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Analytic Number Theory · Wessen Getachew

C_χ(n) — Dirichlet Character-Twisted Density

$$C_\chi(n) = L(2,\chi)\cdot\prod_p\left(1-\chi(p)\cdot\frac{\min(n{+}1,p)}{p^2}\right), \qquad L(s,\chi) = \sum_{n=1}^\infty \frac{\chi(n)}{n^s} = \prod_p\left(1-\frac{\chi(p)}{p^s}\right)^{-1}$$
Every page so far — 1 through 8 — used the trivial character in disguise: gcd(r,·)=1 is exactly χ₀(r), the principal character, which only ever says yes (1) or no (0). Replace that with a genuine Dirichlet character χ mod q — a signed or complex weight on residue classes, not just a filter — and ζ(2) becomes L(2,χ), an actual L-function value. This is the same Euler-product machinery the whole project has used from page 1 onward, now carrying real arithmetic phase information instead of a binary indicator.
What changes here, precisely: every other page on this site asks "does r survive?" — a yes/no question with a density between 0 and 1. This page asks "how does r's residue class interact with χ?" — the answer is a complex number, can exceed 1 in magnitude, and can be negative or rotate through the complex plane entirely. It's the same product architecture, carrying genuinely different arithmetic content.
Verification — Character Orthogonality, Dual L(2,χ), Generalized Jump
Three independent checks, run before any of this was built. Orthogonality: the character table for the modulus you pick satisfies ∑ᵣ χₖ(r)·χₗ(r)‾ = φ(q)·[k=l] exactly (φ(q) = q−1 for prime q; φ(4) = 2) — the fundamental correctness test for a character construction, checked live below for whichever modulus is selected. L(2,χ): computed two independent ways (Euler product vs. direct 3,000,000-term Dirichlet series) for a genuinely complex order-6 character mod 7 — agreed to 12 decimal places. Generalized jump: the local factor's jump at n=p−1 is exactly −χ(p)/p² (a direct algebraic generalization of page 1's own local-factor structure) — verified to 14 decimal places against a complex χ(p), not just the real ±1 case.
CheckMethod AMethod BAgreement
L(2,χ), order-6 char mod 70.90224702530 + 0.23254898129i0.90224702530 + 0.23254898128i12 dp
Jump at p=13, χ mod 7 (k=1)0.00591715976 − 7.25e-19i (measured)0.00591715976 − 7.25e-19i (formula)14 dp
Orthogonality, q=5, q=7diagonal = φ(q) = q−1 exactlyoff-diagonal = 0 exactlyexact
Live orthogonality matrix recomputed for the modulus selected in the Character Explorer below — not a static screenshot of the check above.
Character Explorer
Modulus q
Character χ_k
Order of χ (smallest m with χᵐ=χ₀)
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Real or complex?
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L(2,χ)
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Character Table — χ(r) for r = 0,…,q−1
rgcd(r,q)χ(r) exactχ(r) decimal|χ(r)|
Click any row for the discrete-log derivation of that value. Only prime moduli plus q=4 are offered — those are exactly the cases where (ℤ/qℤ)* is cyclic, so a single primitive root generates every character directly. Composite non-prime-power moduli need the group's full CRT decomposition and are out of scope here.
Live L-Function & Density Calculator
n (block depth) 6
Animates n 0→200 (loops) — everything below updates live, including the per-prime table.
C_χ(n)
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D_χ(n) = C_χ(n)/L(2,χ)
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|C_χ(n)| (magnitude)
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∞ asymptotic product truncated at primes ≤ 2,000,000 — reliable to ~7–8 dp. n's range is 0–2000, uncapped, since the product cost doesn't depend on n's value (same reasoning as Page 7).
Per-Prime Local Factors
First 500 primes (scroll for more) — click any value for its exact derivation. CSV export covers all ≤2,000,000.
pχ(p)f_p^χ(n)cumulative C_χ(n)
Generalized Jump — Signed by χ(p)
trivial character: always −1/p² (uniform decrease) selected χ: −χ(p)/p² (real part shown; can flip sign or vanish)
Bars at each prime p ≤ 61 show the exact local-factor jump at n=p−1. Where χ(p)=−1 the bar flips above the axis relative to the trivial case; where χ(p)=0 (p | q) the bar vanishes entirely — that prime is inert for this character.
The Elementary-to-Advanced Shift, Precisely
1 — What stays exactly the same
The independence-across-primes argument (CRT) that justifies every Euler product on this site, every page of this site, is untouched. C_χ(n) converges absolutely for any fixed n by the same finitely-many-exceptional-primes-plus-summable-tail argument as everywhere else on this site. This page is still fully unconditional — nothing here needs GRH.
2 — What's genuinely new
χ₀ (every other page, implicitly) only ever multiplies by 0 or 1 — pure filtering. A nontrivial χ multiplies by an arbitrary root of unity, so C_χ(n) can be complex, can exceed 1 in magnitude, and the "jump" at n=p−1 can point in any direction in the complex plane rather than always shrinking the density. That's the qualitative shift: from counting survivors to measuring phase-weighted arithmetic structure.
3 — Where GRH would actually enter (and doesn't, here)
This page computes L(2,χ) and C_χ(n) as constants — that's unconditional, same as ζ(2) and C(n). GRH becomes relevant only for a different kind of question: how quickly does a finite character-weighted count (∑_{n≤X} χ(n)·[survives block]) converge to its asymptotic value, as a function of X. That error-term question is genuinely conjectural in general; the constant this page computes is not.
4 — Sanity check against Page 1
The universal trivial character (χ≡1, no modulus, no excluded primes) reduces C_χ(n) to exactly C(n) — verified to 8 decimal places. The principal character mod q (q>1) is different: it reduces to C(n) with the finitely many primes dividing q switched off, which is a related but distinct exact quantity, also directly verifiable.
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