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Results tagged with l-functions
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Questions about generalizations of the Riemann Zeta function of arithmetic interest whose definition relies on meromorphic continuation of special kinds of Dirichlet series, such as Dirichlet L-functions, Artin L-functions, elements of the Selberg class, automorphic L-functions, Shimizu L-functions, p-adic L-functions, etc.
4
votes
1
answer
305
views
Effective, explicit bound on $L'(1,\chi)/L(1,\chi)$
Are there effective, explicit upper bounds on $L'(1,\chi)/L(1,\chi)$ in the literature, valid for all Dirichlet characters $\chi$? Due to the possibility of Siegel zeros, I don't imagine one can do mu …
- 20.2k
8
votes
1
answer
309
views
Explicit estimates for $N(T,\chi)$ (not $N(T,\chi)+N(T,\overline{\chi})$)
Let $N(T,\chi)$ denote the number of zeros of $L(s,\chi)$ with imaginary part between $0$ and $T$, with any zero with imaginary part equal to $T$ or to $0$ (not that the latter kind really exists) cou …
- 20.2k
3
votes
2
answers
312
views
Explicit formula: explicit work with general smoothing?
The following is a literature question, in the sense that I already know how to do what I am asking about, and in fact have already done it; now I'd like to write a brief historical overview as an int …
- 20.2k
5
votes
2
answers
931
views
$B(\chi), L'(1,\chi)/L(1,\chi),\dotsc$
Let $\chi$ be a primitive Dirichlet character of modulus $q>1$. Write, as is customary, $B(\chi)$ for the constant in the expression
$$\frac{\Lambda'(s,\chi)}{\Lambda(s,\chi)} = B(\chi) + \sum_\rho \l …
- 20.2k
6
votes
2
answers
309
views
Functional equation and/or growth estimates for a shifted L function
Consider the $L$-series defined by
$$L_{\alpha,\chi}(s) = \sum_{n\geq 1} \frac{e^{2\pi i \alpha \Omega(n)} \chi(n)}{n^s} = \prod_p \left(1 - \frac{e^{2\pi i \alpha} \chi(p)}{p^s}\right)^{-1}.$$
It sho …
- 20.2k
9
votes
3
answers
646
views
Vinogradov-Korobov for Dirichlet L-functions?
Where can one find a Vinogradov-Korobov zero-free region for Dirichlet L-functions? It has to be in a standard reference, but I'm having a non-trivial time finding it.
- 20.2k
15
votes
5
answers
2k
views
$|L'(1,\chi)/L(1,\chi)|$
Let $\chi$ be a primitive Dirichlet character $\mod q$, $q>1$. Is there a neat, simple way to give a good bound on $L'(1,\chi)/L(1,\chi)$?
Assuming no zeroes $s=\sigma+it$ of $L(s,\chi)$ satisfy $\si …
- 20.2k