Zero detection for the frequency characters

Written by Claude Opus 5.5 (Anthropic), October 2026. Self-checked by the writing AI. Public domain (CC0).

After the Euler product of The Euler product of the probe, the frequency \(u\) enters the probe through \(1/L^S(\xi,\eta\overline{\chi_\bullet(u)})\). Lesson 14 moves the \(\xi\)-contour of every nonprincipal frequency from \(\operatorname{Re}\xi>1\) to \(\operatorname{Re}\xi=0.998\), which is only possible where this reciprocal is under control. This lesson shows that it is under control for all but \(O(U^{1/2})\) of the frequencies of norm about \(U\). A zero of \(L_F(s,\omega_u)\) close to \(\operatorname{Re}s=1\) forces a long Dirichlet polynomial to be large; a large sieve over the frequency characters, ordered by conductor, shows that this happens rarely; and away from the exceptional characters the Borel–Carathéodory theorem and the three circles theorem bound the reciprocal by \(U^\epsilon\).

Prerequisites. S-integers and sixth-power classes, Lemma 2.1 and Lemma 4.1; Sextic reciprocity and Kummer characters, Lemmas 3.1, 3.2 and 4.1; The quadratic large sieve and the reflected moment, Lemma 3.1; The Poisson probe, Lemma 3.1; The Gamma function and Stirling's formula, Corollary 4.1; Entire functions of order one and the Hadamard product of xi, Theorem 2.1; Growth in the critical strip: convexity and the Lindelöf hypothesis, Theorem 2.1; The Riemann hypothesis implies the Lindelöf hypothesis, Theorem 2.1; Tate's global theory: continuation and functional equation, Theorem 9.2; Hecke L-functions and the Dedekind zeta function, (1)–(3) and Theorem 10.1; Dirichlet series and Euler products, Lemma A.3; Schwartz functions and Fourier inversion, Section F4.

1. The frequency characters

For \(u\in\mathcal U\) (Lemma 3.2 of Lesson 3) let \(\omega_u\) be the Hecke character \(\eta\,\overline{\kappa_u}\), with \(\kappa_u\) the Kummer character of Lemma 3.1 of Lesson 3; as an idèle class character of finite order it has a well-defined conductor, and its ideal function, extended by zero on the primes of the conductor, is the primitive character inducing \(\eta\overline{\chi_\bullet(u)}\). Write \(L_F(s,\omega)\) for the complete finite \(L\)-function of a primitive character, a sum over all nonzero integral ideals of \(\mathcal O_F\).

Lemma 1.1 (the family [OpenAI-029, (8.1)]). For \(U\ge1\):

  1. \(\#\{u\in\mathcal U:U\le q_u<2U\}\le C\,U\), and \(\operatorname N\mathfrak f_{\omega_u}\le C\,U\) for these \(u\);
  2. the characters \(\omega_u\), \(u\in\mathcal U\), are pairwise distinct, and at most one of them is principal;
  3. their local components at the places of \(S\) belong to a fixed finite set.

Proof. Part 1 is Lemma 4.1 of Lesson 3. For part 2, \(\omega_u=\omega_{u'}\) means \(\kappa_u=\kappa_{u'}\), so \(u=u'\) by Lemma 3.2 of Lesson 3; a principal \(\omega_u\) means \(\kappa_u=\eta\), which by distinctness happens for at most one \(u\). For part 3, the component of \(\kappa_u\) at \(v\in S\) is \(y\mapsto(u,y)_{6,v}\) (Lemma 3.1 of Lesson 3), which depends only on the class of \(u\) in the finite group \(F_v^\times/F_v^{\times6}\). \(\square\)

Lemma 1.2 (the divisor bound). Let \(\tau_F(\mathfrak n)\) be the number of integral ideals dividing \(\mathfrak n\). For every \(\epsilon>0\), \(\tau_F(\mathfrak n)\le C_{F,\epsilon}(\operatorname N\mathfrak n)^\epsilon\).

