Question

In the paper "exponential sums with multiplicative coefficients", Maier claims that the Chevbychev function in arithmetic progression satisfies

$\theta(x, r, a) = x/(r-1) +O(x^{1-1/r^\epsilon})$ for sufficently large prime $r$ and arbitrary small $\epsilon>0$.

Is this an error?

See: http://projecteuclid.org/euclid.facm/1229442624

Answer 1

I feel like this is an error. It is true that $1-1/r^\epsilon$ is a known upper bound for the real zeros of Dirichlet $L$-functions corresponding to characters $\chi\pmod r$ (for $r$ sufficiently large in terms of $\epsilon$). However, no constant less than $1$ is known to be an upper bound for the real parts of all the zeros; the zero-free region is similar to the classical one for the Riemann $\zeta$ function, so that the best we know for zeros $\beta+i\gamma$ of $L(s,\chi)$ is something like $\beta \le 1-c/\log rt$ for some positive constant $c$. This would give something like $\theta(x;r,a) = x/(r-1) + O_r(x \exp(-c\sqrt{\log x}))$, which is weaker than what is claimed.

Answer 2

Greg Martin is right: Maier's claim is in error. Precisely, the following can be said (cf. Corollaries 11.15 and 11.20 in Montgomery-Vaughan: Multiplicative number theory I):

Theorem (Siegel-Walfisz with a twist). Let $\epsilon>0$ be fixed. Let $q\geq q_0(\epsilon)$ be sufficiently large in terms of $\epsilon$. There is an absolute constant $c>0$ such that $$\theta(x,q,a)=\frac{x}{\varphi(q)}+O(x^{1-cq^{-\epsilon}})\qquad\text{for}\qquad x\leq e^{q^{2\epsilon}},$$ and $$\theta(x,q,a)=\frac{x}{\varphi(q)}+O(x e^{-c\sqrt{\log x}})\qquad\text{for}\qquad x>e^{q^{2\epsilon}}.$$ The implied constants are absolute.

Perhaps Maier only needs the range $x\leq\exp(q^{2\epsilon})$, so his final conclusions are OK.

Answer 3

To Will Jagy: Sorry about the typing errors. The motivation is that it can be used to get a zero-free region of L-function better than anything on the literature, as discussed by Greg martin, and everything is elementary number theory.

Maier claims that it holds for all x => r, and said that the proof is a simple, see the paper. So the answer posted by G. H. should be for all x => r.