4
$\begingroup$

The classical explicit formula for the Riemann Zeta function states that $$ \psi(x)=x-\sum_{\rho} \frac{x^{\rho}}{\rho}+O(1), $$ where $\psi(x)=\sum_{n \leq x} \Lambda(n)$ and the sum is over all non-trivial zeroes of $\zeta(s)$.

Let $L/K$ be a finite Galois extension of number fields and let $V$ be an irreducible complex representation of $\textrm{Gal}(L/K)$. Attached to this piece of data, one can define the Artin $L$-function $L(V, s)$. Does a similar formula exist for $L(V, s)$?

I would also be interested in a similar formula for the slightly less general case of Hecke $L$-functions.

$\endgroup$

2 Answers 2

7
$\begingroup$

One proves explicit formulae essentially by integrating the logarithmic derivative of the $L$-function. For simplicity, let $\psi$ be a Schwartz function with $\psi(1) = 1$, and let $$ \widehat{\psi}(s) = \int_0^\infty \psi(t) t^{s} \frac{dt}{t} $$ be its Mellin transform. Then one can get an explicit formula by considering $$ \sum_{n \geq 1} \Lambda_V(n) \psi(n) = \frac{-1}{2\pi i} \int_{(2)} \frac{L'}{L}(V, s) \widehat{\psi}(s) ds. $$ Typically one would shift the line of integration to $\mathrm{Re}(s) = c < 0$, apply the functional equation, and get an expression of the form $$\begin{align} \sum_{n \geq 1} &\big( \Lambda_V(n) \psi(n) + \Lambda_\overline{V}(n) \tfrac{\psi(n^{-1})}{n} \big) = \\ &\{ \mathrm{polar\;data} \} - \sum_\rho \widehat{\psi}(\rho) + \frac{1}{2\pi i} \int_{(1/2)}\big( \tfrac{\gamma'}{\gamma}(V, s) + \tfrac{\gamma'}{\gamma}(V, 1-s) \big) \widehat{\psi}(s) ds + O(1). \end{align}$$ Here, I take $\overline{V}$ to be the conjugate representation and assume the functional equation is of the form $$ Q^{s/2} \gamma(V, s) L(V, s) = \varepsilon(V) Q^{(1-s)/2} \gamma(V, 1-s) L(\overline{V}, 1-s) $$ for collected gamma factors $\gamma(V, s)$ and a root number $\lvert \varepsilon(V) \rvert = 1$. (Brauer's On Artin's L-series with general group characters details the general functional equations, and this explicit formula is a in $\S$5.5 of Iwaniec–Kowalski).

But a major challenge in the face of being computationally useful is that we don't know the complete polar data for an arbitrary Artin $L$-function. Assuming Artin's conjecture and taking a nontrivial irreducible representation, there should be no polar contribution and we get a classical explicit formula.

$\endgroup$
2
  • $\begingroup$ Thanks for the answer! Can I just clarify what the $\Lambda_V$ appearing in your answer is? $\endgroup$
    – Dekimshita
    Apr 6 at 18:43
  • $\begingroup$ Those are the generalized von Mangoldt numbers associated to $L(V, s)$. They are the coefficients of the Dirichlet series $L'/L(V, s)$. For primes, one has $\Lambda_V(p) = a(p) \log p$ where $a(p)$ is the $p$th coefficient of $L(V, s)$. At prime powers, you'll have a slightly more complicated relation that can be written down in terms of the factors in the Euler product. See 5.26 in Iwaniec&ndash;Kowalski for more. $\endgroup$ Apr 6 at 19:07
1
$\begingroup$

Yes, a similar formula exists for $L(V,s)$. See : Weil André, Sur les formules explicites de la théorie des nombres. Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 3–18.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

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