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Dirichlet L-function

In analytic number theory, the Dirichlet L-function associated to a Dirichlet character \chi modulo a positive integer q is defined for complex numbers s with real part greater than 1 as the Dirichlet series L(s, \chi) = \sum_{n=1}^\infty \frac{\chi(n)}{n^s}, where \chi is a completely multiplicative periodic function with period q that vanishes on integers not coprime to q.^[1] This series converges absolutely in that half-plane and admits an Euler product representation L(s, \chi) = \prod_p \left(1 - \frac{\chi(p)}{p^s}\right)^{-1}, where the product runs over all primes p, reflecting the multiplicative nature of \chi.^[1] Introduced by Peter Gustav Lejeune Dirichlet in his 1837 memoir on primes in arithmetic progressions, these functions generalize the Riemann zeta function \zeta(s), which corresponds to the principal character \chi_1 \equiv 1 modulo q=1.^[2]^[3] For the principal character \chi_1 modulo q, L(s, \chi_1) is meromorphic with a simple pole at s=1 of residue \phi(q)/q, where \phi is Euler's totient function, and equals \zeta(s) \prod_{p \mid q} (1 - p^{-s}) elsewhere.^[1] For non-principal characters, L(s, \chi) is entire, holomorphic everywhere in the complex plane after analytic continuation, and non-vanishing at s=1, a key property Dirichlet used to prove the infinitude of primes in arithmetic progressions congruent to a modulo q for $1 \leq a \leq q with \gcd(a,q)=1.^[1]^[4] Primitive characters (those not induced by characters of smaller modulus) satisfy a functional equation relating L(s, \chi) to L(1-s, \overline{\chi}), involving the Gauss sum G(\chi) = \sum_{r=1}^{q-1} \chi(r) e^{2\pi i r / q} and Gamma factors: \Lambda(s, \chi) = \left( \frac{q}{\pi} \right)^{s/2} \Gamma\left( \frac{s + \kappa}{2} \right) L(s, \chi) = \epsilon(\chi) \Lambda(1-s, \overline{\chi}), where \kappa = 0 if \chi is even and \kappa=1 if odd, and |\epsilon(\chi)|=1.^[1] This symmetry implies zeros symmetric about the critical line \Re s = 1/2, with infinitely many in the critical strip $0 < \Re s < 1, and the generalized Riemann hypothesis posits that all non-trivial zeros lie on this line.^[5] Beyond prime distribution, Dirichlet L-functions underpin class number formulas for quadratic fields; for example, for imaginary quadratic fields with fundamental discriminant d < 0, the class number is h(d) = \frac{w \sqrt{|d|} L(1, \chi_d)}{2\pi} where w is the number of units in the ring of integers (usually w=2), and for real quadratic fields with d > 0, h(d) [R(d)](/page/Regulator) = \frac{\sqrt{d} L(1, \chi_d)}{2} where R(d) is the regulator.^[4]^[6] They more broadly influence the study of automorphic forms, Artin L-functions, and the Langlands program. The placement of all their non-trivial zeros on the critical line \Re s = 1/2, as conjectured by the generalized Riemann hypothesis, has implications for error terms in prime number theorems for arithmetic progressions.^[7]

Fundamentals

Definition

The Dirichlet L-function is a function of a complex variable s and a Dirichlet character \chi modulo an integer q \geq 1, defined by the infinite series

L(s, \chi) = \sum_{n=1}^\infty \frac{\chi(n)}{n^s}

for \operatorname{Re}(s) > 1. This series representation was introduced by Dirichlet in his 1837 memoir on primes in arithmetic progressions.^[8]^[1] The series converges absolutely in this half-plane because |\chi(n)| \leq 1 for all positive integers n, implying that

