Stieltjes constants of L-functions in the extended Selberg class

Let f be an arithmetic function and let $${\mathcal {S}}^\#$$ S # denote the extended Selberg class. We denote by $${\mathcal {L}}(s) = \sum _{n = 1}^{\infty }\frac{f(n)}{n^s}$$ L ( s ) = ∑ n = 1 ∞ f ( n ) n s the Dirichlet series attached to f. The Laurent–Stieltjes constants of $${\mathcal {L}}(s)$$ L ( s ) , which belongs to $${\mathcal {S}}^\#$$ S # , are the coefficients of the Laurent expansion of $${\mathcal {L}}$$ L at its pole $$s=1$$ s = 1 . In this paper, we give an upper bound of these constants, which is a generalization of many known results.

Single-Valued Integration and Superstring Amplitudes in Genus Zero

We study open and closed string amplitudes at tree-level in string perturbation theory using the methods of single-valued integration which were developed in the prequel to this paper (Brown and Dupont in Single-valued integration and double copy, 2020). Using dihedral coordinates on the moduli spaces of curves of genus zero with marked points, we define a canonical regularisation of both open and closed string perturbation amplitudes at tree level, and deduce that they admit a Laurent expansion in Mandelstam variables whose coefficients are multiple zeta values (resp. single-valued multiple zeta values). Furthermore, we prove the existence of a motivic Laurent expansion whose image under the period map is the open string expansion, and whose image under the single-valued period map is the closed string expansion. This proves the recent conjecture of Stieberger that closed string amplitudes are the single-valued projections of (motivic lifts of) open string amplitudes. Finally, applying a variant of the single-valued formalism for cohomology with coefficients yields the KLT formula expressing closed string amplitudes as quadratic expressions in open string amplitudes.

Some Analytic and Geometric Properties of Solution to Skew-Symmetric Elliptic Systems

We study the properties of complex-valued functions of a complex variable, whose real and imaginary parts satisfy a second-order skew-symmetric strongly elliptic system with constant real coefficients in the plane. The behavior of such functions and their dilatations near singular points is investigated and the dependence of the type of the singularity on the form of the Laurent expansion of the function under consideration is established. The principle of the argument is established for the functions with poles under study, analogs of the Ruschet and Hurwitz theorems are proved

Snake Graphs from Triangulated Orbifolds

We give an explicit combinatorial formula for the Laurent expansion of any arc or closed curve on an unpunctured triangulated orbifold. We do this by extending the snake graph construction of Musiker, Schiffler, and Williams to unpunctured orbifolds. In the case of an ordinary arc, this gives a combinatorial proof of positivity to the generalized cluster algebra from this orbifold.

Hilbert series associated to symplectic quotients by SU2

We compute the Hilbert series of the graded algebra of real regular functions on the symplectic quotient associated to an [Formula: see text]-module and give an explicit expression for the first nonzero coefficient of the Laurent expansion of the Hilbert series at [Formula: see text]. Our expression for the Hilbert series indicates an algorithm to compute it, and we give the output of this algorithm for all representations of dimension at most [Formula: see text]. Along the way, we compute the Hilbert series of the module of covariants of an arbitrary [Formula: see text]- or [Formula: see text]-module as well as its first three Laurent coefficients.

The Hilbert series of SL2-invariants

Let [Formula: see text] be a finite-dimensional representation of the group [Formula: see text] of [Formula: see text] matrices with complex coefficients and determinant one. Let [Formula: see text] be the algebra of [Formula: see text]-invariant polynomials on [Formula: see text]. We present a calculation of the Hilbert series [Formula: see text] as well as formulas for the first four coefficients of the Laurent expansion of [Formula: see text] at [Formula: see text].

On certain sums over ordinates of zeta-zeros II

International audience Let γ denote the imaginary parts of complex zeros ρ = β + iγ of ζ(s). The problem of analytic continuation of the function $G(s) :=\sum_{\gamma >0} {\gamma}^{-s}$ to the left of the line $\Re{s} = −1 $ is investigated, and its Laurent expansion at the pole s = 1 is obtained. Estimates for the second moment on the critical line $\int_{1}^{T} {| G (\frac{1}{2} + it) |}^2 dt $ are revisited. This paper is a continuation of work begun by the second author in [Iv01].