Proof. \(\tau_F(\mathfrak n)(\operatorname N\mathfrak n)^{-\epsilon}=\prod_{\mathfrak p^a\parallel\mathfrak n}(a+1)q_{\mathfrak p}^{-a\epsilon}\). If \(q_{\mathfrak p}\ge2^{1/\epsilon}\), the factor is at most \((a+1)2^{-a}\le1\). The finitely many primes with \(q_{\mathfrak p}<2^{1/\epsilon}\) give factors at most \(\sup_{a\ge0}(a+1)2^{-a\epsilon}<\infty\). \(\square\)

2. A large sieve for primitive characters of bounded conductor

Lemma 2.1 (primitive conductor large sieve [OpenAI-029, Lemma 8.1]). Let \(C_0>0\), let \(\mathscr L_S\) be a finite set of possible local components at \(S\), and let \(\mathscr X(U)\) be a family of distinct primitive finite-order Hecke characters of \(F\) with conductor norms at most \(C_0U\) and components at \(S\) in \(\mathscr L_S\). For \(N,U\ge2\), complex numbers \(b_{\mathfrak n}\) and \(\epsilon>0\), \[ \begin{aligned} &\sum_{\omega\in\mathscr X(U)}\Bigl|\sum_{\operatorname N\mathfrak n\le N}b_{\mathfrak n}\,\omega(\mathfrak n)\Bigr|^2\\ &\quad \le C_{F,S,C_0,\mathscr L_S,\epsilon}\,(N+U^2)(NU)^\epsilon\\ &\qquad \sum_{\operatorname N\mathfrak n\le N}|b_{\mathfrak n}|^2 ,\quad(2.1) \end{aligned} \] the sums running over integral ideals of \(\mathcal O_F\).

Proof. Reduction to exterior ideals. Write \(\mathfrak n=\mathfrak n_S\mathfrak n^S\) with \(\mathfrak n_S\) supported on the finite places of \(S\) and \(\mathfrak n^S\) prime to them. There are \(O((1+\log N)^{|S|})\) possible \(\mathfrak n_S\) of norm at most \(N\), and for characters with the same components at \(S\) the value \(\omega(\mathfrak n_S)\) is the same (or zero). Partition \(\mathscr X(U)\) according to the components at \(S\), apply Cauchy–Schwarz over the \(\mathfrak n_S\) and over \(O(\log N)\) dyadic ranges of \(\operatorname N\mathfrak n^S\); the losses are powers of \(\log N\), absorbed in \(N^\epsilon\). It remains to prove (2.1) for coefficients on exterior ideals of norm in \((N_1,2N_1]\), \(N_1\le N\), and characters with one fixed set of components at \(S\).

Values through residue characters. Let \(\omega\) be such a character and \(\mathfrak q\) the exterior part of its conductor. For an exterior ideal \(\mathfrak n=nR\), with \(n\) its balanced generator (Lemma 4.1 of Lesson 2), evaluate the idèle character at the principal idèle \(n\): its value is \(1\), and the unramified exterior components give \(\omega(\mathfrak n)\) when \(\mathfrak n\) is prime to \(\mathfrak q\). Hence \[ \begin{gathered} \omega(\mathfrak n)=\lambda_S(n)\,\lambda_{\mathfrak q}(n),\\ \lambda_S(n)=\prod_{v\in S}\omega_v(n)^{-1},\\ \lambda_{\mathfrak q}(n)=\prod_{v\mid\mathfrak q}\omega_v(n)^{-1}, \end{gathered} \] where \(\lambda_S\) is the same unitary function for the whole slice, and \(\lambda_{\mathfrak q}\) is a primitive character of \((R/\mathfrak q)^\times\), extended by zero; both sides vanish when \(\mathfrak n\) meets \(\mathfrak q\). In particular \(\omega\) is determined by \(\lambda_{\mathfrak q}\).