\sum_{n=1}^\infty \frac{|\chi(n)|}{n^{\operatorname{Re}(s)}} \leq \sum_{n=1}^\infty \frac{1}{n^{\operatorname{Re}(s)}} = \zeta(\operatorname{Re}(s)) < \infty

for \operatorname{Re}(s) > 1, where \zeta denotes the Riemann zeta function.^[1]^[7] As a generalization of the Riemann zeta function, the Dirichlet L-function recovers \zeta(s) when \chi is the principal (trivial) character \chi_0 modulo q=1, since \chi_0(n) = 1 for all n. For q > 1, the principal character \chi_0 modulo q is defined by \chi_0(n) = 1 if \gcd(n,q)=1 and \chi_0(n)=0 otherwise, yielding L(s, \chi_0) = \zeta(s) \prod_{p \mid q} (1 - p^{-s}). Dirichlet characters modulo q may be primitive, meaning they are not induced from any character modulo a proper divisor of q, or imprimitive otherwise.^[1]^[7]

Dirichlet characters

A Dirichlet character \chi modulo q is a completely multiplicative function \chi: \mathbb{Z} \to \mathbb{C} that is periodic with period q, meaning \chi(n + q) = \chi(n) for all integers n, and vanishes on integers not coprime to q, so \chi(n) = 0 whenever \gcd(n, q) > 1. This function extends a group homomorphism from the multiplicative group (\mathbb{Z}/q\mathbb{Z})^\times to the nonzero complex numbers \mathbb{C}^\times, where the values on coprime residues determine the character completely.^[9]^[10] Key properties of a Dirichlet character \chi modulo q include \chi(1) = 1, since multiplicativity implies \chi(1) = \chi(1)^2, and the character is nontrivial at 1. For \gcd(n, q) = 1, |\chi(n)| = 1, reflecting the unitary nature of group characters, while \chi(n) = 0 otherwise. Additionally, characters modulo q may be induced from characters modulo a proper divisor d of q: if \psi is a character modulo d, it induces \chi modulo q by setting \chi(n) = \psi(n \mod d) for all n coprime to q, provided this definition is consistent with the larger modulus.^[9]^[10] Dirichlet characters modulo q are classified into the principal character \chi_0, defined by \chi_0(n) = 1 if \gcd(n, q) = 1 and $0 otherwise, and the non-principal characters, which take values other than 1 on some units modulo q. The set of all Dirichlet characters modulo q forms a group under pointwise multiplication, isomorphic to the dual group of (\mathbb{Z}/q\mathbb{Z})^\times, and thus there are exactly \phi(q) such characters, where \phi is Euler's totient function.^[9]^[10] Examples illustrate these concepts clearly. The trivial character modulo 1 is the constant function \chi(n) = 1 for all n \in \mathbb{Z}, as (\mathbb{Z}/1\mathbb{Z})^\times is the trivial group. A real-valued example is the non-principal character modulo 4, given by \chi(n) = 0 if n even, \chi(n) = 1 if n \equiv 1 \pmod{4}, and \chi(n) = -1 if n \equiv 3 \pmod{4}, which corresponds to the Dirichlet character associated with the sign of quadratic residues modulo 4. Complex characters appear for moduli like 3: one non-principal character takes \chi(1) = 1, \chi(2) = e^{2\pi i / 3}, and $0 on multiples of 3.^[9]^[10] These characters were introduced by Peter Gustav Lejeune Dirichlet in his 1837 paper Recherches sur diverses applications de l'analyse infinitésimale à la théorie des nombres, where they played a central role in establishing the infinitude of primes in arithmetic progressions via associated L-functions.^[11]

Representations

Euler product

The Euler product representation of the Dirichlet L-function arises from the complete multiplicativity of the Dirichlet character \chi, which ensures that \chi(mn) = \chi(m) \chi(n) for all positive integers m and n. For \operatorname{Re}(s) > 1, the Dirichlet series L(s, \chi) = \sum_{n=1}^\infty \chi(n) n^{-s} can be expressed as an infinite product over the primes p:

L(s, \chi) = \prod_p \left(1 - \chi(p) p^{-s}\right)^{-1}.

