TOPICS IN NUMBER THEORY - EXERCISE SHEET I École Polytechnique Fédérale de Lausanne Exercise Non-vanishing of Dirichlet L-functions on the line Rs) = ) Let q and let χ be a Dirichlet character modulo q. Let also Lχ, s) be the Dirichlet L-function defined for Rs) > by Lχ, s) = n χn) n s. The goal of this exercise is to prove that Lχ, s) 0 for Rs) =. This generalizes the result of Dirichlet which asserts that Lχ, ) 0. I) Let ζ q s) be the function defined for Rs) > by ζ q s) = χ mod q) Lχ, s) = n and the opposite of its logarithmic derivative ζ qs) ζ q s) = Λ q n) n n s, Compute Λ q n) and check that Λ q n) 0. a q n) n s, II) Let t R 0 and let H q,it s) be the function defined for Rs) > by H q,it s) = ζ q s) 3 ζ q s + it)ζ q s it)) 2 ζ q s + 2it)ζ q s 2it), and the opposite of its logarithmic derivative H q,it s) H q,it s) = Λ q,it n) n n s. Prove that these two functions extend meromorphically to the open half-plane {s C, Rs) > 0}, and that Λ q,it n) 0. III) We assume that there exists a Dirichlet character χ such that Lχ, + it) = 0 for some t R 0. Check that a q n) R for any n, and deduce that H q,it ) = 0. Prove also that H q,it s)/h q,its) has a simple pole at s = and that res s= H q,it s) ) < 0. H q,it s)
2 ÉCOLE POLYTECHNIQUE FÉDÉRALE DE LAUSANNE IV) Find a contradiction by examining the Laurent expansion of H q,it s)/h q,its) at. Conclude. Exercise 2 The Prime Number Theorem in arithmetic progressions) Let q and a Z coprime to q, and let also ψx; q, a) = Λn). n x n=a mod q) The goal of this exercise is to prove that the results of Exercise imply ψx; q, a) = x ϕq) + o q)). We will also make use of the following lemma. Lemma. Let f : R R be a piecewise continuous function whose absolute value is bounded. Then the function F s) defined by the integral F s) = ft)t s dt is holomorphic in the open half-plane {s C, Rs) > }. Moreover, if F s) extends meromorphically to an open neighborhood of the closed half-plane {s C, Rs) } and if it is holomorphic on the line {s C, Rs) = }, then the integral ft)t dt is convergent and equals F ). I) Let LΛ δ a mod q), s) be the function defined for Rs) > by LΛ δ a mod q), s) = n n=a mod q) Λn) n s. Write LΛ δ a mod q), s) as a linear combination of the logarithmic derivatives L χ, s)/lχ, s) and show that the function LΛ δ a mod q), s) s ϕq) s extends meromorphically to the open half-plane {s C, Rs) > 0} and that it does not have poles in the closed half-plane {s C, Rs) }. II) For t, we set f q,a t) = t ψt; q, a) t ). ϕq) Prove that f q,a t) is a piecewise continuous function whose absolute value is bounded. III) Prove that for Rs) >, we have LΛ δ a mod q), s) s ϕq) s = s f q,a t)t s dt.
TOPICS IN NUMBER THEORY - EXERCISE SHEET I 3 IV) Prove that the integral lim x x t 2 ψt; q, a) t 2 t ) dt ϕq) is convergent and that for every fixed λ >, we have λx ) ψt; q, a) dt = 0. t ϕq) V) Prove that for every ε > 0, there exists x 0 such that for every x x 0, we have ) ψx; q, a) ϕq) + ε x. For this, assume that there exists a sequence x n ) n going to infinity, and such that for every n, ) ψx n ; q, a) > ϕq) + ε x n, and find a contradiction. VI) Conclude. Exercise 3 On the number of zeros of the Riemann zeta function) Let NT ) be the number of zeros s = σ+it of the Riemann zeta function satisfying 0 < σ < and 0 < t < T, counted with multiplicities. The goal of this exercise is to prove that NT ) = T 2π log T 2π T 2π + Olog T ). I) Let ξs) be the function defined by ξs) = s 2 ss )π s/2 Γ ζs). 2) Prove that 2πNT ) = R arg ξs), where R is the rectangle with vertices 2, 2 + it, + it, described in the positive direction, and R denotes the variation of ξs) along the rectangle R. II) Using the functional equation satisfied by ξs), prove that NT ) = T 2π log T 2π T 2π + ST ) + 7 ) 8 + O, T where πst ) = L arg ζs) and L is the path of line segments joining 2 to 2 + it and then to /2 + it. III) Prove that log T, + T I)) 2 where the sum is over the non-trivial zeros of the Riemann zeta function, counted with multiplicities.