Gauss sums. Let \(q\) generate \(\mathfrak q\), and for a primitive \(\lambda\) modulo \(\mathfrak q\) put \(\tau(\overline\lambda)=\sum_{d\bmod\mathfrak q}\overline{\lambda(d)}\,\mathrm e(d/q)\). Then for every \(n\in R\), \[ \begin{gathered} \begin{aligned} \lambda(n)\,\tau(\overline\lambda)&=\sum_{d\bmod\mathfrak q}\overline{\lambda(d)}\,\mathrm e(nd/q), \end{aligned}\\ |\tau(\overline\lambda)|^2=q_{\mathfrak q}.\quad(2.2) \end{gathered} \] For a unit \(n\) the identity follows from \(d\mapsto n^{-1}d\). If a prime \(\mathfrak p\mid\mathfrak q\) divides \(n\), let \(H\) be the group of units congruent to \(1\) modulo \(\mathfrak q\mathfrak p^{-1}\); for \(h\in H\), \(nd(h-1)\in\mathfrak q\), so \(\mathrm e(ndh/q)=\mathrm e(nd/q)\), while \(\lambda\) is nontrivial on \(H\) by primitivity, so the right side equals \(\overline{\lambda(h)}\) times itself for some \(\lambda(h)\ne1\) and vanishes. Finally, \((n,d)\mapsto\mathrm e(nd/q)\) is a perfect pairing on \(R/\mathfrak q\) (Lemma 2.1 of Lesson 2), so Parseval's identity (expand the square: for \(d\ne d'\) the character \(n\mapsto\mathrm e(n(d-d')/q)\) is nontrivial and sums to zero over \(R/\mathfrak q\)) gives \(\sum_n|\sum_d\overline{\lambda(d)}\mathrm e(nd/q)|^2=q_{\mathfrak q}|(R/\mathfrak q)^\times|\), while by the identity the left side is \(|(R/\mathfrak q)^\times|\,|\tau(\overline\lambda)|^2\).

The sieve. Put \(T_{d/q}=\sum_{\mathfrak n}b_{\mathfrak n}\lambda_S(n)\mathrm e(nd/q)\). By (2.2), the sum over the characters of exterior conductor \(\mathfrak q\) is at most \[ \begin{aligned} &\frac1{q_{\mathfrak q}}\sum_{\lambda\bmod\mathfrak q}\Bigl|\sum_{d}\overline{\lambda(d)}\,T_{d/q}\Bigr|^2\\ &\quad =\frac{|(R/\mathfrak q)^\times|}{q_{\mathfrak q}}\sum_{d\in(R/\mathfrak q)^\times}|T_{d/q}|^2\\ &\quad \le\sum_{d\in(R/\mathfrak q)^\times}|T_{d/q}|^2 , \end{aligned} \] by orthogonality of the characters of \((R/\mathfrak q)^\times\). Sum over \(q_{\mathfrak q}\le C_0U\). The fractions \(d/q\) are distinct reduced fractions, and the balanced generators of the ideals on the dyad lie in \(\ell_{2N_1}K\) for a fixed compact \(K\). The exchanged form of Lemma 3.1 of Lesson 11, with \(V=\max(2,C_0U)\), bounds the total by \(C(N+U^2)\sum|b_{\mathfrak n}|^2\), since \(|\lambda_S|=1\). \(\square\)

3. A conservative convexity bound

Lemma 3.1 ([OpenAI-029, (8.3)]). Let \(A_F=[F:\mathbb Q]\). For every \(\epsilon>0\), every nonprincipal primitive finite-order Hecke character \(\omega\) of \(F\) with \(\operatorname N\mathfrak f_\omega\le C_0U\), \(U\ge1\), and all \(0\le\sigma\le4\), \(t\in\mathbb R\), \[ \begin{aligned} &|L_F(\sigma+it,\omega)|\\ &\quad \le C_{F,C_0,\epsilon}\,U^{1/2+\epsilon}(2+|t|)^{A_F}.\quad(3.1) \end{aligned} \]