^[12] This form follows from the fundamental theorem of arithmetic, which decomposes every positive integer n uniquely into prime powers, allowing the series to factor into local Euler factors at each prime.^[12] Each local factor expands as a geometric series: \left(1 - \chi(p) p^{-s}\right)^{-1} = \sum_{k=0}^\infty \chi(p^k) p^{-k s}, where \chi(p^k) = [\chi(p)]^k due to complete multiplicativity. This mirrors the Euler product for the Riemann zeta function, which corresponds to the principal character modulo 1, but generalizes it to incorporate the oscillatory behavior of \chi(p) on primes. The absolute convergence of the series for \operatorname{Re}(s) > 1 guarantees the uniform convergence of the infinite product.^[12] For a non-principal character \chi modulo q > 1, the local factor at any prime p dividing q simplifies because \chi(p) = 0, yielding \left(1 - 0 \cdot p^{-s}\right)^{-1} = 1. Thus, such primes contribute trivially to the product, effectively excluding them and ensuring that L(s, \chi) remains holomorphic at s=1 with no pole, unlike the zeta function.^[12] The Euler product uniquely characterizes L(s, \chi) among all Dirichlet series with completely multiplicative coefficients, as the values of the coefficients at prime powers are directly recoverable from the local factors, determining the entire series via multiplicativity.^[12]

Relation to the Hurwitz zeta function

The Dirichlet L-function associated to a Dirichlet character \chi modulo q admits the following representation in terms of the Hurwitz zeta function:

L(s, \chi) = q^{-s} \sum_{k=1}^{q} \overline{\chi}(k) \, \zeta\left(s, \frac{k}{q}\right),

where the Hurwitz zeta function is given by

\zeta(s, a) = \sum_{n=0}^{\infty} (n + a)^{-s}

for \operatorname{Re}(s) > 1 and $0 < a \leq 1. This formula arises from grouping the terms of the defining Dirichlet series \sum_{n=1}^{\infty} \chi(n) n^{-s} according to the residue classes of n modulo q, expressing each subsum as a Hurwitz zeta function shifted by the residue, and then invoking the orthogonality relation for Dirichlet characters \sum_{\chi \bmod q} \overline{\chi}(k) \chi(m) = \phi(q) if m \equiv k \pmod{q} and $0$ otherwise to isolate the desired character.^[13] The representation implies that L(s, \chi) extends to a holomorphic function on \operatorname{Re}(s) > 0 when \chi is non-principal, since each constituent Hurwitz zeta function \zeta(s, k/q) with $1 \leq k < q is holomorphic there (the term for k=q vanishes due to \overline{\chi}(q) = 0), avoiding the pole of the Riemann zeta function at s=1.^[14] For the principal character \chi_0 modulo q, the relation yields L(s, \chi_0) = \zeta(s) \prod_{p \mid q} (1 - p^{-s}) for \operatorname{Re}(s) > 1, reflecting the exclusion of arithmetic progressions divisible by primes dividing q.

