Abstract: This paper is closely related to the recent work [BW17] of the same authors and our purpose is to elaborate more on some of the results and methods from [BW17]. More specifically our goal is two-fold. Firstly, we will indicate how a simple variant related to Section 4 in [BW17] leads to the following improvements of Theorem 3 in [BW17]

Abstract: We show in this paper that after proper scalings, the characteristic polynomial of a random unitary matrix converges to a random analytic function whose zeros, which are on the real line, form a determinantal point process with sine kernel. Our scaling is performed at the so-called “microscopic” level, that is we consider the characteristic polynomial at points whose distance to 1 has order 1 / n. We prove that the rescaled characteristic polynomial does not even have a moment of order one, hence making the classical techniques of random matrix theory difficult to apply. In order to deal with this issue, we couple all the dimensions n on a single probability space, in such a way that almost sure convergence occurs when n goes to infinity. The strong convergence results in this setup provide us with a new approach to ratios: we are able to solve open problems about the limiting distribution of ratios of characteristic polynomials evaluated at points of the form $$\exp (2 i \pi \alpha /n)$$ and related objects (such as the logarithmic derivative). We also explicitly describe the dependence relation for the logarithm of the characteristic polynomial evaluated at several points on the microscopic scale. On the number theory side, inspired by the work by Keating and Snaith, we conjecture some new limit theorems for the value distribution of the Riemann zeta function on the critical line at the level of stochastic processes.

Abstract: For k ≤ n, let E(2n,k) be the sum of all multiple zeta values with even arguments whose weight is 2n and whose depth is k. Of course E(2n, 1) is the value ζ(2n) of the Riemann zeta function at 2n, and it is well known that E(2n, 2) = 3 4ζ(2n). Recently Shen and Cai gave formulas for E(2n, 3) and E(2n, 4) in terms of ζ(2n) and ζ(2)ζ(2n − 2). We give two formulas for E(2n,k), both valid for arbitrary k ≤ n, one of which generalizes the Shen–Cai results; by comparing the two we obtain a Bernoulli-number identity. We also give explicit generating functions for the numbers E(2n,k) and for the analogous numbers E⋆(2n,k) defined using multiple zeta-star values of even arguments.

Abstract: We consider the one-parameter generalization S 4 of 4-sphere with a conical singularity due to identification τ = τ +2πq in one isometric angle. We compute the value of the spectral zeta-function at zero $$ \widehat{\zeta}(q)=\zeta \left(0;q\right) $$ that controls the coefficient of the logarithmic UV divergence of the one-loop partition function on S 4 . While the value of the conformal anomaly a-coefficient is proportional to $$ \widehat{\zeta}(1) $$ , we argue that in general the second c ∼ C T anomaly coefficient is related to a particular combination of the second and first derivatives of $$ \widehat{\zeta}(q) $$ at q = 1. The universality of this relation for C T is supported also by examples in 6 and 2 dimensions. We use it to compute the c-coefficient for conformal higher spins finding that it coincides with the “r = −1” value of the one-parameter Ansatz suggested in arXiv:1309.0785 . Like the sums of a s and c s coefficients, the regularized sum of $$ {\widehat{\zeta}}_s(q) $$ over the whole tower of conformal higher spins s = 1, 2,… is found to vanish, implying UV finiteness on S 4 and thus also the vanishing of the associated Renyi entropy. Similar conclusions are found to apply to the standard 2-derivative massless higher spin tower. We also present an independent computation of the full set of conformal anomaly coefficients of the 6d Weyl graviton theory defined by a particular combination of the three 6d Weyl invariants that has a (2, 0) supersymmetric extension.

Abstract: In this paper a functional equation for the fractional derivative of the Riemann zeta function is presented. The fractional derivative of the zeta function is computed by a generalization of the Gr ...

Abstract: For any m,n ∈ ℕ we first give new proofs for the following well-known combinatorial identities Sn(m) =∑k=1nn k (–1)k–1 km =∑n≥r1≥r2≥⋯≥rm≥1 1 r1r2⋯rm and ∑k=1n(–1)n–kn kkn = n!, and then we produce the generating function and an integral representation for Sn(m). Using them we evaluate many interesting finite and infinite harmonic sums in closed form. For example, we show that ζ(3) = 1 9∑n=1∞Hn3 + 3H nHn(2) + 2H n(3) 2n , and ζ(5) = 2 45∑n=1∞Hn4 + 6H n2H n(2) + 8H nHn(3) + 3(H n(2))2 + 6H n(4) n2n , where Hn(i) are generalized harmonic numbers defined below.

