Nonzero summable totient kernels do not vanish on a progression

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_proof: Divisor counting gives exact progression means; two fresh-prime refinements separate an isolated nonzero coefficient from the l1 tail.

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Statement

Let $\varphi$ be Euler's totient, with $\varphi(1)=1$, and write $\rho(n)=\varphi(n)/n$ for $n\ge1$. For a real sequence $(c_s)_{s\ge1}$ with $0<\sum_s|c_s|<\infty$, set

$$F(N)=\sum_{s\ge1}c_s\rho(N+s),\qquad N\ge1.$$

For every integer $Q\ge1$ and $a\in\mathbb Z$,

$$\limsup_{\substack{N\to\infty\\N>0,\ N\equiv a\ (Q)}}|F(N)|>0.$$

The series is absolutely and uniformly convergent in $N$ since $0<\rho\le1$. The following explicit refinement statement is part of the claim. Define, for integer $r$,

$$\kappa_Q(r)=\frac1{\zeta(2)} \prod_{p\mid Q}(1-p^{-2})^{-1} \prod_{p\mid\gcd(Q,r)}(1-p^{-1}),\qquad d_s=c_s\kappa_Q(a+s),$$

where products are over primes and $\zeta(2)=\sum_{n\ge1}n^{-2}$. Take any $s_*\ge1$ with $d_{s_*}\ne0$, any integer $K\ge s_*$ with $\sum_{s>K}|d_s|<|d_{s_*}|/4$, and any prime $\ell>K$ not dividing $Q$. Let $a_0,a_1$ satisfy

$$a_0\equiv a_1\equiv a\pmod Q,\qquad a_0\equiv0\pmod\ell,\qquad a_1\equiv-s_*\pmod\ell.$$

The Cesàro means of $F$ along the two progressions modulo $Q\ell$ exist and differ. At least one has magnitude greater than

$$\eta=\frac{|d_{s_*}|}{4\ell(1-\ell^{-2})}>0,$$

and that progression contains infinitely many positive $N$ with $|F(N)|>\eta/2$. Means omit any finitely many nonpositive indices. The choices exist because $\kappa_Q(r)>0$, $\sum_s|d_s|<\infty$, and there are arbitrarily large primes outside the finite set dividing $Q$. The proof establishes those facts as well as the claim.

Scope and application

The kernel and the original progression are fixed before choosing the anchor, cutoff, refining prime and limit. No lower bound uniform over all nonzero kernels or growing moduli is asserted. The claim concerns every fixed subprogression as well, but it does not prohibit vanishing on a specially selected subsequence.

The research applications give a finite-window error bound and apply this claim to fixed dilation grids. The combined sketch explains its proposed use for Problem 249. Rationality of $\sum_{n\ge1}\varphi(n)/2^n$ has not been shown to force a nonzero kernel of this kind to tend to zero. The theorem therefore does not prove or disprove that target.

Current standing and verification

The complete proof is author-recorded at tier 0. It has no identified gap, independent whole-claim acceptance, formal verification, or novelty claim. Internal research challenge supplied no independent acceptance. No native L-claim is consumed. The proof uses the elementary totient divisor identity, congruence counting, absolutely convergent series and Euler products, the Chinese remainder theorem, and infinitude of primes. It uses neither Dirichlet's theorem on primes in progressions nor an uninspected source theorem.

The entire mathematical subject is this statement and its self-contained proof; no program or computational input is required. The finite evidence and subsequent tail applications in the research home are separate results and do not certify this claim.