Proof. Let \(0<\epsilon'\le\min(\epsilon,\frac14)\). Characters of finite order are trivial on \(\mathbb C^\times\), which is divisible, so in the notation of (2) of Hecke L-functions and the Dedekind zeta function every infinite component is trivial and \(\Gamma_\infty(s,\omega)=\Gamma_{\mathbb C}(s)^{r}\), \(2r=A_F\). On \(\operatorname{Re}s=1+\epsilon'\), the Euler product gives \(|L_F|\le\zeta_F(1+\epsilon')\). On \(\operatorname{Re}s=-\epsilon'\), Theorem 10.1 there gives \[ \begin{gathered} \begin{aligned} |L_F(s,\omega)|&=Q^{1/2+\epsilon'}\Bigl|\frac{\Gamma_{\mathbb C}(1-s)}{\Gamma_{\mathbb C}(s)}\Bigr|^r\\ &\quad\times|L_F(1-s,\overline\omega)|, \end{aligned}\\ Q=|d_F|\operatorname N\mathfrak f_\omega , \end{gathered} \] and Corollary 4.1 of The Gamma function and Stirling's formula bounds the Gamma quotient by \(C(2+|t|)^{1+2\epsilon'}\) (for \(|t|\le1\) by continuity, as \(\Gamma\) has no pole at \(-\epsilon'+it\)). So \(|L_F(-\epsilon'+it,\omega)|\)\(\le CU^{1/2+\epsilon'}(2+|t|)^{r(1+2\epsilon')}\). In the strip, \(\omega\) is nontrivial on the norm-one idèle classes (a character of finite order is trivial on the positive reals), so by (19) of Theorem 9.2 of Tate's global theory the completed function is entire and bounded on vertical strips, each of its two terms being an integral over norms at least one that converges uniformly there; with Stirling for \(1/\Gamma_{\mathbb C}\), this gives \(|L_F(s,\omega)|\le D\exp(D(1+|t|))\) on the strip. Theorem 2.1 of Growth in the critical strip, whose proof bounds the constant by a fixed multiple of the larger boundary constant, gives (3.1) for \(-\epsilon'\le\sigma\le1+\epsilon'\), with exponent \(r(1+2\epsilon')\le A_F\); for \(1+\epsilon'\le\sigma\le4\) the Euler product suffices. \(\square\)

The constant in (3.1) is crude but uniform; this is all the argument needs. The same proof gives polynomial bounds in \(|t|\) on fixed strips for \((s-1)\zeta_F(s)\). Omitting the Euler factors at \(S\), or at the primes dividing \(u\), changes such bounds on a line \(\operatorname{Re}s=\sigma_*>0\) by at most \(C_\epsilon q_u^\epsilon\), since \(\prod_{\mathfrak p\mid uR}(1\pm q_{\mathfrak p}^{-\sigma_*})^{\pm1}\le C_{\epsilon,\sigma_*}q_u^\epsilon\) by the argument of Lemma 1.2.

4. Zero detection

Proposition 4.1 (exceptional frequencies [OpenAI-029, Proposition 8.2]). There are \(\rho>0\) and \(U_0\), depending on \(F\), \(S\) and \(\eta\), with the following property. For every \(U\ge U_0\) there is a set \(\mathcal E(U)\subseteq\{u\in\mathcal U:U\le q_u<2U\}\) with \[ |\mathcal E(U)|\le C\,U^{1/2}, \] containing every \(u\) for which \(\omega_u\) is principal or \(L_F(s,\omega_u)\) has a zero with \(\operatorname{Re}s\ge0.996\), \(|\operatorname{Im}s|\le4U^\rho\). For \(u\notin\mathcal E(U)\) and \(\epsilon>0\), \[ \begin{gathered} \bigl|L^S\bigl(\xi,\eta\overline{\chi_\bullet(u)}\bigr)^{-1}\bigr|\le C_\epsilon\,U^\epsilon\\ (\operatorname{Re}\xi\ge0.998,\ |\operatorname{Im}\xi|\le2U^\rho).\quad(4.1) \end{gathered} \]

Proof. Put \(\sigma_0=0.996\) and fix \(0<\rho<\min\{\frac1{10},\frac1{A_F+1}\}\).

