Dirichlet character

A Dirichlet character modulo $m$ is a homomorphism $(\mathbb{Z}/m\mathbb{Z}{)}^{\times }\to {\mathbb{C}}^{\times }$, extended periodically to integers by zero on numbers not coprime to $m$.

A real Dirichlet character takes values in $\{0,1,-1\}$. Its restriction to the unit group has order at most two; if it is nonprincipal, the order is exactly two. A real nonprincipal primitive Dirichlet character is the character to which a possible Siegel zero belongs.

// Target: analytic-number-theory.bigb

The parity determines the gamma factor and theta transformation in the completed Dirichlet L-function. It is the sign of the character at minus one.

A Dirichlet character with odd parity satisfies $\chi (-n)=-\chi (n)$, so its extension to integer arguments is an odd function.

A Dirichlet character with even parity satisfies $\chi (-n)=\chi (n)$, so its extension to integer arguments is an even function.

The least divisor of a character's modulus through whose unit-group reduction the Dirichlet character factors. The Chinese remainder theorem reduces its construction to choosing the least exponent at each prime. Surjectivity of reduction on units makes the descended primitive Dirichlet character unique.

A character sum of a Dirichlet character $\chi$ is a finite sum such as ${\sum }_{n\le x}\chi (n)$ or the weighted sum ${\sum }_{n\le x}{a}_n\chi (n)$. Orthogonality of Dirichlet characters converts residue class restrictions to these sums.

A nonprincipal Dirichlet character differs from the principal Dirichlet character. Equivalently, it is a nontrivial character on the unit group, so its sum over one full residue period is zero by character orthogonality. Its Dirichlet L-function is entire by Mellin continuation of a nonprincipal Dirichlet L-function.

The principal Dirichlet character modulo $N$ is one on integers coprime to $N$ and zero on all other integers. Its Dirichlet L-function is $L({\chi }_0,s)=\zeta (s){\prod }_{p\mid N}(1-p^{-s})$, so it has a simple pole at one. For $N>1$ it is induced from the character of modulus one and is not a primitive Dirichlet character.

Extend a Dirichlet character by zero on nonunits. Its Gauss sum is $g(\chi )={\sum }_{a\mathrm{mod}N}\chi (a)e^{2\pi ia/N}$, and its finite Fourier transform at $r$ is $S_{\chi }(r)={\sum }_{a\mathrm{mod}N}\chi (a)e^{2\pi ira/N}$.

For a nontrivial Dirichlet character modulo a prime $p$ and $a\not\equiv 0(\mathrm{mod}p)$, set $\tau ={\sum }_{y\ne 0}\chi (y)e(ay/p)$. Substituting $y=tz$ in $|\tau {|}^2$ gives ${\sum }_{t\ne 0}\chi (t){\sum }_{z\ne 0}e(a(t-1)z/p)$. The inner sum is $p-1$ at $t=1$ and $-1$ otherwise. Since character orthogonality gives ${\sum }_{t\ne 0}\chi (t)=0$, the result is $p$.

For a character of modulus $p^k$, a primitive Dirichlet character attains the bound by finite Plancherel theorem. An imprimitive character with $k\ge 2$ has periodic values modulo $p^{k-1}$, so its frequency-one sum vanishes. The principal Dirichlet character modulo $p$ has unnormalized sum $-1$. The sign convention of the exponential does not affect its magnitude.

For a primitive Dirichlet character, $S_{\chi }(r)=\overline{\chi }(r)g(\chi )$ for every integer $r$, with zero on nonunits. For a unit $r$ this follows by substitution. If a prime $p$ divides both $r$ and $N$, primitivity supplies a unit $u\equiv 1(\mathrm{mod}N/p)$ with $\chi (u)\ne 1$; substitution by $u$ fixes the additive exponential and forces $S_{\chi }(r)=0$. For an imprimitive Dirichlet character this vanishing can fail.

A Dirichlet character modulo $N$ is primitive if its character on $(\mathbb{Z}/N\mathbb{Z}{)}^{\times }$ does not factor through reduction to the unit group for any proper divisor of $N$. Its least inducing modulus is called its conductor. Primitivity is essential in the usual formula expressing its finite Fourier transform as a conjugate character times a Gauss sum of a Dirichlet character.

For reduced residue classes $a,n$ modulo $q$,This is the orthogonality relation for the characters of the finite abelian group $(\mathbb{Z}/q\mathbb{Z}{)}^{\times }$.

The Kronecker symbol extends the Jacobi symbol to all integer denominators. For a fundamental discriminant $D$, the function ${\chi }_D(n)=\left(\frac{D}{n}\right)$ is the primitive quadratic Dirichlet character associated with $\mathbb{Q}(\sqrt{D})$.

For $\mathrm{Re}s>1$,

For real $s>1$, Orthogonality of Dirichlet characters gives $\log {\mathcal{P}}_q(s)=\phi (q){\sum }_{p\nmid q}{\sum }_{j\ge 1,p^j\equiv 1(\mathrm{mod}q)}1/(j{p}^{js})\ge 0$. Thus ${\mathcal{P}}_q(s)\ge 1$. Only the principal Dirichlet character contributes a pole at one, and that pole is simple. A zero of a nonreal Dirichlet L-function at one brings a zero of its conjugate too; their two zeros overpower that pole and would make the product tend to zero. This proves nonvanishing for nonreal Dirichlet characters and is a useful complement to arguments for real Dirichlet characters.

