Hybrid moments of the Riemann zeta-function

The "hybrid" moments $$\int_T^{2T}|\zeta(1/2+it)|^k{(\int_{t-G}^{t+G}|\zeta(1/2+ix)|^\ell dx)}^m dt$$ of the Riemann zeta-function $\zeta(s)$ on the critical line $\Re s = 1/2$ are studied. The expected upper bound for the above expression is $O_\epsilon(T^{1+\epsilon}G^m)$. This is shown to be true for certain specific values of the natural numbers $k,\ell,m$, and the explicitly determined range of $G = G(T;k,\ell,m)$. The application to a mean square bound for the Mellin transform function of $|\zeta(1/2+ix)|^4$ is given.Comment: 27 page

A new invariant that's a lower bound of LS-category

Let $X$ be a simply connected CW-complex of finite type and $\mathbb{K}$ any field. A first known lower bound of LS-category $cat(X)$ is the Toomer invariant $e_{\mathbb{K}} (X)$ (\cite{Too}). In $1980$'s F\'elix et al. introduced the concept of {\it depth} in algebraic topology and proved the depth theorem: $depth (H_*(\Omega X, \mathbb{K})) \leq cat(X)$. In this paper, we use the Eilenberg-Moore spectral sequence of $X$ to introduce a new numerical invariant, denoted by \textsc{r}(X, \mathbb{K}), and show that it has the same properties as those of $e_{\mathbb{K}} (X)$. When the evaluation map (\cite{FHT88}) is non-trivial and $char(\mathbb{K})\not = 2$, we prove that \textsc{r}(X, \mathbb{K}) interpolates $depth(H_*(\Omega X, \mathbb{K}))$ and $e_{\mathbb{K}} (X)$. Hence, we obtain an improvement of L. Bisiaux theorem (\cite{Bis99}) and then of the depth theorem. Motivated by these results, we associate to any commutative differential graded algebra $(A,d)$, a purely algebraic invariant \textsc{r}(A,d) and, via the theory of minimal models, we relate it with our previous topological results. In particular, if $(\Lambda V,d)$ is a Sullivan minimal algebra such that $d=\sum_{i\geq k}d_i$ and $d_i(V)\subseteq \Lambda ^iV$, a greater lower bound is obtained, namely e_0(\Lambda V, d)\geq \textsc{r}(\Lambda V, d) + (k-2).Comment: 21 page

Formulas vs. Circuits for Small Distance Connectivity

We give the first super-polynomial separation in the power of bounded-depth boolean formulas vs. circuits. Specifically, we consider the problem Distance $k(n)$ Connectivity, which asks whether two specified nodes in a graph of size $n$ are connected by a path of length at most $k(n)$. This problem is solvable (by the recursive doubling technique) on {\bf circuits} of depth $O(\log k)$ and size $O(kn^3)$. In contrast, we show that solving this problem on {\bf formulas} of depth $\log n/(\log\log n)^{O(1)}$ requires size $n^{\Omega(\log k)}$ for all $k(n) \leq \log\log n$. As corollaries: (i) It follows that polynomial-size circuits for Distance $k(n)$ Connectivity require depth $\Omega(\log k)$ for all $k(n) \leq \log\log n$. This matches the upper bound from recursive doubling and improves a previous $\Omega(\log\log k)$ lower bound of Beame, Pitassi and Impagliazzo [BIP98]. (ii) We get a tight lower bound of $s^{\Omega(d)}$ on the size required to simulate size-$s$ depth-$d$ circuits by depth-$d$ formulas for all $s(n) = n^{O(1)}$ and $d(n) \leq \log\log\log n$. No lower bound better than $s^{\Omega(1)}$ was previously known for any $d(n) \nleq O(1)$. Our proof technique is centered on a new notion of pathset complexity, which roughly speaking measures the minimum cost of constructing a set of (partial) paths in a universe of size $n$ via the operations of union and relational join, subject to certain density constraints. Half of our proof shows that bounded-depth formulas solving Distance $k(n)$ Connectivity imply upper bounds on pathset complexity. The other half is a combinatorial lower bound on pathset complexity