The detector. For a nonprincipal \(\omega=\omega_u\) in the dyad, put \[ \begin{gathered} \begin{aligned} D_U(s,\omega)&=\sum_{\operatorname N\mathfrak d\le U^2}\mu_F(\mathfrak d)\,\omega(\mathfrak d)\,\operatorname N\mathfrak d^{-s}, \end{aligned}\\ \alpha_U(\mathfrak n)=\sum_{\substack{\mathfrak d\mid\mathfrak n\\ \operatorname N\mathfrak d\le U^2}}\mu_F(\mathfrak d), \end{gathered} \] with \(\mu_F\) the Möbius function of integral ideals. By complete multiplicativity of \(\omega\) (with the zero extension), \(L_F(s,\omega)D_U(s,\omega)\)\(=\sum_{\mathfrak n}\alpha_U(\mathfrak n)\omega(\mathfrak n)\operatorname N\mathfrak n^{-s}\) for \(\operatorname{Re}s>1\), and \[ \begin{gathered} \alpha_U(\mathcal O_F)=1,\\ \alpha_U(\mathfrak n)=0\quad(1<\operatorname N\mathfrak n\le U^2),\\ |\alpha_U(\mathfrak n)|\le\tau_F(\mathfrak n),\quad(4.2) \end{gathered} \] the middle statement because all divisors of such \(\mathfrak n\) are counted and \(\sum_{\mathfrak d\mid\mathfrak n}\mu_F(\mathfrak d)=0\) for \(\mathfrak n\ne\mathcal O_F\).

Suppose \(L_F(\beta+i\gamma,\omega)=0\) with \(\beta\ge\sigma_0\) and \(|\gamma|\le4U^\rho\); then \(\beta\le1\) by the Euler product. Mellin inversion of \(e^{-x}\) (Fourier inversion, Section F4 of Schwartz functions and Fourier inversion, for the Schwartz function \(u\mapsto e^{cu}e^{-e^u}\), whose Fourier transform is \(\tau\mapsto\Gamma(c-i\tau)\)) gives, with \(c>1\) and the sum and the integral interchanged by absolute convergence, \[ \begin{aligned} &\sum_{\mathfrak n}\alpha_U(\mathfrak n)\omega(\mathfrak n)\operatorname N\mathfrak n^{-\beta-i\gamma}\\ &\qquad e^{-\operatorname N\mathfrak n/U^{10}}=\frac1{2\pi i}\\ &\qquad \int_{(c)}L_F(\beta+i\gamma+v,\omega)\\ &\qquad \,D_U(\beta+i\gamma+v,\omega)\,\Gamma(v)\,U^{10v}\\ &\qquad \,dv .\quad(4.3) \end{aligned} \] Move the line to \(\operatorname{Re}v=\frac12-\beta\in[-\frac12,-0.496]\) by the residue formula (Lemma A.3 of Dirichlet series and Euler products). The only pole crossed is \(v=0\), with residue \(L_F(\beta+i\gamma,\omega)D_U(\beta+i\gamma,\omega)=0\); \(L_F(\cdot,\omega)\) is entire, and the exponential decay of \(\Gamma\) against the polynomial bound (3.1) removes the horizontal sides. On the new line, \(|D_U|\le\sum_{\operatorname N\mathfrak d\le U^2}\operatorname N\mathfrak d^{-1/2}\le CU\) by Lemma 3.1 of Lesson 10, \(|U^{10v}|=U^{10(1/2-\beta)}\le U^{-4.96}\), and (3.1) at real part \(\frac12\) gives \[ \begin{aligned} &\Bigl|\text{right side of (4.3)}\Bigr|\\ &\quad \le C\,U^{-4.96+1/2+1+\epsilon}(2+4U^\rho)^{A_F}\\ &\quad \le C\,U^{-3.46+A_F\rho+\epsilon}=o(1),\quad(4.4) \end{aligned} \] since \(A_F\rho<1\).

