gpt 6 astra ultra attempt on Math-299

Montgomery's Large Value Conjecture for Dirichlet Polynomials

Definitions and mathematical statement

Let $D(t)=\sum_{N<n\le2N}b_n n^{it}$ with $|b_n|\le1$, and let $W\subset[0,T]$ be one-separated: distinct elements differ by at least one. For fixed $\sigma>1/2$, assume $|D(t)|>N^\sigma$ on $W$. The target is \[ |W|\le T^{o(1)}N^{2-2\sigma},\qquad N\le T\le N^{O(1)}. \] Explicitly, with a fixed exponent range $T\le N^A$, the subpower notation asks for every $\eta>0$ a bound $|W|\le C_{\sigma,A,\eta}T^\eta N^{2-2\sigma}$ uniformly in the coefficients and the separated set. An exponent range growing with $N$ is not the fixed-polynomial-range convention. The statement counts large values, not only an integral mean value.

Formulation and concept references: [S1].

Short English statement

Can a bounded-coefficient Dirichlet polynomial take very large values at only the conjectured number of well-separated times in every polynomial-length interval?

Research attempt

Scope and primary-source check.

This attempt was prepared by GPT-6 Astra Ultra on 27 September 2026. Conjecture 1.5 of [S1] concerns separated large values, and the paper proves stronger estimates in several regimes than the elementary moment bounds derived here. We do not describe those elementary bounds as current records. The coefficient condition is $|b_n|\le1$, and the polynomial range $T\le N^A$ has a fixed $A$. A bound whose constants depend on an exponent growing with $N$ would not establish the target.

Take $N$ to be a positive integer, as in the standard formulation. If $\sigma\ge1$, the strict inequality $|D(t)|>N^\sigma$ is impossible, because $|D(t)|\le N$. The substantive range is $1/2<\sigma<1$. We prove the conjectured order, up to subpowers, when $T$ is comparable to $N$, and give a family of higher-moment estimates for longer intervals. The transition from integral means to separated samples is proved explicitly instead of assuming that point values have positive measure.

An elementary mean-square estimate.

Let $F(t)=\sum_{m\le M}a_m m^{it}$, with arbitrary complex coefficients, and let $I$ be an interval of length $L$. Expanding and integrating, the diagonal contributes $L\sum|a_m|^2$. For $m\ne l$, \[ \left|\int_I(m/l)^{it}\,dt\right| \le\frac2{|\log(m/l)|} \le\frac{2M}{|m-l|}. \tag{1} \] The last inequality follows by integrating $1/x$ between $m$ and $l$. For each positive difference $h$, Cauchy--Schwarz gives $\sum_m|a_ma_{m+h}|\le\sum_m|a_m|^2$, extending coefficients by zero. Summing $1/h$ and both orders of the pair yields \[ \int_I|F(t)|^2\,dt \le\left(L+C M\log(2M)\right)\sum_m|a_m|^2. \tag{2} \] The bound is uniform in the location of the interval: translating it only multiplies coefficients by unimodular factors. The logarithmic loss is harmless for subpower purposes but is not suppressed before it has been accounted for.

A separated-sample inequality.

For a continuously differentiable complex function $f$ on $J=[t_0-1/2,t_0+1/2]$, the fundamental theorem of calculus applied to $|f|^2$ gives, for each $s\in J$, \[ |f(t_0)|^2\le|f(s)|^2+2\int_J|f f'|. \] Integrating over $s$ proves \[ |f(t_0)|^2\le\int_J|f|^2+2\int_J|f f'|. \tag{3} \] If $W$ is one-separated, the interiors of these length-one intervals are disjoint. Summing $3$ and using Cauchy--Schwarz gives \[ \sum_{t\in W}|f(t)|^2 \le\int_{-1/2}^{T+1/2}|f|^2 +2\left(\int_{-1/2}^{T+1/2}|f|^2 \int_{-1/2}^{T+1/2}|f'|^2\right)^{1/2}. \tag{4} \] The slight enlargement of the interval handles samples at zero and $T$. Separation is used here, and cannot be dropped.

Apply this to $f(t)=N^{-it}D(t)$, which has the same modulus as $D$. Its derivative has coefficients $i b_n\log(n/N)$, bounded in modulus by $\log2$. Estimate $2$ applies equally to $f$ and $f'$, since the common phase $N^{-it}$ cancels in the expanded integral. We obtain \[ \sum_{t\in W}|D(t)|^2\le C(T+N\log(2N))N, \] and hence \[ |W|\le C(T+N\log(2N))N^{1-2\sigma}. \tag{5} \] In particular, for $N\le T\le C_0N$, this is $O_{C_0}(N^{2-2\sigma}\log(2N))$, which implies the requested $O_{\sigma,C_0,\eta}(T^\eta N^{2-2\sigma})$ for every $\eta>0$. This proves a genuine interval-range special case.

