Mathematics

We consider sums of the form ∑ϕ(γ) , where ϕ is a given function, and γ ranges over the ordinates of nontrivial zeros of the Riemann zeta-function in a given interval. We show how the numerical estimation of such sums can be accelerated by a simple device, and give examples involving both convergent and divergent infinite sums.

Let g?? be an integer. A natural number is said to be a base- g Niven number if it is divisible by the sum of its base- g digits. Assuming Hooley's Riemann Hypothesis, we prove that the set of base- g Niven numbers is an additive basis, that is, there exists a positive integer C g such that every natural number is the sum of at most C g base- g Niven numbers.

For a set A of non-negative integers, let R A (n) denote the number of solutions to the equation n=a+ a ′ with a , a ′ ∈A . Denote by χ A (n) the characteristic function of A . Let b n >0 be a sequence satisfying lim sup n→∞ b n <1 . In this paper, we prove some Erd\H os--Fuchs-type theorems about the error terms appearing in approximation formulæ for R A (n)= ∑ n k=0 χ A (k) χ A (n−k) and ∑ N n=0 R A (n) having principal terms ∑ n k=0 b k b n−k and ∑ N n=0 ∑ n k=0 b k b n−k , respectively.

The degree sequence of the algebraic numbers in an algebraic linear recurrence sequence is shown to be virtually periodic. This is proved using the Skolem-Mahler-Lech theorem. It has applications to the degree sequence and the minimal polynomial sequence for exponential sums over finite fields. The degree periodicity also holds for some more complicated non-linear recurrence sequences. We give one example from the iterations of a polynomial map. This depending on the dynamic Mordell-Lang conjecture which has been proved in some cases.

We prove an algebraicity result for certain critical value of adjoint L -functions for GSp 4 over a totally real number field in terms of the Petersson norm of normalized generic cuspidal newforms on GSp 4 . This is a generalization of our previous result arXiv:1902.06429.

We study the algebraicity of the central critical values of twisted triple product L -functions associated to motivic Hilbert cusp forms over a totally real étale cubic algebra in the totally unbalanced case. The algebraicity is expressed in terms of the cohomological period constructed via the theory of coherent cohomology on quaternionic Shimura varieties developed by Harris. As an application, we generalize our previous result on Deligne's conjecture for certain automorphic L -functions for GL 3 × GL 2 . We also establish a relation for the cohomological periods under twisting by algebraic Hecke characters.

Arithmetical structures on graphs were first mentioned in \cite{Lorenzini89} by D. Lorenzini. Later in \cite{arithmetical} they were further studied on square non-negative integer matrices. In both cases, necessary and sufficient conditions for the finiteness of the set of arithmetical structures were given. Therefore, it is natural to ask for an algorithm that compute them. This article is divided in two parts. In the first part we present an algorithm that computes arithmetical structures on a square integer non-negative matrix L with zero diagonal. In order to do this we introduce a new class of Z-matrices, which we call quasi M -matrices. We recall that arithmetical structures on a matrix L are solutions of the polynomial Diophantine equation f L (X):=det(Diag(X)?�L)=0. In the second part, the ideas developed to solve the problem over matrices are generalized to a wider class of polynomials, which we call dominated. In particular the concept of arithmetical structure is generalized on this new setting. All this leads to an algorithm that computes arithmetical structures of dominated polynomials. Moreover, we show that any other integer solution of a dominated polynomial is bounded by a finite set and we explore further methods to obtain them.

In this paper, we define some weighted sums of the alternating multiple T -values (AMTVs), and study several duality formulas for them by using the tools developed in our previous papers. Then we introduce the alternating version of the convoluted T -values and Kaneko-Tsumura ψ -function, which are proved to be closely related to the AMTVs. At the end of the paper, we study the $\Q$-vector space generated by the AMTVs of any fixed weight w and provide some evidence for the conjecture that their dimensions { d w } w≥1 form the tribonacci sequence 1, 2, 4, 7, 13, ....

We construct a family of examples of harmonic Maass forms of polynomial growth for any level whose shadows are Eisenstein series of integral weight. We further consider Dirichlet series attached to a harmonic Maass form of polynomial growth, study its analytic properties, and prove an analogue of Weil's converse theorem.

In [2], the author claims that the fields Q( D ??4 ) defined in the paper and the compositum of all D 4 extensions of Q coincide. The proof of this claim depends on a misreading of a celebrated result by Shafarevich. The purpose is to salvage the main results of [2]. That is, the classification of torsion structures of E defined over Q when base changed to the compositum of all D 4 extensions of Q main results of [2]. All the main results in [2] are still correct except that we are no longer able to prove that these two fields are equal.