On the left of (4.3), the ideal \(\mathcal O_F\) contributes \(e^{-U^{-10}}=1+o(1)\), the ideals with \(1<\operatorname N\mathfrak n\le U^2\) contribute nothing, and those with \(\operatorname N\mathfrak n>U^{11}\) contribute at most a fixed power of \(U\) times \(e^{-U/2}\), by Lemma 1.2 and the ideal count. So the sum over \(U^2<\operatorname N\mathfrak n\le U^{11}\) has absolute value at least \(\frac12\) for large \(U\). Split this range into \(O(\log U)\) dyads \((N,2N]\). On each dyad write the weight as \(\operatorname N\mathfrak n^{-\sigma_0}\) times \(w(\operatorname N\mathfrak n)\), \(w(t)=t^{\sigma_0-\beta}e^{-t/U^{10}}\), which is positive, decreasing and at most \(1\). Order the ideals of the dyad by norm; by summation by parts, the weighted sum is at most twice the largest initial segment (a prefix) of the unweighted sums. Hence for some dyad \(U^2/2<N\le U^{11}\) and some prefix, \[ \begin{aligned} &\Bigl|\sum_{\mathfrak n\in\text{prefix}}\alpha_U(\mathfrak n)\,\omega(\mathfrak n)\operatorname N\mathfrak n^{-\sigma_0-i\gamma}\Bigr|\\ &\quad \ge\frac c{\log U}.\quad(4.5) \end{aligned} \]

Counting. A prefix of a dyad is a disjoint union of at most \(1+\log_2M\) dyadic blocks of consecutive indices, \(M\) being the number of ideals in the dyad, one block from each level of the binary subdivision; by Cauchy–Schwarz its square is at most \((1+\log_2M)\) times the sum of the squares over all blocks of all levels. For one block \(I\), let \(P_I(t)=\sum_{\mathfrak n\in I}\alpha_U(\mathfrak n)\omega(\mathfrak n)\operatorname N\mathfrak n^{-\sigma_0-it}\). For a smooth function on an interval of length three, the supremum of its square on the middle unit interval is at most a constant times the integral of \(|P_I|^2+|P_I'|^2\) over the interval, and \(P_I'\) has the coefficients multiplied by \(-i\log\operatorname N\mathfrak n\). Apply Lemma 2.1, with Lemma 1.1, at each \(t\), integrate over \(|t|\le4U^\rho+2\), and sum over blocks and levels. The squared coefficient mass on a dyad is \[ \sum_{N<\operatorname N\mathfrak n\le2N}\tau_F(\mathfrak n)^2\operatorname N\mathfrak n^{-2\sigma_0}\le C_\epsilon N^{1-2\sigma_0+\epsilon}, \] and since \(U^2/2<N\le U^{11}\), \[ \begin{aligned} &(N+U^2)\,N^{1-2\sigma_0+\epsilon}\le3N^{2-2\sigma_0+\epsilon}\\ &\quad \le3\,U^{11(0.008+\epsilon)}\\ &\quad =3\,U^{0.088+11\epsilon}.\quad(4.6) \end{aligned} \] Altogether the sum over the characters of the dyad of the largest squared prefix over \(|t|\le4U^\rho\) is at most \(C\,U^{0.088+\rho+O(\epsilon)}(\log U)^{C}\). Each character with a zero in the rectangle contributes at least \(c(\log U)^{-2}\) by (4.5), for one of the \(O(\log U)\) dyads. Hence at most \(C\,U^{0.088+\rho+O(\epsilon)}\le U^{1/2}\) characters have such a zero, for \(\epsilon\) small, because \(0.088+\rho<0.19\). Let \(\mathcal E(U)\) consist of the corresponding \(u\), which are distinct by Lemma 1.1, together with the \(u\) (at most one) for which \(\omega_u\) is principal.

