bibkey: radziwill2012mollifying
authors: Maksym Radziwiłł
year: 2012
title: "Limitations to mollifying ζ(s)"
doi: null
url: https://arxiv.org/abs/1207.6583v1
claim: "Theorem 1 gives an unconditional positive lower bound for normalized dyadic critical-line mollifier error at fixed polynomial length; the actual Selberg polynomial satisfies its conditions at both active height scales, so a vanishing unweighted error is unavailable as a joint-term supplier."
strata_touched: []
license: citation-only
triage: anchor

A lower bound for normalized mollifier error

The inspected author version is arXiv:1207.6583v1, submitted 27 July 2012. Its 17-page PDF has SHA-256 57b3135287f911a1f8ee9e9e96e8df03ae079704c5f29291a5afca2a2047d621. Equation (1) and Theorem 1, printed p.2, were also checked in the versioned HTML. This note pins that preprint; it makes no claim about a later journal version and supplies no Lean verification.

The source’s exact contract

Fix $\theta >0$. Let $$M_T(s)=\sum _{n\le T^{\theta }}{a}_T(n)n^{-s},a_T(1)=1,a_T(n){\ll }_{\epsilon }{n}^{\epsilon }.$$ For such polynomial families, Theorem 1 gives an absolute $c>0$ and, for sufficiently large $T$, $$\frac{1}{T}{\int }_T^{2T}|1-\zeta (1/2+it)M_T(1/2+it){|}^{2}dt\ge c/\theta .\text{\hspace{1em}(RL1)}$$ The exponent $\theta$ is fixed; this note does not assume uniform starting thresholds when it varies. The coefficient condition is uniform in the family. No RH, simple-zero or inverse-${\zeta }^{\prime }$ moment condition is assumed in Theorem 1. Theorems 2–3 impose RH and additional conditions to obtain stronger lower bounds; they are not used for (RL1).

Exact maps for the original Selberg polynomial

The actual residual application uses $$V_N(s)=\sum _{n\le N}\mu (n)\left(1-\frac{\log n}{\log N}\right)n^{-s}.$$ Its first coefficient equals one and every coefficient has absolute value at most one, so it meets the source’s uniform growth condition. At $T=\sqrt{N}$, take $\theta =2$, giving directly $$\frac{1}{\sqrt{N}}{\int }_{\sqrt{N}}^{2\sqrt{N}}|1-\zeta (1/2+it)V_N(1/2+it){|}^{2}dt\ge c/2.\text{\hspace{1em}(RL2)}$$ This is the height scale of the residual smoothing $\nu =N^{-1/2}$ in (I84)–(I86), rather than the much larger contour height $T_N\asymp N^4$ used only to pay horizontal edges.

For the smooth-frequency height in (I82), $T\asymp (\log N{)}^2/h$ with $h\asymp N^{-1/4}{e}^{\lambda \Omega }$, one has $\log N/\log T\to 4$. Choose the fixed exponent $\theta =5$ and pad the coefficient list by zero for $N<n\le T^5$. Eventually $N\le T^5$, so (RL1) yields $$\frac{1}{T}{\int }_T^{2T}|1-\zeta (1/2+it)V_N(1/2+it){|}^{2}dt\ge c/5.\text{\hspace{1em}(RL3)}$$ This avoids substituting a varying exponent into a fixed-parameter theorem. Padding changes neither the actual polynomial nor its first coefficient.

What this excludes and what it leaves open

Equations (RL2)–(RL3) exclude a supplier requiring these unweighted, normalized dyadic mean-square errors to tend to zero. They do not supply an upper bound or sign for the project’s jointly signed functional, whose arithmetic coefficients and weights differ.

After restoring the Nyman–Beurling weight $(1/4+t^2{)}^{-1}$, the corresponding window lower bound furnished by (RL1) is only of order $1/T$, which tends to zero. Thus (RL1) is compatible with the full weighted norm tending to zero. A positive mean-square lower bound also remains compatible with a particular signed interaction tending to zero. Neither a contradiction to RH nor a noncancellation theorem for the project’s full residue sum follows from this source.