Analytic continuation

Convergence

The Dirichlet L-function L(s, \chi) = \sum_{n=1}^\infty \chi(n) n^{-s} associated to a Dirichlet character \chi modulo q converges absolutely in the half-plane \operatorname{Re}(s) > 1. This follows from the bound |\chi(n)| \leq 1, which implies \sum_{n=1}^\infty |\chi(n) n^{-s}| \leq \zeta(\sigma) where \sigma = \operatorname{Re}(s) > 1 and \zeta is the Riemann zeta function, known to converge absolutely there.^[15] For non-principal characters \chi, the series converges conditionally in the larger half-plane \operatorname{Re}(s) > 0. The proof relies on partial summation: let S(N) = \sum_{n=1}^N \chi(n), then for non-principal \chi, the partial sums satisfy |S(N)| \ll \phi(q) and are bounded independently of N due to the orthogonality of characters over complete residue systems modulo q. Applying partial summation to the tail of the series then yields convergence for \sigma > 0, as the integrated term vanishes and the remaining sum is controlled.^[15]^[16] The L-function admits an analytic continuation to the entire complex plane. For non-principal \chi, this continuation is holomorphic everywhere, while for the principal character \chi_0, L(s, \chi_0) is meromorphic with a single simple pole at s=1 and residue \phi(q)/q. One method to obtain this continuation ties into the relation with the Hurwitz zeta function, where the combination of terms cancels the pole at s=1 for non-principal \chi; alternatively, contour integration techniques applied to the Dirichlet series achieve the same result.^[7]^[17] A key consequence of this analytic continuation is that L(s, \chi) has no zeros on the line \operatorname{[Re](/page/Re)}(s) = 1 for non-principal \chi. The function is holomorphic in a neighborhood of this line, and Dirichlet established L(1, \chi) \neq 0, ensuring non-vanishing there; this property underpins the density of primes in arithmetic progressions.^[15] In the critical strip $0 < \operatorname{[Re](/page/Re)}(s) < 1, growth estimates for |L(s, \chi)| follow from Landau's theorem applied to auxiliary Dirichlet series with non-negative coefficients, such as the sum \sum_{\chi} L(s, \chi) over characters modulo q, which converges only for \operatorname{[Re](/page/Re)}(s) > 1 and forces bounded growth for individual non-principal factors after accounting for the principal term's pole. The convexity bound yields |L(\sigma + it, \chi)| \ll_\epsilon (q(2 + |t|))^{(1 - \sigma)/2 + \epsilon} for $1/2 \leq \sigma \leq 1. For \sigma \geq 1, a simpler bound is |L(\sigma + it, \chi)| \ll \log(q(2 + |t|)).^[15]

Functional equation

The completed Dirichlet L-function, often denoted \Lambda(s, \chi), incorporates the L-function L(s, \chi) with Gamma factors to symmetrize its behavior under the transformation s \to 1-s. For an even primitive Dirichlet character \chi modulo q, it is defined as

\Lambda(s, \chi) = \left( \frac{q}{\pi} \right)^{s/2} \Gamma\left( \frac{s}{2} \right) L(s, \chi),

while for an odd primitive character, it takes the form

\Lambda(s, \chi) = i \left( \frac{q}{\pi} \right)^{(s+1)/2} \Gamma\left( \frac{s+1}{2} \right) L(s, \chi).

^[18]^[19] This completed function satisfies the functional equation

\Lambda(s, \chi) = \varepsilon(\chi) \Lambda(1-s, \overline{\chi}),

where \overline{\chi} is the complex conjugate character and \varepsilon(\chi) is the root number with |\varepsilon(\chi)| = 1.^[18]^[19] The root number is given by \varepsilon(\chi) = \tau(\chi) / (i^\kappa \sqrt{q}), where \kappa = 0 for even characters and \kappa = 1 for odd characters, and \tau(\chi) denotes the Gauss sum

\tau(\chi) = \sum_{k=1}^q \chi(k) e^{2\pi i k / q}.

^[18]^[19] The magnitude of the Gauss sum satisfies |\tau(\chi)| = \sqrt{q} for primitive \chi.^[18] The functional equation relates the values of the completed L-function in the upper and lower half-planes and holds primarily for primitive characters. Its derivation typically employs the Poisson summation formula applied to associated theta functions. For even characters, one considers the theta series \theta_\chi(iy) = \sum_n \chi(n) e^{-\pi n^2 y} and derives an integral representation \pi^{-s/2} \Gamma(s/2) L(s, \chi) = \int_0^\infty y^{s/2 - 1} \theta_\chi(iy) \, dy; applying Poisson summation to the sum over residues modulo q yields the transformation \theta_\chi(iy) = \tau(\overline{\chi}) q^{-1/2} y^{-1/2} \theta_{\overline{\chi}}(i q^2 / y), leading to the functional equation upon splitting the integral at y=1.^[19] A similar approach for odd characters uses the adjusted theta series \tilde{\theta}_\chi(iy) = \sum_n \chi(n) n \sqrt{y} \, e^{-\pi n^2 y} and incorporates an extra factor from the odd parity.^[19] For the principal character \chi_0 modulo q=1, corresponding to the Riemann zeta function \zeta(s), the completed form \Lambda(s, \chi_0) = \pi^{-s/2} \Gamma(s/2) \zeta(s) satisfies the symmetric equation \Lambda(s, \chi_0) = \Lambda(1-s, \chi_0) due to the pole of \zeta(s) at s=1, which introduces an asymmetry absent in the non-principal primitive cases. For principal characters modulo q>1, the equation adapts to the imprimitive nature, with L(s, \chi_0) holomorphic but related to \zeta(s) via Euler factors.^[1]