High pseudomoments of the Riemann zeta function

The pseudomoments of the Riemann zeta function, denoted $\mathcal{M}_k(N)$, are defined as the $2k$th integral moments of the $N$th partial sum of $\zeta(s)$ on the critical line. We improve the upper and lower bounds for the constants in the estimate $\mathcal{M}_k(N) \asymp_k (\log{N})^{k^2}$ as $N\to\infty$ for fixed $k\geq1$, thereby determining the two first terms of the asymptotic expansion. We also investigate uniform ranges of $k$ where this improved estimate holds and when $\mathcal{M}_k(N)$ may be lower bounded by the $2k$th power of the $L^\infty$ norm of the $N$th partial sum of $\zeta(s)$ on the critical line.Comment: This paper has been accepted for publication in Journal of Number Theor

New zero free regions for the derivatives of the Riemann zeta function

The main aim of this paper is twofold. First we generalize, in a novel way, most of the known non-vanishing results for the derivatives of the Riemann zeta function by establishing the existence of an infinite sequence of regions in the right half-plane where these derivatives cannot have any zeros; and then, in the rare regions of the complex plane that do contain zeros of the k-th derivative of the zeta function, we describe a unexpected phenomenon, which implies great regularities in their zero distributions. In particular, we prove sharp estimates for the number of zeros in each of these new critical strips, and we explain how they converge, in a very precise, periodic fashion, to their central, critical lines, as k increases. This not only shows that the zeros are not randomly scattered to the right of the line Re(s)=1, but that, in many respects, their two-dimensional distribution eventually becomes much simpler and more predictable than the one-dimensional behavior of the zeros of the zeta function on the line Re(s)=1/2

Necessary and sufficient conditions for existence of bound states in a central potential

We obtain, using the Birman-Schwinger method, a series of necessary conditions for the existence of at least one bound state applicable to arbitrary central potentials in the context of nonrelativistic quantum mechanics. These conditions yield a monotonic series of lower limits on the "critical" value of the strength of the potential (for which a first bound state appears) which converges to the exact critical strength. We also obtain a sufficient condition for the existence of bound states in a central monotonic potential which yield an upper limit on the critical strength of the potential.Comment: 7 page

Lower bounds on Ricci curvature and quantitative behavior of singular sets

Let Yn denote the Gromov-Hausdorff limit M[superscript n][subscript i][d[subscript GH] over ⟶]Y[superscript n] of v-noncollapsed Riemannian manifolds with Ric[subscript M[superscript n][subscript i]] ≥ −(n−1). The singular set S ⊂ Y has a stratification S[superscript 0] ⊂ S[superscript 1] ⊂ ⋯ ⊂ S, where y ∈ S[superscript k] if no tangent cone at y splits off a factor ℝ[superscript k+1] isometrically. Here, we define for all η > 0, 0 < r ≤1, the k-th effective singular stratum S[superscript k][subscript η,r] satisfying ⋃[subscript η]⋂[subscript r]S[superscript k][subscript η,r] = S[superscript k]. Sharpening the known Hausdorff dimension bound dim S[superscript k] ≤ k, we prove that for all y, the volume of the r-tubular neighborhood of S[superscript k][subscript η,r] satisfies Vol(T[subscript r](S[superscript k][subscript η,r])∩B[subscript 1/2](y)) ≤ c(n,v,η)r[superscript n−k−η]. The proof involves a quantitative differentiation argument. This result has applications to Einstein manifolds. Let B[subscript r] denote the set of points at which the C[superscript 2]-harmonic radius is ≤ r. If also the M[superscript n][subscript i] are Kähler-Einstein with L[subscript 2] curvature bound, ∥Rm∥[subscript L2] ≤ C, then Vol(B[subscript r]∩B[subscript 1/2](y)) ≤ c(n,v,C)r[superscript 4] for all y. In the Kähler-Einstein case, without assuming any integral curvature bound on the M[superscript n][subscript i], we obtain a slightly weaker volume bound on B[subscript r] which yields an a priori L[subscript p] curvature bound for all p < 2. The methodology developed in this paper is new and is applicable in many other contexts. These include harmonic maps, minimal hypersurfaces, mean curvature flow and critical sets of solutions to elliptic equations