + m P ms P T ) monic, irreducible P T ) m 2 Let us bound the second term for s, ): which is bounded as s + Thus, as s +, we have P T ) m 2 m P ms P T ) P T ) monic, irreducible P 2s ) 3ζ2s), log ζs) + O) log s + O) This is Euler s proof of the infinitude of irreducible polynomials This also gives us some quantitative measure of the number of irreducible polynomials, which will be useful when we study irreducible polynomials in arithmetic progressions 2 An exact formula Slightly different considerations can give us an exact formula and asymptotics for the number of irreducible polynomials of a given degree Let n d be the number of monic irreducible polynomials of degree d log ζs) log ) P T ) F q[t ] monic, irreducible m P ms P T ) F q[t ] monic, irreducible m n d m q mds d m On the other hand:: log ζs) log q s ) n q n q ns If we let U q s, this gives us an equality of power series in U, d m n d m U md q n U n n 2

3 This implies that: q n d n dn d By Mobius inversion, we get the exact formula: dn d k d µd/k)q k q d + k d,k<d µd/k)q k q d ± Oq d/2 ) This also gives us nice asymptotics for the number of degree d irreducibles: Two remarkable features of this asymptotic: n d qd d + Oqd/2 d ) We have an error-term which is about the square root of the main term, showing that the irreducible polynomials have a very random-like distribution This kind of error is what you would get if each polynomial independently decided to become irreducible with probability d 2 This is very strongly analogous to the prime number theorem We showed that the number of irreducibles P such that P x is about x log q x + O x) The main term is like we have in the usual prime number theorem, but the error term is far better than what we know for the primes In fact, achieving such an error term in the usual prime number theorem is equivalent to the original Riemann Hypothesis In the F q [T ] case, we just happened to know that the analogous Riemann Hypothesis is true! 2 L-functions Let : F q [T ] C be a totally multiplicative function: Here are some interesting examples of such functions: F G) F ) G) Let MT ) F q [T ] Consider the ring F q [T ]/MT ) which is a field if MT ) is irreducible, in general it is only a ring) Let G F q [T ]/MT )) be the multiplicative group of invertible elements in that ring Let χ be a character of G Define F ) χf mod M) if F is relatively prime to M, and F ) 0 otherwise These functions are called Dirichlet characters, and will be important in showing that the 2 Let ψ be an additive character of F q Let gx) F q [X] Define degf ) F ) ψ gα i ), where F T ) β degf ) i T α i ) is the factorization of F T ) in F q [T ] These characters will play a crucial role in bounding the character sums x F q ψgx)) 3 Let χ be a multiplicative character of F q Let gx) F q [X] Define degf ) F ) χ gα i ), i i 3

4 where F T ) β degf ) i T α i ) is the factorization of F T ) in F q [T ] These characters are in fact special cases of Dirichlet characters They will play a crucial role in bounding the character sums x F q χgx)) The L-function associated to is defined by: Ls, ) F T ) F q[t ] monic Similar to ζ, we have the following Euler-like factorization: Ls, ) P T ) F q[t ] monic, irreducible F ) F s 2 Irreducible polynomials in fixed residue classes P ) ) We will now use L-functions to determine which residue classes mod MT ) of F q [T ] contain infinitely many irreducible polynomials It is clear that if AT ), MT ) have nontrivial GCD, then there can be at most one irreducible that is AT ) mod MT ) We will show that all the remaining residue classes those relatively prime to MT )) contain infinitely many irreducible polynomials This is known as Kornblum s theorem, and is the polynomial analogue of Dirichlet s theorem for the usual primes Define φm) to be the number of such classes Note that if G F q [T ]/MT )), then we have φm) G Let Ĝ be the dual group of G For each χ Ĝ, let χ be the Dirichlet character obtained by extending χ to all of F q [T ] by 0 Lemma For each nontrivial χ Ĝ: Ls, χ) is a polynomial in q s of degree at most degm) Proof We need to show that for each d degm), F T ) F q[t ] monic,degf )d since the LHS is the coefficient of q ds in Ls, χ ) We get this from the following calculation: F ) F T ) F q[t ] monic,degf )d F ) 0, F T ) F q[t ] monic,degf )d q d degm) χa) 0 A G χf mod M) For the trivial character χ 0 Ĝ, we have: Ls, χ0 ) F T ) monic,gcdf,m) We will now study the behaviour of Ls, χ ) as s + F s ζs) P T ) MT ) ) 4