Zeros

Trivial zeros

The trivial zeros of the Dirichlet L-function L(s, \chi) are located at negative integers and arise from the poles of the gamma function factors in the completed L-function \Lambda(s, \chi), which is defined via the functional equation to ensure holomorphy. For a primitive even character \chi (satisfying \chi(-1) = 1), these poles occur in the factor \Gamma(s/2) at s = 0, -2, -4, \dots, forcing L(s, \chi) to have simple zeros at these points since the poles of the gamma function are simple.^[1] Similarly, for a primitive odd character \chi (satisfying \chi(-1) = -1), the poles of \Gamma((s+1)/2) occur at s = -1, -3, -5, \dots, yielding simple zeros of L(s, \chi) there.^[1] For the principal character \chi_0 modulo q \geq 1, which is even, the L-function L(s, \chi_0) behaves analogously to the Riemann zeta function \zeta(s) (the case q=1), with simple trivial zeros at the negative even integers s = -2, -4, -6, \dots, but no zero at s=0. In this case, \Lambda(s, \chi_0) is meromorphic with simple poles at s=0 and s=1, rather than entire, so the pole of \Gamma(s/2) at s=0 is not canceled by a zero in L(s, \chi_0).^[20] These trivial zeros parallel those of the Riemann zeta function, which lie at negative even integers due to the same \Gamma(s/2) factor, but for Dirichlet L-functions the locations shift according to the parity of the character: even characters align with zeta's pattern (adjusted for the principal case at s=0), while odd characters produce zeros at negative odd integers.^[1]

Non-trivial zeros

The non-trivial zeros of the Dirichlet L-function L(s, \chi), for a non-principal Dirichlet character \chi modulo q, lie in the critical strip $0 < \operatorname{Re}(s) < 1. This follows from the analytic continuation of L(s, \chi) to the entire complex plane (except possibly at s=1 for the principal character, where it has a pole) and after accounting for the trivial zeros at non-positive integers (depending on the parity of \chi), all lying on or to the left of the line \operatorname{Re}(s) = 0. Moreover, L(s, \chi) has no zeros on the line \operatorname{Re}(s) = 1, a result proven by de la Vallée Poussin in 1899 using methods analogous to those for the Riemann zeta function, ensuring the non-vanishing necessary for the prime number theorem in arithmetic progressions.^[21]^[22] The generalized Riemann hypothesis (GRH) posits that all non-trivial zeros of L(s, \chi) lie on the critical line \operatorname{Re}(s) = 1/2. This conjecture extends the Riemann hypothesis for the zeta function, proposed by Riemann in 1859, to the family of Dirichlet L-functions introduced by Dirichlet in 1837. While unproven, GRH has profound implications for analytic number theory, including optimal error bounds in the distribution of primes. Known results include zero-free regions near \operatorname{Re}(s) = 1, such as the classical region \operatorname{Re}(s) \geq 1 - \frac{c}{\log(q(2 + |t|))} for some absolute c > 0, established by de la Vallée Poussin and refined in explicit forms for computational purposes. Additionally, von Mangoldt-type explicit formulas relate the zeros to the distribution of primes in arithmetic progressions; for instance, the Chebyshev function \psi(x; q, a) = \sum_{n \leq x, n \equiv a \pmod{q}} \Lambda(n) satisfies \psi(x; q, a) = \frac{x}{\phi(q)} + O\left( \sum_{\rho} \frac{x^{\rho}}{\rho} \right), where the sum is over non-trivial zeros \rho of L(s, \chi) for characters \chi modulo q, linking prime gaps to zero locations.^[20]^[23]^[24] Numerical computations support the alignment of zeros with the critical line for small q. For example, as of 2013, extensive calculations for primitive characters modulo q \leq 400{,}000 verified over $3.8 \times 10^{13} zeros lying on \operatorname{Re}(s) = 1/2, with no counterexamples to GRH found as of 2025. These computations often reveal statistical patterns in zero spacings resembling those of eigenvalues from random matrix theory, particularly the Gaussian Unitary Ensemble (GUE), as conjectured by Montgomery and extended by Katz and Sarnak to families of L-functions. Under GRH, the prime number theorem in arithmetic progressions achieves its strongest error term: \pi(x; q, a) = \frac{\operatorname{Li}(x)}{\phi(q)} + O\left( \sqrt{x} \log(qx) \right) for \gcd(a, q) = 1, enabling precise asymptotic estimates for primes in progressions.^[25]^[26]^[27]^[20]