For a primitive nonprincipal Dirichlet character of conductor $q$ and parity $a$, the completed Dirichlet L-function is entire of order one. Its genus-one Hadamard factorization gives the displayed logarithmic derivative, with $B_{\chi }$ the logarithmic derivative of the completion at zero and zeros counted with multiplicity. Combined summands converge locally normally away from zeros. For an imprimitive character induced by ${\chi }^{\ast }$, add ${\sum }_{p\mid Q,p\nmid q}{\chi }^{\ast }(p)\log p/(p^s-{\chi }^{\ast }(p))$ for the removed Euler factors. The primitive principal case is the global partial-fraction expansion of the zeta logarithmic derivative.

// Target: algebraic-number-theory.bigb

For a primitive nonprincipal Dirichlet character with parity $a$, the Gamma function poles in its completed Dirichlet L-function force zeros of $L(s,\chi )$ at $s=-a-2m$, $m=0,1,\dots$. The functional equation and nonvanishing of nonprincipal Dirichlet L-functions at one show that these zeros are simple and cancel the poles, rather than remaining zeros of the completion. For the Dirichlet beta function, the odd-parity examples are $\beta (-1)=\beta (-3)=\cdots =0$. The primitive principal conductor-one case is different: its zeta trivial zeros start at $-2$ and the completion has poles at zero and one.

The coefficients of $\zeta (s)L(s,\chi )$ are multiplicative. On $p^k$, their values are $k+1$ when $\chi (p)=1$, alternating one and zero when $\chi (p)=-1$, and one when $\chi (p)=0$. In particular $a_{m^2}\ge 1$. Their logarithmic Euler coefficients $\Lambda (n)(1+\chi (n))$ are also nonnegative, supporting uniqueness of a possible exceptional real Dirichlet zero.

For a nonprincipal Dirichlet character, the line is zero-free. Euler product positivity for L-function nonvanishing proves this for nonreal characters at every height, and for all characters at nonzero height, where a possibly principal squared-character factor has no pole. The point of height zero is the nonvanishing of nonprincipal Dirichlet L-functions at one. A principal L-function has a pole at one, and no zeros at the other points of this line.

The inducing primitive Dirichlet character has conductor of a Dirichlet character $q_0$. Increasing the modulus removes Euler factors only at new prime divisors. The correction creates zeros on the imaginary axis; raising the exponents of primes already dividing the conductor of a Dirichlet character changes no Euler factor. Primitive gamma factors continue to use $q_0$. For real-even characters this includes principal primitive conductor of a Dirichlet character one, with its Riemann zeta function pole treated separately.

For a primitive nonprincipal Dirichlet character, the theta Mellin integral makes this an entire function satisfying $\Lambda (s,\chi )={\epsilon }_{\chi }\Lambda (1-s,\overline{\chi })$. The gamma poles cancel the trivial zeros of a Dirichlet L-function. For the principal Dirichlet character whose conductor of a Dirichlet character is one the completion is meromorphic, with poles at zero and one.

For a primitive Dirichlet character whose conductor of a Dirichlet character is $q>1$ and whose character parity is $a$, Poisson summation gives ${\theta }_{\chi }(x)={\epsilon }_{\chi }{x}^{-a-1/2}{\theta }_{\overline{\chi }}(1/x)$, where ${\epsilon }_{\chi }=\tau (\chi )/(i^a\sqrt{q})$. Its rapid decay at both ends yields the entire Mellin representation of the completed Dirichlet L-function. The case of conductor of a Dirichlet character equal to one uses the ordinary Jacobi theta function and a subtracted constant term.

For a primitive Dirichlet character modulo $N>1$ of parity $a\in \{0,1\}$, put ${\Theta }_{\chi }(\tau )={\sum }_n{n}^a\chi (n)e^{\pi i{n}^2\tau }$, extending the character by zero on nonunits. The square root is analytic with positive value when $-i\tau$ is positive real. Apply Poisson summation separately in each residue class modulo $N$, and use the finite Fourier transform of a primitive Dirichlet character. The Fourier transforms of $e^{\pi i\tau x^2}$ and $x{e}^{\pi i\tau x^2}$ are $(-i\tau {)}^{-1/2}{e}^{-\pi i{t}^2/\tau }$ and $(t/\tau )(-i\tau {)}^{-1/2}{e}^{-\pi i{t}^2/\tau }$. These give the formula, including its odd-character phase.

Take the even primitive Dirichlet character ${\chi }_{12}$ whose values at $1,5,7,11$ modulo twelve are $1,-1,-1,1$. Its Gauss sum of a Dirichlet character is $\sqrt{12}$. The displayed series $A$ has $A(-1/\tau )=\sqrt{-i\tau }A(\tau )$ and $A(\tau +1)=e^{\pi i/12}A(\tau )$. Thus $A^{24}$ is the normalized weight-twelve modular discriminant. It has no interior zeros by the valence formula for the modular group. Its logarithmic derivative equals $E_2/24$, so integration gives the Dedekind eta function product. The odd character modulo four similarly produces a series $B$ with $B^8=A^{24}$ and leading term $q^{1/8}$; consequently $B=A^3$.

For a nonprincipal Dirichlet character modulo $N$, setThe numerator vanishes at $t=0$, so $H_{\chi }$ is analytic there and exponentially decreasing as $t\to \infty$. Initially for $\mathrm{Re}s>1$, $\Gamma (s)L(\chi ,s)={\int }_0^{\infty }{H}_{\chi }(t)t^{s-1}dt$. This integral already extends holomorphically to $\mathrm{Re}s>0$. Subtracting finite Taylor polynomials near zero, as in meromorphic continuation of a Mellin transform from an asymptotic expansion, continues it with possible simple poles only at nonpositive integers. Zeros of the reciprocal Gamma function cancel these, proving that $L(\chi ,s)$ is an entire function. This does not require a primitive Dirichlet character.