4
$\begingroup$

Let me first provide context for this question.

There is a series of four exercises in Ireland & Rosen's book (in second edition it's exercises 14-17 in chaprer 16), aim of which is (although this is not relevant to my question) to establish that in the interval $0<x<\frac{p}{2}$ there are more quadratic residues than nonresidues $\mod p$ for $p\equiv 3\pmod 4$.

The content of exercise 17, when restated, is essentially the following:

Let $\chi$ be the Dirichlet character $\mod 2p$ such that $\chi(n)=\left(\frac{n}{p}\right)$ for odd $n$. Given that $L(1,\chi)\neq 0$, show $L(1,\chi)>0$.

Since I am asking this question in the context of mentioned book, let me briefly go through what was established in the chapter (for the same of people who don't own the book and want to give an answer at the adequate level):

  • Definitions of: zeta function, Dirichlet characters and L-functions was given and their basic properties were established
  • Nonvanishing of L-functions has been proven by standard method for complex characters and for real characters using Landau's lemma (concerning Dirichlet series with nonnegative coefficients)
  • I suppose these aren't related to my question, but just in case I mention these: Dirichlet's theorem was proven, it was shown that L-functions can be continued to the whole (punctured) complex plane, and L-functions have been evaluated at negative integers with help of generalized Bernoulli numbers.

The exercise so restated seems to be quite difficult, since sign of L-function value was at no point dealt with in the chapter. At this point you can guess what my question:

How to solve this exercise without using tools outside the ones provided by Ireland & Rosen in their book?

I know from other places that the value of $L(1,\chi)$ for real $\chi$ is always positive. So, extending my question:

Is there an easy way to show, for general real Dirichlet character $\chi$, that $L(1,\chi)>0$, given that we know it's not zero?

Let me also note that I am aware of a proof of nonvanishing of L-functions which at the same time proves that the value is positive, but this wouldn't be a valid answer to my question - as far as I can recall, the proof was by itself pretty nontrivial, and I'm looking for something that, in theory, could've been figured out by a reader of the book who, say, has not encountered analytic number theory before.

Thanks in advance.

$\endgroup$
  • $\begingroup$ At least we need the definition of L function, some theory of Dirichlet series, analytic continuation. I would not expect someone without analytic number theory background to solve this. $\endgroup$ – Sungjin Kim Jul 4 '16 at 18:48
  • $\begingroup$ @i707107 Definition of L-functions is within scope of the chapter. As for theory of Dirichlet series and analytic continuation, some basics of it have been developed throughout the chapter (indeed, enough to prove nonvanishing of L-functions). $\endgroup$ – Wojowu Jul 4 '16 at 18:58
3
$\begingroup$

The Euler product for $L(s,\chi)$ will tell you that $L(\sigma,\chi)>0$ for real $\sigma>1$. Moreover, $L(s,\chi)$ is continuous....

$\endgroup$
  • $\begingroup$ Damn, I can't believe the answer is this simple! Thank you very much. $\endgroup$ – Wojowu Jul 4 '16 at 20:09
  • $\begingroup$ I'm pretty sure figuring out this answer took you far less time than it tooke me to write up the question :P $\endgroup$ – Wojowu Jul 4 '16 at 20:16
  • $\begingroup$ (+1) Great! Much simpler than the method I thought about. $\endgroup$ – Sungjin Kim Jul 4 '16 at 20:24

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.