Special cases and values

Principal character

The principal character \chi_0 modulo q is defined by \chi_0(n) = 1 if \gcd(n, q) = 1 and \chi_0(n) = 0 otherwise.^[1] The corresponding Dirichlet L-function is given by

L(s, \chi_0) = \sum_{\substack{n=1 \\ \gcd(n,q)=1}}^\infty \frac{1}{n^s}

for \Re(s) > 1.^[1] This series equals \zeta(s) \prod_{p \mid q} (1 - p^{-s}), where \zeta(s) is the Riemann zeta function, reflecting the exclusion of terms divisible by primes dividing q.^[4] The Euler product for L(s, \chi_0) is

L(s, \chi_0) = \prod_{p \nmid q} (1 - p^{-s})^{-1},

valid for \Re(s) > 1, which omits the factors for primes p dividing q.^[1] This function admits an analytic continuation to a meromorphic function on the complex plane, with a single simple pole at s = 1 and residue \phi(q)/q, where \phi is Euler's totient function.^[1] The pole arises from the corresponding pole of \zeta(s) at s=1, scaled by the product \prod_{p \mid q} (1 - p^{-1}).^[7] At negative integers, the values of L(s, \chi_0) are expressed using generalized Bernoulli numbers B_{n, \chi_0}, defined via the generating function \sum_{a=1}^q \chi_0(a) \frac{t e^{a t}}{e^{q t} - 1} = \sum_{n=0}^\infty B_{n, \chi_0} \frac{t^n}{n!}.^[28] Specifically, for integers n \geq 1,

L(1 - n, \chi_0) = -\frac{B_{n, \chi_0}}{n}.

^[28] For the principal character, these generalized Bernoulli numbers relate to the classical Bernoulli numbers B_n by B_{n, \chi_0} = \frac{\phi(q)}{q} B_n for n \geq 2 even, with adjustments for odd indices and the case n=1.^[28] Dirichlet introduced the L-function for the principal character in 1837 as part of his proof of the infinitude of primes in arithmetic progressions coprime to q, where the residue at s=1 provides the asymptotic density \phi(q)/q of such primes among all primes.^[1]^[16]

