Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 23 / 4

For $\sigma >1$, the Euler product positivity for L-function nonvanishing givesIndeed the logarithm expands into terms proportional to $3+4\cos \theta +\cos 2\theta =2(1+\cos \theta {)}^2\ge 0$. At primes dividing $q$, the character terms vanish and the remaining zeta term is positive. For a nonreal character, ${\chi }^2$ is nonprincipal, so its L-function is entire, even if imprimitive.

If $L(1+it,\chi )$ vanished to order $m\ge 1$, the product would be $O((\sigma -1{)}^{4m-3})$ as $\sigma \downarrow 1$: zeta has a simple pole, the last factor is bounded, and the middle factor has the asserted vanishing. The product would tend to zero, contradicting its lower bound one. This proves nonvanishing for every real $t$, including zero.

The Dirichlet-series coefficients areThey are multiplicative. At a prime power, their values are $k+1$ if ${\chi }_D(p)=1$, one for even $k$ and zero for odd $k$ if ${\chi }_D(p)=-1$, and one if ${\chi }_D(p)=0$. Thus every $a_n$ is nonnegative. In particular $a_{m^2}\ge 1$. This is the nonnegative zeta-times-real-L coefficients identity. The same Euler expansion gives $-{\zeta }^{\prime }/\zeta (\sigma )-L^{\prime }/L(\sigma ,{\chi }_D)\ge 0$ for $\sigma >1$, with coefficients $\Lambda (n)(1+{\chi }_D(n))\ge 0$.

Put $D_0=|D|$. The reference to part (c) in the printed hint is a reference to the positive-coefficient function from part (b). For $\sigma >1$ close to one, the preceding positivity and the supplied partial-fraction expansion giveAll omitted zero terms have nonnegative real parts because their real parts are at most one. The $O(1)$ zeta-pole remainder is included in $C\log D_0$, increasing the absolute constant if needed; a nonprincipal primitive real conductor of a Dirichlet character is at least three.

Suppose there were two real zeros, counted with multiplicity, with $\beta \ge 1-c/\log D_0$. Set $\sigma =1+a/\log D_0$. Division by $\log D_0$ givesChoose $C\ge 1$, $a=1/(4C)$ and $c=a/4$. The right side is strictly negative. ThusThis is the uniqueness of a possible exceptional real Dirichlet zero. It proves uniqueness, rather than existence of such a zero.