Dirichlet's Theorem on Primes

• Dirichlet's Theorem is a foundational result in analytic number theory that asserts every arithmetic progression with coprime initial term and modulus contains infinitely many primes.
• It employs analytic techniques using Dirichlet characters and L-series to reveal the divergence in prime distributions within these progressions.
• The theorem has far-reaching extensions, including multidimensional generalizations, combinatorial proofs, and applications in Diophantine approximation and function fields.

Dirichlet's Theorem provides a foundational result in analytic number theory, establishing that every arithmetic progression with coprime initial term and modulus contains infinitely many primes. The theorem and its generalizations form the bedrock of arithmetic aspects of prime distribution, underpinning both classical analytic approaches via Dirichlet L-functions and modern elementary proofs, as well as generalizations to higher-dimensional Diophantine approximation and function fields.

1. Statement and Classical Formulation

Dirichlet's Theorem asserts that for integers $a, d$ with $\gcd(a,d)=1$, the progression $\{a+nd : n\ge 0\}$ contains infinitely many prime numbers. Formally, for every pair $(a,d)$ of coprime integers,

$$|\{p \text{ prime}: p \equiv a \pmod{d}\}| = \infty.$$

This result was originally proved in 1837 and represents the first substantial use of analytic techniques in number theory to demonstrate arithmetic infinitude beyond divisibility arguments (Avigad et al., 2012, Carneiro, 2016).

2. Analytic Proof via Dirichlet Characters and L-Series

The standard analytic proof introduces the machinery of Dirichlet characters as group homomorphisms $\chi: (\mathbb{Z}/d\mathbb{Z})^* \to \mathbb{C}^*$, extended to all integers by periodicity and vanishing off units. The main ingredients include:

• Dirichlet L-series: For each character $\chi$ and $\Re(s)>1$,

$$L(s,\chi) = \sum_{n=1}^\infty \frac{\chi(n)}{n^s} = \prod_p (1 - \chi(p)p^{-s})^{-1}.$$

• Orthogonality relations: Sums over residues and sums over characters enforce selective amplification of primes in a fixed congruence (Avigad et al., 2012).
• Divergence argument: The principal $L(s,\chi_0)$ has a simple pole at $s=1$; all nonprincipal $L(s,\chi)$ remain finite as $s\rightarrow 1^+$. By summing suitably twisted logarithms,

$$\sum_{p\equiv a \pmod d} p^{-s} \to \infty \quad \text{as } s\to 1^+,$$

implying infinitude of such primes.

This method, reliant on the analytic properties of $L$-functions, provides the distributional backbone for arithmetic equidistribution results (e.g., prime number theorem for arithmetic progressions) (Carneiro, 2016).

3. Elementary and Combinatorial Proofs

Several more elementary or combinatorial proofs have been developed, avoiding the full power of complex analysis. Approaches include:

• Periodic sieving and combinatorial arguments: Construct explicit sieved sets $M_{p_n}$ by removing multiples of the first $n$ primes, analyze the periodicity and gap patterns, and show that no single column (arithmetic progression) can be exhaustively sieved out without running afoul of known prime gap properties (Xu, 2015).
• Jacobsthal function and covering arguments: By bounding the minimal gap $h(k)$ between numbers coprime to the product of the first $k$ primes, Mercer shows that a combinatorial proof of Dirichlet’s theorem is possible for all moduli provided conjecturally small bounds for $h(k)$ (specifically, $h(k)=o(p_{k+1})$) (Mercer, 2017).

These combinatorial methods achieve the infinitude result for primes in arithmetic progressions (and, in some cases, for small moduli unconditionally), but require deep input from the distribution of small prime gaps for generality.

4. Extensions: Metric and Multidimensional Dirichlet's Theorem

Dirichlet's theorem admits far-reaching generalizations to metric Diophantine approximation and higher dimensions:

• Classical Approximation (one-variable): For every real $x$ and $t>1$, there exist integers $p, q \neq 0$ with

$|x - p/q| \leq 1/(|q| t), \quad |q| < t$

(Kleinbock et al., 2016).

• Multidimensional Form: For every $A\in\mathbb{R}^{m\times n}$ and $t>1$, there exist $p\in\mathbb{Z}^m$, $q\in\mathbb{Z}^n\setminus\{0\}$ such that

$\|Aq-p\|^m \leq 1/t, \quad \|q\|^n < t$

(Fishman et al., 2015).

• Zero-one laws: Measure-theoretic zero-one laws have been established for sets of matrices or real numbers permitting uniform improvements to Dirichlet's bound, conditional on the decay of certain critical series involving the approximation function (Strömbergsson et al., 18 Feb 2026, Kleinbock et al., 2016). These results leverage shrinking target problems in homogeneous dynamics and effective mixing estimates.
• Function fields: Analogous forms of Dirichlet's theorem exist over local fields of positive characteristic, with effective results on "Dirichlet improvability" for submanifolds in such settings (Ganguly et al., 2015).

5. Generalizations and Special Cases

The theorem admits noteworthy special cases and generalizations:

• Special progressions: There exist elementary (non-analytic) proofs in the case of modulus $p$ prime (Macías, 2023) and for residual classes congruent to $1 \pmod{q^v}$ using combinatorial properties derived from Midy's theorem (Castillo et al., 2012).
• S-unit theorem: In algebraic number theory, the Dirichlet $S$-unit theorem generalizes the construction of units in rings of $S$-integers of number fields, and more recently, to infinite rank $\mathbb{Q}$-vector spaces of algebraic $S$-units modulo torsion, with the structure analyzed via the Weil height and the product formula (Fili et al., 2012).
• Sharpness and exceptional sets: For generalizations to Diophantine approximation, the sets of exceptions where the Dirichlet property fails—e.g., for certain exponent pairs $(a,A)$—form comeager sets and can be described via explicit Diophantine exponents or continued fraction properties (Fishman et al., 2015, Kleinbock et al., 2016).

6. Impact, Limitations, and Ongoing Directions

Dirichlet's theorem established the analytic methodology for subsequent results in prime distribution (notably the prime number theorem), the analytic theory of $L$-functions, and advanced sieve methods. However, the classical result is insensitive to effective bounds for the least prime in an arithmetic progression, for which no sufficiently strong unconditional bounds are known. Recent combinatorial/sieve approaches (e.g., leveraging the Jacobsthal function) suggest new routes but depend on deep conjectures about prime patterns (Mercer, 2017).

The theorem's metric generalizations, especially in homogeneous dynamics and Diophantine approximation, continue to evolve rapidly and now underpin zero-one laws for improvements and "shrinking target" phenomena with substantial implications across number theory and ergodic theory (Strömbergsson et al., 18 Feb 2026, Ganguly et al., 2015, Kleinbock et al., 2016).

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References

• Classical and character-theoretic approach: (Avigad et al., 2012, Carneiro, 2016)
• Periodic sieving and analytic-free proofs: (Xu, 2015, Macías, 2023)
• Jacobsthal function and elementary bounds: (Mercer, 2017)
• Zero-one laws and multidimensional results: (Fishman et al., 2015, Strömbergsson et al., 18 Feb 2026, Kleinbock et al., 2016)
• Function field analogues: (Ganguly et al., 2015)
• $S$-unit theorem and generalizations: (Fili et al., 2012)
• Special cases and combinatorial methods: (Castillo et al., 2012)