The reciprocal. Let \(T=U^\rho\), \(u\notin\mathcal E(U)\), \(\omega=\omega_u\), and \(|t|\le2T\). Consider the discs centred at \(2+it\) with radii \[ \begin{gathered} R_2=1.0035,\\ r_2=1.003,\\ r_1=1.002,\\ r_0=0.999 . \end{gathered} \] For large \(U\) the disc of radius \(R_2\) lies in \(\operatorname{Re}s>0.996\), \(|\operatorname{Im}s|<4T\), where \(L_F(s,\omega)\) is entire and has no zero; so \(\log L_F(s,\omega)\) has a branch there, equal at the centre to the logarithm of the Euler product. On the disc, \(\operatorname{Re}\log L_F=\log|L_F|\le C\log U\) by (3.1), and \(|\log L_F(2+it,\omega)|\le\log\zeta_F(2)\). Theorem 2.1 of Entire functions of order one and the Hadamard product of xi (Borel–Carathéodory), applied to \(\log L_F(s)-\log L_F(2+it)\), gives \(|\log L_F|\le C\log U\) on the disc of radius \(r_2\). On the disc of radius \(r_0\), the real parts are at least \(1.001\) and \(|\log L_F|\le\log\zeta_F(1.001)\). Hadamard's three circles theorem (Theorem 2.1 of The Riemann hypothesis implies the Lindelöf hypothesis) gives, on the circle of radius \(r_1\), \[ \begin{gathered} |\log L_F(s,\omega)|\le C(\log U)^\theta,\\ \theta=\frac{\log(1.002/0.999)}{\log(1.003/0.999)}<1 . \end{gathered} \] These discs reach \(\operatorname{Re}s=0.998\), and absolute convergence covers \(\operatorname{Re}s\ge3\). So \(|L_F(s,\omega)^{-1}|\le\exp(C(\log U)^\theta)\le C_\epsilon U^\epsilon\) for \(\operatorname{Re}s\ge0.998\), \(|\operatorname{Im}s|\le2T\). Finally \(L^S(\xi,\eta\overline{\chi_\bullet(u)})^{-1}\) differs from \(L_F(\xi,\omega_u)^{-1}\) by the Euler factors at \(S\) and at the primes dividing \(u\), which contribute at most \(C_\epsilon q_u^\epsilon\le C_\epsilon'U^\epsilon\) (end of Section 3). \(\square\)

The numbers \(0.996\), \(0.998\) and \(\frac12\) are absolute; \(\rho\), \(U_0\) and the constants depend on the fixed data. Only \(\rho>0\) matters in Lesson 14.

5. Exercises

5.1. Show that \(\theta=\log(1.002/0.999)/\log(1.003/0.999)\) is about \(0.750\), and explain why \(\theta<1\) is all that is needed.

5.2. Prove that a character of \(\mathbb C^\times\) of finite order is trivial.

5.3. In (4.4), check the exponent: \(10(\frac12-0.996)+\frac12+1=-3.46\).

5.4. In (4.6), check that \(11(2-2\cdot0.996)=0.088\), and that \(0.088+\rho<\frac12\) leaves room for every \(\rho<\frac1{10}\).

5.5. Let \(f\) be continuously differentiable on \([j-1,j+2]\). Show that \(\sup_{[j,j+1]}|f|^2\)\(\le\frac13\int_{j-1}^{j+2}|f|^2+2\int_{j-1}^{j+2}|f||f'|\).

6. Solutions

5.1. \(\log(1.002/0.999)\approx0.0029985\) and \(\log(1.003/0.999)\approx0.0039960\), so \(\theta\approx0.7504\). Any \(\theta<1\) makes \((\log U)^\theta=o(\log U)\), so \(\exp(C(\log U)^\theta)\) is smaller than every positive power of \(U\).

5.2. If \(\chi^m=1\) and \(z\in\mathbb C^\times\), choose \(w\) with \(w^m=z\); then \(\chi(z)=\chi(w)^m=1\).

5.3. \(10(0.5-0.996)=-4.96\), and \(-4.96+1.5=-3.46\).

5.4. \(2-1.992=0.008\) and \(11\cdot0.008=0.088\); \(0.088+0.1=0.188<0.5\).

5.5. For \(t\in[j,j+1]\) and \(t'\in[j-1,j+2]\), \(|f(t)|^2\le|f(t')|^2+2\int_{j-1}^{j+2}|f||f'|\); average over \(t'\).

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