Integer moments and their coefficient growth.

For a fixed positive integer $k$, expand \[ D(t)^k=\sum_m a_m m^{it},\qquad a_m=\sum_{\substack{N<n_1,\ldots,n_k\le2N\\n_1\cdots n_k=m}} b_{n_1}\cdots b_{n_k}. \] The support lies in $N^k<m\le(2N)^k$. Let $r_m$ count the ordered tuples in this sum. Then $|a_m|\le r_m\le d_k(m)$ and $\sum_mr_m=N^k$, so \[ \sum_m|a_m|^2\le\max_{m\le(2N)^k}d_k(m)\,N^k. \tag{6} \] For fixed $k$ and every $\epsilon>0$, the elementary divisor estimate $d_k(m)\le C_{k,\epsilon}m^\epsilon$ follows from \[ d_k(m)=\prod_{p^a\parallel m}\binom{a+k-1}{k-1}. \] For sufficiently large primes, $\binom{a+k-1}{k-1}\le k^a\le p^{\epsilon a}$. For the finitely many smaller primes, the polynomial in $a$ is at most a prime-dependent constant times $p^{\epsilon a}$; their constants have a finite product. Renaming $\epsilon$ to absorb the fixed factor $k$ in $m\le(2N)^k$, $6$ gives \[ \sum_m|a_m|^2\le C_{k,\epsilon}N^{k+\epsilon}. \tag{7} \] This is where fixed $k$ matters. Constants in this proof are not uniform when $k$ is allowed to grow with $N$.

Apply $4$ to $f(t)=N^{-ikt}D(t)^k$. Its derivative has multiplier $\log(m/N^k)$, bounded by $k\log2$. Equations $2$, $4$, and $7$ yield, after absorbing logarithms, \[ |W|\le C_{k,\epsilon} (T+N^k)N^{k-2k\sigma+\epsilon}. \tag{8} \] All coefficients and all separated sets allowed by the problem are covered. The argument has not assumed random signs or an average over the coefficients.

The exponent loss remains polynomial.

Let $T=N^A$ with fixed $A\ge1$. Formula $8$ has exponent \[ \max\{A,k\}+k(1-2\sigma)+\epsilon.\tag{9} \] At $A=1$, $k=1$ gives the conjectured $2-2\sigma$. At $A=2$, $\sigma=3/4$, even the best integer choices in this family give exponent one, attained at $k=2$; the target exponent is $1/2$. For $k=1$ the exponent is $3/2$, and for $k\ge2$ it is $k/2$. Thus merely taking higher integer moments in this manner does not close that representative gap. An $N^{1/2}$ loss cannot be renamed $T^{o(1)}$: for $T=N^2$, it is $T^{1/4}$, which fails the requested bound for sufficiently small fixed $\eta$.

This calculation diagnoses this particular moment argument, not the limits of the stronger methods in [S1]. It also shows why invoking an unspecified "high moment" without tracking its constants and length can conceal the main difficulty.

A phase-coherence countertest.

Choose $b_n=n^{-it_0}$. Then $D(t_0)=N$ exactly. For $|s|\le1/2$, after removing the common phase $N^{is}$, \[ \operatorname{Re}\sum_{N<n\le2N}(n/N)^{is} \ge N\cos((\log2)/2). \] Thus a high peak persists on an interval of fixed width. The coefficients are fully admissible, so a proof cannot assume pointwise cancellation at each time or treat all phases as independent. This example gives only a bounded number of one-separated samples in that interval and does not contradict the conjectured count. Constructing many mutually compatible peaks is a separate issue.

Verification and final status.

The local checks expand small powers exactly, count the product multiplicities in $6$, and verify the algebra in $9$. Numerical phase samples are labeled only as sanity checks of the explicit example. The proof of $4$ supplies the otherwise missing link between integral and pointwise large-value statements.

UNRESOLVED. The result proved here is the requested estimate for $T\asymp N$, together with the uniform fixed-moment family $8$. The first missing step is a separated-sample estimate eliminating the extra polynomial dependence on $T/N$ in the longer ranges. Concrete targets are correlations among large-value times, cancellation in the off-diagonal moment sum before absolute values, and estimates for product coefficients sharper than their maximum multiplicity. The arbitrary fixed-polynomial-range conjecture is not established.

Sources

• S1 Larry Guth and James Maynard, New large value estimates for Dirichlet polynomials, Annals of Mathematics 203 $2026$, 623-675, Conjecture 1.5. https://arxiv.org/html/2405.20552. Formulation source.