Abstract: Montgomery’s pair correlation conjecture predicts the asymptotic behavior of the function N(T, β) defined to be the number of pairs γ and γ′ of ordinates of nontrivial zeros of the Riemann zetafunction satisfying 0 0, using Montgomery’s formula and some extremal functions of exponential type. These functions are optimal in the sense that they majorize and minorize the characteristic function of the interval [−β, β] in a way to minimize the L1 ( R, { 1− ( sinπx πx )2} dx ) -error. We give a complete solution for this extremal problem using the framework of reproducing kernel Hilbert spaces of entire functions. This extends previous work of P. X. Gallagher [18] in 1985, where the case β ∈ 1 2 N was considered using non-extremal majorants and minorants.

Abstract: We consider the integer QH state on Riemann surfaces with conical singularities, with the main objective of detecting the effect of the gravitational anomaly directly from the form of the wave function on a singular geometry. We suggest the formula expressing the normalisation factor of the holomorphic state in terms of the regularized zeta determinant on conical surfaces and check this relation for some model geometries. We also comment on possible extensions of this result to the fractional QH states.

Abstract: In this paper, we work out some explicit formulae for double nonlinear Euler sums involving harmonic numbers and alternating harmonic numbers. As applications of these formulae, we give new closed form representations of several quadratic Euler sums through Riemann zeta function and linear sums. The given representations are new.

Abstract: Assuming the Riemann Hypothesis, Soundararajan [Ann. of Math. (2) 170 (2009), 981–993] showed that . His method was used by Chandee [Q. J. Math. 62 (2011), 545–572] to obtain upper bounds for shifted moments of the Riemann Zeta function. Building on these ideas of Chandee and Soundararajan, we obtain, conditionally, upper bounds for shifted moments of Dirichlet -functions which allow us to derive upper bounds for moments of theta functions.

Abstract: Let $G$ be a finitely generated torsion-free nilpotent group. The representation zeta function $\zeta_G(s)$ of $G$ enumerates twist isoclasses of finite-dimensional irreducible complex representations of $G$. We prove that $\zeta_G(s)$ has rational abscissa of convergence $a(G)$ and may be meromorphically continued to the left of $a(G)$ and that, on the line $\{s\in\mathbb{C} \mid \textrm{Re}(s) = a(G)\}$, the continued function is holomorphic except for a pole at $s=a(G)$. A Tauberian theorem yields a precise asymptotic result on the representation growth of $G$ in terms of the position and order of this pole. We obtain these results as a consequence of a more general result establishing uniform analytic properties of representation zeta functions of finitely generated nilpotent groups of the form $\mathbf{G}(\mathcal{O})$, where $\mathbf{G}$ is a unipotent group scheme defined in terms of a nilpotent Lie lattice over the ring $\mathcal{O}$ of integers of a number field. This allows us to show, in particular, that the abscissae of convergence of the representation zeta functions of such groups and their pole orders are invariants of $\mathbf{G}$, independent of $\mathcal{O}$.

Abstract: The zeta function of a curve C over a finite field may be expressed in terms of the characteristic polynomial of a unitary matrix Θ C . We develop and present a new technique to compute the expected value of tr(Θ C n ) for various moduli spaces of curves of genus g over a fixed finite field in the limit as g is large, generalising and extending the work of Rudnick [ Rud10 ] and Chinis [ Chi16 ]. This is achieved by using function field zeta functions, explicit formulae, and the densities of prime polynomials with prescribed ramification types at certain places as given in [ BDF + 16 ] and [ Zha ]. We extend [ BDF + 16 ] by describing explicit dependence on the place and give an explicit proof of the Lindelof bound for function field Dirichlet L -functions L (1/2 + it , χ). As applications, we compute the one-level density for hyperelliptic curves, cyclic l-covers, and cubic non-Galois covers.