4 ÉCOLE POLYTECHNIQUE FÉDÉRALE DE LAUSANNE IV) Let s = σ + it with σ 2 and t not equal to an ordinate of a zero. Prove that ζ s) ζs) = + Olog t ), s where the dash on the summation indicates that it is restricted to those for which t I) <. V) Use the last step to conclude that ST ) log T. To go further. Using a similar method, one can prove an analog result for Dirichlet L-functions. Let q and let χ be a primitive Dirichlet character modulo q. Let also NT, χ) be the number of zeros s = σ + it of Ls, χ) satisfying 0 < σ < and T < t < T, counted with multiplicities. We have 2 NT, χ) = T qt log 2π 2π T + Olog qt ). 2π Exercise 4 Weil s explicit formula). Let ϕ Cc R >0 ) and let ϕ be its Mellin transform. We define the function ˇϕ by ˇϕx) = x ϕx ). We have the identity ϕn) + ˇϕn)) Λn) = ϕ)+ ϕ0) ϕ)+ ζ ) s) n 2πi /2) ζ s) + ζ s) ϕs)ds, ζ s) where the sum in the right-hand side is over all non-trivial zeros of the Riemann ζ function, counted with multiplicities, and where s ζ s) = π s/2 Γ. 2) that 2πi 3/2) I) Recall the definition of the function ξs) given in Exercise 3. Prove ξ s) ξs) ϕs)ds = 2πi 3/2) s + ) s + ζ s) ϕs)ds Λn)ϕn). ζ s) n II) Let T and let R T /2 ± it. Prove that 2πi and also that ϕ) = 2πi R T 3/2) be the rectangle whose vertices are 3/2 ± it and ξ s) ξs) ϕs)ds = I) T ϕ), ξ s) ξs) ϕs)ds ξ s) 2πi /2) ξs) ϕs)ds. III) Use the functional equation to prove that ϕ) = 2πi s + ) s + ζ s) ϕs) + ϕ s)) ds Λn) ϕn) + ˇϕn)). ζ s) n IV) Conclude. 3/2)
TOPICS IN NUMBER THEORY - EXERCISE SHEET I 5 Exercise 5 Consequence of the Generalized Riemann Hypothesis) Let q and a Z coprime to q, and recall the definition of ψx; q, a) given in Exercise 2. The goal of this exercise is to prove that the Generalized Riemann Hypothesis implies that ψx; q, a) x ϕq) x/2 log x) 2, where the constant involved in the notation does not depend on q. I) Assuming the Riemann Hypothesis, prove that where, as usual, ψx) x x /2 log x) 2, ψx) = n x Λn). To do this, use the following approximate explicit formula, which is valid if the distance from x to any prime power is, say, at least /2, x ψx) = x ζ 0) ζ0) + ) 2 log x 2) xlog T x) 2 + O, T I) T where the sum is over the non-trivial zeros of the Riemann zeta function, counted with multiplicities. II) Let χ be a non-trivial character modulo q. Assuming the Generalized Riemann Hypothesis, prove that where, as usual, ψx, χ) x /2 log qx) 2, ψx, χ) = n x Λn)χn). To do this, use the following approximate explicit formula, which is valid if the distance from x to any prime power is, say, at least /2, x ψx, χ) = l)log x + bχ)) + x l 2m ) xlog qt x) 2 2m l + O, T I) T where the sum is over the non-trivial zeros of the L-function Ls, χ), counted with multiplicities, and l = 0 if χ ) =, l = if χ ) =, and finally L s, χ) bχ) = lim s 0 Ls, χ) ). s III) Conclude. To go further. On average over the residue classes modulo q, one expects much more to be true. Assuming the Generalized Riemann Hypothesis, one can prove that ψx; q, a) x ) 2 xlog qx) 4, ϕq) a mod q) where the summation is over a coprime to q. What does it imply on the average size of ψx; q, a) x/ϕq) as a varies? m=
6 ÉCOLE POLYTECHNIQUE FÉDÉRALE DE LAUSANNE Exercise 6 On the least prime in an arithmetic progression) Let q and a Z coprime to q. Assuming the Generalized Riemann Hypothesis, prove that there exists a prime number p = a mod q) satisfying p q 2 log q) 4. Exercise 7 On the least quadratic non-residue). Let q be prime and let nq) be the least quadratic non-residue modulo q. Assuming the Generalized Riemann Hypothesis, prove that nq) log q) 4. Exercise 8 The Class Number Formula for imaginary quadratic fields) Let K be an imaginary quadratic field and let O K be the ring of integers of K. We let ω K be the number of roots of unity in K, and DiscK) be the discriminant of K. We also let h K be the class number of K, that is the cardinality of the group CK) = IK)/P K), where IK) is the free abelian group generated by prime ideals of O K, and P K) is the subgroup of principal ideals. If I is an ideal of O K, recall that its norm NI) is defined by NI) = #O K /I. We define the Dedekind zeta function of the field K by the formula ζ K s) = I O K NI) s, which is valid for Rs) >, and where the sum is over all ideals I of O K. Prove that ζ K s) has a simple pole at s = and that res s= ζ K s) = 2πh K ω K DiscK) /2. Pierre Le Boudec - Fall 204 École Polytechnique Fédérale de Lausanne