Mathematics

In this paper we provide an explicit bound for |ζ(1+it)| in the form of |ζ(1+it)|≤min(logt, 1 2 logt+1.93, 1 5 logt+44.02) . This improves on the current best-known explicit bound of |ζ(1+it)|≤62.6(logt ) 2/3 up until t of the magnitude 10 10 7 .

We prove the existence of the M-function, by which we can state the limit theorem for the value-distribution of the main term in the asymptotic formula for the summatory function of the Goldbach generating function.

In a seminal 2007 paper, Andrews introduced a class of combinatorial objects that generalize partitions called k -marked Durfee symbols. Multivariate rank generating functions for these objects have been shown by many to have interesting modularity properties at certain vectors of roots of unity. Motivated by recent studies of rank generating functions for strongly unimodal sequences, we apply methods of Andrews to define an analogous class of combinatorial objects called k -marked strongly unimodal symbols that generalize strongly unimodal sequences. We establish a multivariate rank generating function for these objects, which we study combinatorially. We conclude by discussing potential quantum modularity properties for this rank generating function at certain vectors of roots of unity.

We find an asymptotic expansion of Selberg's central limit theorem for the Riemann zeta function on ?= 1 2 +(logT ) ?��?and t?�[T,2T] , where 0<θ< 1 2 is a constant.

We describe and illustrate the local neighbourhoods of vertices and edges in the (2, 2)-isogeny graph of principally polarized abelian surfaces, considering the action of automorphisms. Our diagrams are intended to build intuition for number theorists and cryptographers investigating isogeny graphs in dimension/genus 2, and the superspecial isogeny graph in particular.

Let P n ( y 1 ,?? y n ):= ??1?�i<j?�n (1??y i y j ) and P n := sup ( y 1 ,?? y n ) P n ( y 1 ,?? y n ) where the supremum is taken over the n -ples ( y 1 ,?? y n ) of real numbers satisfying 0<| y 1 |<| y 2 |<??| y n | . We prove that P n ??2 ?�n/2??for every n , i.e., we extend to all n the bound that Pohst proved for n??1 . As a consequence, the bound for the absolute discriminant of a totally real field in terms of its regulator is now proved for every degree of the field.

We prove that any smooth projective geometrically connected non-isotrivial curve of genus g?? over a function field of any characteristic has at most 112 g 2 +240g+380 torsion points for any Abel-Jacobi embedding of the curve into its Jacobian. The proof basically uses Zhang's admissible pairing on curves, the arithmetic Hodge index theorem for function fields and the metrized graph analogue of Elkie's lower bound for the Green function.

Let F/ Q p be a non-Archimedean local field, and G F be the absolute Galois group of F . Let ? 1 and ? 2 be two finite dimensional complex representations of G F . Let ? be a nontrivial additive character of F . Then question is: What is the twisting formula for the root number W( ? 1 ??? 2 ,?) ? In general, answer of this question is not known yet. But if one of ? i (i=1,2) is one-dimensional with "sufficiently" large conductor, then Deligne gave a twisting formula for W( ? 1 ??? 2 ,?) . Later Deligne and Henniart give a general twisting formula for a zero dimensional virtual representation twisted by a finite dimensional representation of G F . In this paper, first we extend Deligne's twisting formula for Heisenberg representation of dimension prime p , then we further extend Deligne-Henniart's result. Finally, we give two very important applications of our twisting formula -- invariant formula of local root numbers for U-isotropic Heisenberg representations and a converse theorem in the Galois side.

For a certain function J(s) we prove that the identity ζ(2s) ζ(s) ??s??1 2 )J(s)= ζ(2s+1) ζ(s+1/2) , holds in the half-plane Re (s)>1/2 and both sides of the equality are analytic in this half-plane.

We prove an improved spectral large sieve inequality for the family of S L 3 (Z) Hecke-Maass cusp forms. The method of proof uses duality and its structure reveals unexpected connections to Heath-Brown's large sieve for cubic characters.