science

1. Dirichlet characters
   • A Dirichlet character modulo m is a completely multiplicative arithmetic function
   χ : ℤ → ℂ such that
   – χ(n+m) = χ(n) for all n,
   – χ(n) = 0 whenever gcd(n, m) > 1,
   – χ(n) ≠ 0 (and in fact |χ(n)| = 1) whenever gcd(n, m)=1.
   • There are exactly φ(m) such characters (including the “principal character” χ₀, which is 1 on all n coprime to m, and 0 otherwise).

2. Dirichlet L-series
   For each character χ mod m one forms the Dirichlet L-series
   L(s, χ) = ∑_{n=1}^∞ χ(n) n^(−s) , Re(s) > 1.
   This generalizes the Riemann zeta function (which is the special case χ = χ₀ with m=1).

3. Euler product factorization
   Because χ is completely multiplicative, one shows for Re(s)>1:
   L(s, χ) = ∏_{p prime} (1 − χ(p) p^(−s))^(−1).
   This factorization links L(s, χ) to the distribution of primes in each residue class mod m.

4. Nonvanishing at s = 1
   A key analytic fact (Dirichlet’s theorem) is that for every nonprincipal character χ (i.e. χ≠χ₀), the L-series L(s, χ) extends to a holomorphic, nonzero function at s=1. Meanwhile L(s, χ₀) has a simple pole at s=1, just like the Riemann zeta function ζ(s).

5. Summing over characters
   Use the orthogonality relations of characters: for any integer n with gcd(n,m)=1,
   (1/φ(m)) ∑_{χ mod m} χ(n) over all φ(m) characters = 1,
   and it’s 0 if n is not coprime to m.
   This lets you isolate those primes in a particular residue class a mod m by forming
   ∑_{χ mod m} χ(a)̄ · L(s,χ)
   (where χ(a)̄ is the complex conjugate of χ(a)). In the Euler product this picks out exactly the primes p with p≡a mod m.

6. Logarithmic differentiation and Tauberian theorems
   One takes logarithms of the Euler products and looks at
   –(L′/L)(s,χ) = ∑_{n=1}^∞ Λ(n) χ(n) n^(−s),
   where Λ is the von Mangoldt function. Summing these expressions over characters and using orthogonality isolates
   ∑_{p ≡ a (mod m)} p^(−s) ,
   plus lower‐order terms. One then applies complex‐analytic Tauberian theorems (or elementary partial summation arguments) to deduce that as x→∞
   #{ p ≤ x : p ≡ a (mod m) } ∼ (1/φ(m)) x / log x ,
   in particular infinitely many such primes exist.

Conclusion
Dirichlet’s theorem on arithmetic progressions (proven by exactly the steps sketched above) implies that for any m and any a with gcd(a,m) = 1, there are infinitely many primes p ≡ a mod m. Setting m = 35 and a = 6 shows there are infinitely many primes that leave remainder 6 upon division by 35.