# Name of important number-theoretical function $\lambda_p$

Volovich states in P-adic Analysis and Mathematical Physics (pages 54, 55) that the following function:

$$\lambda_p(a) = \begin{cases}1 & v_p(a) \mbox{ is even} \\ \Bigl(\frac{a_0}p \Bigr) & v_p(a) \mbox{ is odd and } p \equiv 1 \mod 4 \\ i \Bigl(\frac{a_0}p \Bigr) & v_p(a) \mbox{ is odd and } p \equiv 3 \mod 4 \end{cases}$$

is of great importance but refuses to name it implicitly. $\Bigl(\frac ab\Bigr)$ denotes the Legendre symbol and $v_p$ is the $p$-adic valuation. What is its name?

Here $a = p^n(a_0 + a_1p + a_2p^2+ \ldots)$ is a $p$-adic number.

• Is $a$ a number or a sequence? Because you have used $a_0$ in the definition. – Mark Fischler Sep 21 '16 at 14:36
• @MarkFischler it's a $p$-adic number. – Santiago Sep 21 '16 at 14:59
• Who is Volovich ? Could you give a reference ? – Jean Marie Sep 21 '16 at 16:09
• @JeanMarie I've added a hyperlink. – Santiago Sep 22 '16 at 10:37