5 Lemma 2 As s +, we have: logls, χ )) P T ) monic, irreducible χp ) + O) The proof is by taking log of both sides of the Euler identity for Ls, χ ), expanding log in Taylor series, and bounding the error term the way we did for ζ We now show the infinitude of irreducibles AT ) mod MT ) The trick is to isolate that residue class mod MT ) using the characters via the following identity: { Ĝ P A mod M χ P )χa) 0 otherwise χ Ĝ We have: P T ) AT ) mod MT ),P monic, irreducible φm) φm) φm) P monic, irreducible χ Ĝ χ P )χa) χ P )χa) χ Ĝ P monic, irreducible χa) logls, χ )) + O) χ Ĝ φm) log s + χ χ 0 χa) logls, χ )) + O) To analyze this, we need to understand the behavior of Ls, χ ) as s Theorem 3 For all χ Ĝ, L, χ)) 0 In fact, much more is true Theorem 4 The Riemann Hypothesis for Dirichlet L-functions) For all χ Ĝ, all zeroes of Ls, χ) lie in the half-plane Rs) /2 In fact, all the zeroes lie either on the line Rs) /2 or on the line Rs) 0 We will prove the Riemann Hypothesis for several interesting classes of L-functions in the coming weeks Then we will be able to vaguely outline how a proof of Theorem 4 goes unfortunately, a proper proof of this requires more advanced tools, such as the Riemann-Roch theorem and some class field theory) Wrapping up, Theorem 3 gives us: P T ) AT ) mod MT ),P monic, irreducible φm) log s + O) Thus, in a quantitative sense Dirichlet density ), the irreducible polynomials are equidistributed in the φm) invertible residue classes mod MT ) 2 Asymptotics Now we will see how to derive an asymptotic for the number of irreducibles of degree d in a given residue class mod MT ) 5

6 Let χ be a nontrivial character of G By the previous section, we can write: Ls, χ ) t α i q s ), i where t degm) We have Ls, χ ) 0 if and only if q s α i for some s Thus Theorem 4 implies that α i q for each i [t] Let U q s We then have the following two expressions for Ls, χ ): t i α i U) P χ P )U degp )) Take log of both sides, and then differentiate: t i α i α i U P monic irreducible degp ) χ P ) degp ) U χ P )U degp ) t i α i U α i U χ P ) degp ) U degp ) χ P )U P monic irreducible degp ) χ P ) m degp ) U m degp ) P m χ P ) d/ degp ) degp ) U d d P monic, irreducible:degp ) d Equating coefficients of U d in either side, we get: P monic, irreducible:degp ) d χ P ) d/ degp ) degp ) Observe that the number of P whose degree divides d is at most q d/2 Thus: χ P ) d Oq d/2 ) P monic, irreducible:degp )d t αi d Oq d/2 ) As before, we can use this to count the number of P of degree d which are A mod M {P T ) monic, irreducible : degp ) d, P A mod M} χ P )χa) φm) χ Ĝ P monic irreducible,degp )d φm) {P T ) monic, irreducible : degp ) d} + χ χ 0 φm) {P T ) monic, irreducible : degp ) d} + Oq d/2 /d) This gives us the desired asymptotics, along with a square-root error bound i P monic irreducible,degp )d ) χ P )χa) 6

7 22 A character sum We end by noting an interesting character sum bound that arises from the Riemann Hypothesis above this bound is due to Katz/Lenstra) Let MT ) be an irreducible polynomial of degree d Note that F q [T ]/MT ) is the field F q d Let χ be a nontrivial character mod MT ), which we can treat as a multiplicative character of F q d Let U q s, and consider t Ls, χ ) α i U) + χ T + α) U + + c t U t, α Fq i where t d By Theorem 4, we have that α i q Thus we get the bound on the multiplicative character sum: χt + α) mod MT )) α F q t q d ) q This shows that if we sum a multiplicative character χ of the field F q d over a certain explicit -dimensional affine subspace S namely S {T + α) mod MT ) α F q } ), then we get very strong cancellation with the sum being about square-root of the number of terms What makes this remarkable is the fact that this cancellation is achieved over a set that is much much smaller than the size of the field F q d our earler results derived from the Gauss sums said nothing about subspaces of size < q d/2 ) This is particularly strong when d very large in relation to q 7