2 added 14 characters in body

This might not be exactly what you have in mind, but it is an old result of Paley that for an infinite (but very sparse) set of real Dirichlet characters the inequality

$max_{N} |\sum_{n}^{N} \chi(n)| > c Q \sqrt{Q} \ln\ln(Q)$

holds, where Q is the modulus of the character. Montgomery and Vaughan have shown the reverse inequality (on RH) for all Dirichlet characters, that is to say,

$max_{N} |\sum_{n}^{N} \chi(n)| < c Q \sqrt{Q} \ln\ln(Q)$.

Recently Paley's theorem was improved by Granville and Soundararajan in their work on Pretentious characters (among other things, they prove the lower bound holds for a larger set of characters).

references: R.E.A.C. Paley, A theorem on characters, J. London Math. Soc 7 (1932), 28-32

H.L. Montgomery and R.C. Vaughan, Exponential sums with multiplicative coefficients, Invent. Math 43 (1977), 69-82.

LARGE CHARACTER SUMS: PRETENTIOUS CHARACTERS AND THE POLYA-VINOGRADOV THEOREM: http://arxiv.org/abs/math.NT/0503113.

1

This might not be exactly what you have in mind, but it is an old result of Paley that for an infinite (but very sparse) set of real Dirichlet characters the inequality

$max_{N} |\sum_{n}^{N} \chi(n)| > c Q \ln\ln(Q)$

holds, where Q is the modulus of the character. Montgomery and Vaughan have shown the reverse inequality (on RH) for all Dirichlet characters, that is to say,

$max_{N} |\sum_{n}^{N} \chi(n)| < c Q \ln\ln(Q)$.

Recently Paley's theorem was improved by Granville and Soundararajan in their work on Pretentious characters (among other things, they prove the lower bound holds for a larger set of characters).

references: R.E.A.C. Paley, A theorem on characters, J. London Math. Soc 7 (1932), 28-32

H.L. Montgomery and R.C. Vaughan, Exponential sums with multiplicative coefficients, Invent. Math 43 (1977), 69-82.

LARGE CHARACTER SUMS: PRETENTIOUS CHARACTERS AND THE POLYA-VINOGRADOV THEOREM: http://arxiv.org/abs/math.NT/0503113.