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| − | Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130160/d1301601.png" /> be an algebraic number field (cf. also [[Algebraic number|Algebraic number]]) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130160/d1301602.png" /> be a set of prime ideals (of the ring of integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130160/d1301603.png" />) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130160/d1301604.png" />. If an equality of the form | + | Let $K$ be an algebraic number field (cf. also [[Algebraic number|Algebraic number]]) and let $A$ be a set of prime ideals (of the ring of integers $A_K$) of $K$. If an equality of the form |
| − | | + | $$ |
| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130160/d1301605.png" /></td> </tr></table>
| + | \sum_{\mathfrak{p}\in A} N(\mathfrak{p})^{-s} = a \log\frac{1}{1-s} +g(s) |
| − | | + | $$ |
| − | holds, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130160/d1301606.png" /> is regular in the closed half-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130160/d1301607.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130160/d1301608.png" /> is a regular set of prime ideals and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130160/d1301609.png" /> is called its Dirichlet density. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130160/d13016010.png" /> is the norm of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130160/d13016011.png" />, i.e. the number of elements of the residue field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130160/d13016012.png" />. | + | holds, where $g(s)$ is regular in the closed half-plane $\mathrm{Re}(s) \ge 1$, then $A$ is a regular set of prime ideals and $a$ is called its Dirichlet density. Here, $N(\mathfrak{p})$ is the norm of $\mathfrak{p}$, i.e. the number of elements of the residue field $A_k/\mathfrak{p}$. |
| | | | |
| | ===Examples.=== | | ===Examples.=== |
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| − | i) The set of all prime ideals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130160/d13016013.png" /> is regular with Dirichlet density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130160/d13016014.png" />. | + | i) The set of all prime ideals of $K$ is regular with Dirichlet density $1$. |
| | | | |
| − | ii) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130160/d13016015.png" /> be a finite extension and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130160/d13016016.png" /> the set of all prime ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130160/d13016017.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130160/d13016018.png" /> that are of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130160/d13016019.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130160/d13016020.png" /> (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130160/d13016021.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130160/d13016022.png" /> is the prime ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130160/d13016023.png" /> under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130160/d13016024.png" />). Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130160/d13016025.png" /> is regular with Dirichlet density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130160/d13016026.png" />. | + | ii) Let $L/K$ be a finite extension and $A$ the set of all prime ideals $\mathfrak{P}$ in $L$ that are of degree $1$ over $K$ (i.e. $[A_L/\mathfrak{P} : A_K/\mathfrak{p}] = 1$, where $\mathfrak{p}$ is the prime ideal $\mathfrak{P} \cap A_K$ under $\mathfrak{P}$). Then $A$ is regular with Dirichlet density $1$. |
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| − | iii) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130160/d13016027.png" /> be a finite normal extension and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130160/d13016028.png" /> the set of all prime ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130160/d13016029.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130160/d13016030.png" /> that split in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130160/d13016031.png" /> (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130160/d13016032.png" /> is a product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130160/d13016033.png" /> prime ideals in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130160/d13016034.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130160/d13016035.png" />). Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130160/d13016036.png" /> is regular with Dirichlet density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130160/d13016037.png" />. | + | iii) Let $L/K$ be a finite normal extension and $A$ the set of all prime ideals $\mathfrak{p}$ in $K$ that split in $L$ (i.e. $\mathfrak{p}A_L$ is a product of $[L:K]$ prime ideals in $L$ of degree $1$). Then $A$ is regular with Dirichlet density $[L:K]^{-1}$. |
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| − | The notion of a Dirichlet density can be extended to not necessarily regular sets of prime ideals. Such a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130160/d13016038.png" /> has Dirichlet density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130160/d13016039.png" /> if | + | The notion of a Dirichlet density can be extended to not necessarily regular sets of prime ideals. Such a set $A$ has Dirichlet density $a$ if |
| | + | $$ |
| | + | \lim_{s \searrow 1} \frac{ \sum_{\mathfrak{p}\in A} N(\mathfrak{p})^{-s} }{ a \log\frac{1}{1-s} } \ . |
| | + | $$ |
| | | | |
| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130160/d13016040.png" /></td> </tr></table> | + | ====References==== |
| | + | <table> |
| | + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> W. Narkiewicz, "Elementary and analytic theory of algebraic numbers" , PWN/Springer (1990) pp. Sect. 7.2 (Edition: Second)</TD></TR> |
| | + | </table> |
| | | | |
| − | ====References====
| + | {{TEX|done}} |
| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> W. Narkiewicz, "Elementary and analytic theory of algebraic numbers" , PWN/Springer (1990) pp. Sect. 7.2 (Edition: Second)</TD></TR></table>
| |
Let $K$ be an algebraic number field (cf. also Algebraic number) and let $A$ be a set of prime ideals (of the ring of integers $A_K$) of $K$. If an equality of the form
$$
\sum_{\mathfrak{p}\in A} N(\mathfrak{p})^{-s} = a \log\frac{1}{1-s} +g(s)
$$
holds, where $g(s)$ is regular in the closed half-plane $\mathrm{Re}(s) \ge 1$, then $A$ is a regular set of prime ideals and $a$ is called its Dirichlet density. Here, $N(\mathfrak{p})$ is the norm of $\mathfrak{p}$, i.e. the number of elements of the residue field $A_k/\mathfrak{p}$.
Examples.
i) The set of all prime ideals of $K$ is regular with Dirichlet density $1$.
ii) Let $L/K$ be a finite extension and $A$ the set of all prime ideals $\mathfrak{P}$ in $L$ that are of degree $1$ over $K$ (i.e. $[A_L/\mathfrak{P} : A_K/\mathfrak{p}] = 1$, where $\mathfrak{p}$ is the prime ideal $\mathfrak{P} \cap A_K$ under $\mathfrak{P}$). Then $A$ is regular with Dirichlet density $1$.
iii) Let $L/K$ be a finite normal extension and $A$ the set of all prime ideals $\mathfrak{p}$ in $K$ that split in $L$ (i.e. $\mathfrak{p}A_L$ is a product of $[L:K]$ prime ideals in $L$ of degree $1$). Then $A$ is regular with Dirichlet density $[L:K]^{-1}$.
The notion of a Dirichlet density can be extended to not necessarily regular sets of prime ideals. Such a set $A$ has Dirichlet density $a$ if
$$
\lim_{s \searrow 1} \frac{ \sum_{\mathfrak{p}\in A} N(\mathfrak{p})^{-s} }{ a \log\frac{1}{1-s} } \ .
$$
References
| [a1] | W. Narkiewicz, "Elementary and analytic theory of algebraic numbers" , PWN/Springer (1990) pp. Sect. 7.2 (Edition: Second) |