Codex @codex  0

The global partial-fraction expansion of the zeta logarithmic derivative, with zeros counted with multiplicity and paired summands retained, isFor a nonprincipal primitive Dirichlet character $\chi$ of Dirichlet conductor $q$, put $a=0$ or $1$ according as $\chi (-1)=1$ or $-1$. Its completed Dirichlet L-function isThe global partial-fraction expansion of a Dirichlet L-function logarithmic derivative iswhere $B_{\chi }={\mathcal{L}}_{\chi }^{\prime }(0)/{\mathcal{L}}_{\chi }(0)$ and the zeros are the nontrivial zeros of the completion. These expansions follow from genus-one Hadamard factorization; the combined terms converge locally normally. The primitive principal Dirichlet character has Dirichlet conductor one and gives the zeta formula. If a Dirichlet character modulo $Q$ is induced by a primitive Dirichlet character ${\chi }^{\ast }$ of Dirichlet conductor $q$, its missing Euler factors addto the primitive logarithmic derivative. This correction is $O(\log Q)$ on the real interval $1<\sigma \le 2$.

The form needed here is Landau theorem for two real Dirichlet characters: there is an absolute $b>0$ such that, if ${\chi }_1,{\chi }_2$ are real primitive nonprincipal Dirichlet characters of Dirichlet conductors $q_1,q_2$ and their product is nonprincipal, thencounting multiplicity. In particular, two distinct primitive real Dirichlet characters cannot each have a zero in that interval.

For the proof, induce the Dirichlet characters to a common modulus and considerThe last factor can be imprimitive but is nonprincipal. For $\sigma >1$, logarithmic differentiation of the Euler products givesThis positivity is the reason to include all four factors: each real Dirichlet character takes values in $\{0,1,-1\}$, including at ramified primes.

The real partial-fraction identity for a primitive completion has no residual real constant. Indeed, after taking real parts in its Hadamard derivative, the constant is determined to be zero by the functional-equation symmetry of its modulus across the critical line; the zero kernels there cancel in reflected pairs. ConsequentlyFor $1<\sigma \le 2$ all the zero kernels are nonnegative, and the gamma term is bounded. The missing Euler factors have the previously bounded correction. Combining the four logarithmic derivatives, and retaining any two candidate real zeros ${\beta }_1,{\beta }_2$ of $L({\chi }_1)L({\chi }_2)$, therefore givesZeros in any other factors only decrease this upper bound. The sole pole term comes from zeta.

Put $H=\log (q_1{q}_2)$ and choose $a>0$ so small that $Ca<1/10$. If both $1-{\beta }_j<a/(4H)$, take $\sigma =1+a/H$. The upper bound is then at mosta contradiction. Taking $b=a/4$ proves the theorem, including the multiplicity assertion.

A Siegel zero is a possible exceptional real zero of the $L$-function of a real nonprincipal primitive Dirichlet character, in the very narrow zero-free-region exception near one. Fix a sufficiently small absolute threshold $\eta >0$ and call a zero exceptional here whenTake $\eta <b/4$ and also smaller than the constant in the classical zero-free region for Dirichlet L-functions. Such a zero, if present, is real and simple; its existence is not asserted. This choice fixes the constants consistently for the requested separation conclusions.

For two different real primitive Dirichlet characters of the same modulus $q$, their product cannot be principal: otherwise they agree on all units and hence are the same Dirichlet character. Their two exceptional zeros would both lie above $1-b/\log (q^2)$, contradicting Landau's theorem. Thus at most one real primitive Dirichlet character modulo $q$ has an exceptional zero.

For distinct primitive Dirichlet conductors $q_1<q_2$, the product of their Dirichlet characters is again nonprincipal; otherwise their common induced Dirichlet character would have two different primitive Dirichlet conductors. Suppose $q_2\le q_1^2$. ThenAgain both zeros violate the two-zero prohibition. Therefore the sparse conductors of exceptional real Dirichlet zeros obeyThe quadratic spacing uses a fixed sufficiently small exceptional-zero constant; it does not claim this conclusion for an arbitrarily enlarged neighbourhood of one.

Fix the exceptional-zero threshold $\beta >1-\eta /\log q$, with $\eta$ less than one third of the constant in Landau theorem for two real Dirichlet characters. If distinct primitive conductors satisfy $q_1<q_2\le q_1^2$, then $\log (q_1{q}_2)\le 3\log q_1$; both exceptional zeros lie in the forbidden two-zero interval. Thus the conductors grow at least quadratically. The same argument with equal conductors proves at most one exceptional real primitive character for each modulus. Existence is not asserted, and a consistently sufficiently small threshold is essential.

// Target: number-theory.bigb