Primitive characters

A Dirichlet character \chi modulo q is primitive if its conductor equals q, meaning it is not induced from any character modulo a proper divisor q' of q.^[29] Equivalently, \chi is primitive if and only if its Gauss sum \tau(\chi) = \sum_{a=1}^{q} \chi(a) e^{2\pi i a / q} \neq 0, or if q is the minimal period of \chi.^[30] The primitive characters modulo q form a basis for the vector space of all Dirichlet characters modulo q, in the sense that every character modulo q can be uniquely expressed as an induction from a primitive character of conductor dividing q.^[9] The number of primitive Dirichlet characters modulo q is \sum_{d \mid q} \mu(d) \phi(q/d), where \mu is the Möbius function and \phi is Euler's totient function; this count equals q \prod_{p \parallel q} (1 - 2/p) \prod_{p^2 \mid q} (1 - 1/p)^2.^[31] For a primitive character \chi modulo q, the Gauss sum satisfies |\tau(\chi)| = \sqrt{q}, and orthogonality relations among characters imply that |\tau(\chi)|^2 = q.^[30] These properties underpin decompositions and estimates in analytic number theory. In the functional equation of the associated L-function, primitive characters yield a simplified symmetric form: the root number \varepsilon(\chi) = \tau(\chi) / (i^{\kappa} \sqrt{q}), where \kappa = 0 if \chi is even and \kappa=1 if odd, satisfies |\varepsilon(\chi)| = 1, eliminating asymmetric scaling factors present for imprimitive characters.^[18] For even primitive \chi, the equation reads

\Lambda(s, \chi) = \left( \frac{q}{\pi} \right)^{s/2} \Gamma\left(\frac{s}{2}\right) L(s, \chi) = \varepsilon(\chi) \Lambda(1-s, \overline{\chi}),

with an analogous form \left( \frac{q}{\pi} \right)^{(s+1)/2} \Gamma\left(\frac{s+1}{2}\right) L(s, \chi) = \varepsilon(\chi) \Lambda(1-s, \overline{\chi}) for odd \chi.^[1] Primitive L-functions feature prominently in advanced applications, such as the Artin conjecture, which posits that Artin L-functions factor as products of Dirichlet L-functions for primitive characters, and in the analytic class number formula for imaginary quadratic fields K = \mathbb{Q}(\sqrt{-d}), where the class number h_K is given by h_K = \frac{w_K \sqrt{d}}{2\pi} L(1, \chi_d) with \chi_d the primitive quadratic character modulo d.^[6] For example, all non-principal characters modulo a prime q are primitive, as their conductors equal q.^[9]

References

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[22]
[PDF] Nonvanishing of L-functions on <(s)=1.
Jan 21, 2004 · The advantage of Poussin's method here is that it yields a standard zero free region (4). According to the general functoriality conjectures of ...
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[PDF] An explicit formula for Dirichlet's L-Function - UTC Scholar
Apr 9, 2018 · The purpose of this thesis is to obtain a generalization of Landau's and Gonek's explicit formulas in terms of the zeros of the Dirichlet L- ...
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[PDF] Counting zeros of Dirichlet 𝐿-functions - UBC Math
Jan 26, 2021 · We give explicit upper and lower bounds for N(T,χ), the number of zeros of a Dirichlet L-function with character χ and height at most T. Suppose ...
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[PDF] Notes on L-functions and Random Matrix Theory
The mean-value theorem of Montgomery and Vaughan asserts that ... ps ! −1 . Note that this L-function has a degree three Euler product associated with it.<|control11|><|separator|>
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[PDF] 3 Special values of L(s, χ)
The values of L(s, χ) at negative integers may be expressed in terms of generalized Bernoulli numbers: 1 Page 2 Theorem 3.3. For any integer n ≥ 1, L(1 − n, χ) ...Missing: principal Dirichlet
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[PDF] ON PRIMITIVITY OF DIRICHLET CHARACTERS 1. Introduction Let ...
Abstract. Recall that a Dirichlet character is called imprimitive if it is induced from a character of smaller level, and otherwise it is called primitive.
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[PDF] 13. Gauss sums Let χ be a Dirichlet character mod q. Definition
Theorem 13.4. Suppose χ is a primitive Dirichlet character mod q. Then τ(n, χ) = ¯χ(n)τ(χ) for all n ∈ Z. Moreover |τ(χ)| = √ q.
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[PDF] 11. Dirichlet characters
Theorem 11.13. Let χ be a Dirichlet character mod q with conductor d. Then there exists a unique primitive character χ∗ mod d that induces χ.
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[PDF] 19 The analytic class number formula
Nov 13, 2017 · In the previous lecture we proved Dirichlet's theorem on primes in arithmetic progressions modulo the claim that the L-function L(s, ...