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For starters, see Montgomery & Vaughan's "Multiplicative Number Theory", Theorem 11.3 and 11.4. For example, in the zero free region of 11.3, $$ \frac{1}{L(s,\chi)} \ll \log(q(|t|+4)). $$

Not an answer, but long for a comment. Siegel's theorem referenced above does not see the individual forms, it instead uses the Dirichlet class number formula for $L(1,\chi)$ and estimates the asympt …

In answer to #1, I'm not quite sure what you're asking but I'll point out the Epstein zeta function $\zeta_Q(s)=\sum^\prime_{(m,n)}Q(m,n)^{-s}$ attached to, say, a positive definite binary quadratic f …

The normal order of $\log(d(n))$ is $\log(2)\log\log(n))$. So, for every $\epsilon>0$, $$ \log(d(n))<(1+\epsilon)\log(2)\log(\log(n)) $$ hold for almost all n: that is, if the proportion of $n\le x$ …

This is not an answer to what the best known upper bound is, but rather a comment that the (known) average distribution of divisors indicates you might not expect to do any better than the bounds on t …

In answer to #2, the Riemann-Siegel formula was developed by Siegel based on unpublished posthumous notes of Riemann, the Nachlass. It writes the Hardy function $Z(t)$ as a sum of two finite series o …

Actually, the Riemann Hypothesis (in the stronger version which says all the zeros are simple) implies that all zeros of $\zeta^\prime(s)$ have real part $> 1/2$. This is proved in "Zeros of the deri …

The reason that $\mu(n)$ and $\lambda(n)$ have such expressions is that the corresponding Dirichlet generating functions can be expressed in terms of the Riemann zeta function: $$ \sum_n \frac{\mu(n)} …

The answer above by KConrad above gives a very good 'big picture' explanation. I can think of three different 'nuts and bolts' explanations, which might be of interest. The first looks at the clas …

The third approach looks at a known zero free region. The simplest such theorem is that there is some $A$ such that $$ \zeta(s)\ne0\quad\text{for}\quad \sigma>1-A/\log(t). $$ What prevents us from 'st …

For a second approach, we could look at what is known about zeros of $\zeta(s)$ to the right of a vertical line $\text{Re}(s)=\sigma_0$. These are known as 'zero density results' and take the form o …

In skimming through Narkiewicz "The Development of Prime Number Theory", one sees a reference on p. 155 (footnote 38) to a R. Lipschitz, who in Crelle volume 54 in 1857 "studied the series $\sum_{n=1} …

In Real zeros of real odd Dirichlet $L$-functions, Mark Watkins showed in 2003 that $L(s,\chi_d)$ (as in the title) has no positive real zeros for $d<300,000,000$. (I think this is still state of the …

I am now very dubious that one can deduce anything about the sign of $L(\sigma,\chi)$ for $0<\sigma<1$ from the positivity of $$ \sum_{n<x}\frac{\chi(n)}{n}(1-n/x)^k. $$ Here's why. For simplicity c …

An example which is almost trivial: take the sequence $\{1,4,8,16,32\ldots\}$, i.e. powers of $2$ omitting two itself. For real part of $s>0$, the Dirichlet series is a convergent geometric series, s …