Let $f$ be a monotone decreasing, continuously differentiable function with $\lim_{x\rightarrow \infty}f(x)=0$. Let $\chi$ be a non-principal Dirichlet character. It is standard to show that $\sum_{n\geq x}\chi(n)f(n)=O(f(x))$, where the big-O constant is easily computable and depends only on $\chi$. In particular, we have $\sum_{n\leq x}\chi(n)f(n)=A+O(f(x))$ where $A=\sum_{n\in \mathbb{N}}\chi(n)f(n)$ is a constant.
When $f(x)=\log(x)/x$, Mertens used the fact that $\log(ab)=\log(a)+\log(b)$ to re-express the sum in terms of a sum over primes. He showed that $\sum_{p\leq x}\chi(p)\log(p)/p$ is bounded, in absolute value, by a computable constant. Then, by partial summation techniques, he removed the $\log(p)$ from the numerator and obtained bounds of the form $$\left|\sum_{p\leq x}\chi(p)/p- C \right| < D/\log(x)$$ where $C$ and $D$ are easily computable constants (possibly depending on $\chi$).
My question is two-fold. First, what conditions on a function $f$ (satisfying any of the nice properties above, or more) guarantees that $\sum_{p\in \mathbb{N}}\chi(p)f(p)$ exists?
Second, since $-L'(s;\chi)/L(s;\chi)$ is analytic in a neighborhood of $s=1$, we know that $\sum_{p\in\mathbb{N}}\chi(p)\log(p)/p$ converges, say to a constant E. Is there an easy way to obtain explicit bounds of the form $$\left|\sum_{p\leq x}\chi(p)\log(p)/p – E \right| < o(1)$$ where the $o$-function is fairly simple, etc…?
The reason I ask is that I want to find an effective formula for $\sum_{p\equiv a\pmod{k},\, p\leq x}\log(p)/p$, where the error term is small. If anyone has such a reference that would also be appreciated.
Best Answer
An update on this problem:
I found out how to compute effective (and asymptotically accurate) bounds for $\sum_{p\leq x,\, p\equiv a\pmod{k}}\log(p)/p$. Basically it boils down to the usual analytic techniques [see Rosser and Schoenfeld's paper "Formulas for some functions of prime numbers"], and effective bounds on $\theta(x;k,a)$ [e.g. see a 2002 paper of Pierre Dusart], which can be found by transferring information about zero-free regions of Dirichlet L-functions to this context. I'm still interested in the original question though.