Abstract: For a fixed $b\in\mathbb{N}=\{1,2,3,\ldots\}$ we say that a point $(r,s)$ in the integer lattice $\mathbb{Z} \times \mathbb{Z}$ is $b$-visible from the origin if it lies on the graph of a power function $f(x)=ax^b$ with $a\in\mathbb{Q}$ and no other integer lattice point lies on this curve (i.e., line of sight) between $(0,0)$ and $(r,s)$. We prove that the proportion of $b$-visible integer lattice points is given by $1/\zeta(b+1)$, where $\zeta(s)$ denotes the Riemann zeta function. We also show that even though the proportion of $b$-visible lattice points approaches $1$ as $b$ approaches infinity, there exist arbitrarily large rectangular arrays of $b$-invisible lattice points for any fixed $b$. This work specialized to $b=1$ recovers original results from the classical lattice point visibility setting where the lines of sight are given by linear functions with rational slope through the origin.

Abstract: We study motivic zeta functions of degenerating families of Calabi-Yau varieties. Our main result says that they satisfy an analog of Igusa's monodromy conjecture if the family has a so-called Galois-equivariant Kulikov model; we provide several classes of examples where this condition is verified. We also establish a close relation between the zeta function and the skeleton that appeared in Kontsevich and Soibelman's non-archimedean interpretation of the SYZ conjecture in mirror symmetry.

Abstract: Two results related to the mixed joint universality for a polynomial Euler product and a periodic Hurwitz zeta function , when is a transcendental parameter, are given. One is the mixed joint functional independence and the other is a generalised universality, which includes several periodic Hurwitz zeta functions.

Abstract: By a “generalized Calabi–Yau hypersurface” we mean a hypersurface in ℙn of degree d dividing n + 1. The zeta function of a generic such hypersurface has a reciprocal root distinguished by minimal p-divisibility. We study the p-adic variation of that distinguished root in a family and show that it equals the product of an appropriate power of p times a product of special values of a certain p-adic analytic function ℱ. That function ℱ is the p-adic analytic continuation of the ratio F(Λ)∕F(Λp), where F(Λ) is a solution of the A-hypergeometric system of differential equations corresponding to the Picard–Fuchs equation of the family.

Abstract: Assume G is a finite abstract simplicial complex with f-vector (v0,v1, ), and generating function f(x) = sum(k=1 v(k-1) x^k = v0 x + v1 x^2+ v2 x^3 + , the Euler characteristic of G can be written as chi(G)=f(0)-f(-1) We study here the functional f1'(0)-f1'(-1), where f1' is the derivative of the generating function f1 of G1 The Barycentric refinement G1 of G is the Whitney complex of the finite simple graph for which the faces of G are the vertices and where two faces are connected if one is a subset of the other Let L is the connection Laplacian of G, which is L=1+A, where A is the adjacency matrix of the connection graph G', which has the same vertex set than G1 but where two faces are connected they intersect We have f1'(0)=tr(L) and for the Green function g L^(-1) also f1'(-1)=tr(g) so that eta1(G) = f1'(0)-f1'(-1) is equal to eta(G)=tr(L-L^(-1) The established formula tr(g)=f1'(-1) for the generating function of G1 complements the determinant expression det(L)=det(g)=zeta(-1) for the Bowen-Lanford zeta function zeta(z)=1/det(1-z A) of the connection graph G' of G We also establish a Gauss-Bonnet formula eta1(G) = sum(x in V(G1) chi(S(x)), where S(x) is the unit sphere of x the graph generated by all vertices in G1 directly connected to x Finally, we point out that the functional eta0(G) = sum(x in V(G) chi(S(x)) on graphs takes arbitrary small and arbitrary large values on every homotopy type of graphs

Abstract: The fractional derivative of the Dirichlet eta function is computed in order to investigate the behavior of the fractional derivative of the Riemann zeta function on the critical strip. Its converg ...

Abstract: Making use of topological periodic cyclic homology, we extend Grothendieck's standard conjectures of type C and D (with respect to crystalline cohomology theory) from smooth projective schemes to smooth proper dg categories in the sense of Kontsevich. As a first application, we prove Grothendieck's original conjectures in the new cases of linear sections of determinantal varieties. As a second application, we prove Grothendieck's (generalized) conjectures in the new cases of "low-dimensional" orbifolds. Finally, as a third application, we establish a far-reaching noncommutative generalization of Berthelot's cohomological interpretation of the classical zeta function and of Grothendieck's conditional approach to "half"' of the Riemann hypothesis. Along the way, following Scholze, we prove that the topological periodic cyclic homology of a smooth proper scheme X agrees with the crystalline cohomology theory of X (after inverting the characteristic of the base field).