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Text Let K be a complete, algebraically closed, non-Archimedean valued field, and let ϕ ∈ K ( z ) with deg ⁡ ( ϕ ) ≥ 2 . In this paper we consider the family of functions ord Res ϕ n ( x ) , which measure the resultant of ϕ n at points x in P K 1 , the Berkovich projective line, and show that they converge locally uniformly to the diagonal values of the Arakelov–Green's function g μ ϕ ( x , x ) attached to the canonical measure of ϕ . Following this, we are able to prove an equidistribution result for Rumely's crucial measures ν ϕ n , each of which is a probability measure supported at finitely many points whose weights are determined by dynamical properties of ϕ . Video For a video summary of this paper, please visit https://youtu.be/YCCZD1iwe00 . [ABSTRACT FROM AUTHOR]

Text Two infinite sequences A and B of nonnegative integers are called additive complements , if their sum contains all sufficiently large integers. We also say that B is an additive complement of A if A and B are additive complements. In this paper, we consider a problem of Ben Green on additive complements of the squares: S = { 1 2 , 2 2 , … } . The following result is proved: if B = { b n } n = 1 ∞ with b n ≥ π 2 16 n 2 − 0.57 n 1 2 log ⁡ n − β n 1 2 for all positive integers n and any given constant β , then B is not an additive complement of S . In particular, B = { ⌊ π 2 16 n 2 ⌋ | n = 1 , 2 , … } is not an additive complement of S . Video For a video summary of this paper, please visit https://youtu.be/cVXWCP4Igp8 . [ABSTRACT FROM AUTHOR]

In this paper, we prove that the expected number of points in F q 2 of multiplicity m , for 0 ≤ m ≤ q + 1 , with respect to a randomly chosen arrangement of q + 1 lines with different slopes, is ( 1 / ( m ! e ) ) q 2 + O ( q ) , as q → ∞ . We further state that the distance between the number of such points in a randomly chosen arrangement and ( 1 / ( m ! e ) ) q 2 is lower than q ln ⁡ q with probability close to 1 for large q . [ABSTRACT FROM AUTHOR]

Text We develop geometric methods to study the generating weights of free modules of vector-valued modular forms of half-integral weight, taking values in a complex representation of the metaplectic group. We then compute the generating weights for modular forms taking values in the Weil representation attached to cyclic quadratic modules of order 2 p r , where p ≥ 5 is a prime. We also show that the generating weights approach a simple limiting distribution as p grows, or as r grows and p remains fixed. Video For a video summary of this paper, please visit https://youtu.be/QNbPSXXKot4 . [ABSTRACT FROM AUTHOR]

We investigate level p Eisenstein congruences for GSp 4 , generalisations of level 1 congruences predicted by Harder. By studying the associated Galois and automorphic representations we see conditions that guarantee the existence of a paramodular form satisfying the congruence. This provides theoretical justification for computational evidence found in the author's previous paper. [ABSTRACT FROM AUTHOR]

In this paper, we study the polynomial Gauss sums over finite fields, and present an analogue of Davenport–Hasse's theorem for the polynomial Gauss sums, which is a generalization of the previous result obtained by Hayes. [ABSTRACT FROM AUTHOR]

In this paper we focus on finding all the factorials expressible as a product of a fixed number of 2 k -nacci numbers with k ≥ 2 . We derive the 2-adic valuation of the 2 k -nacci sequence and use it to establish bounds on the solutions of the initial equation. In addition, we specify a more general family of sequences, for which we can perform a similar procedure. We also investigate a possible connection of these results with p -regular sequences. [ABSTRACT FROM AUTHOR]

A new class of partitions, partitions with designated summands, was introduced by Andrews, Lewis and Lovejoy. Let PD ( n ) denote the number of partitions of n with designated summands. Andrews, Lewis and Lovejoy established many congruences modulo 3 and powers of 2 for PD ( n ) by using the theory of modular forms. In this paper, we prove several infinite families of congruences modulo 9 and 27 for PD ( n ) by employing the generating functions of PD ( 3 n ) and PD ( 3 n + 1 ) which were discovered by Chen, Ji, Jin and Shen. For example, we prove that for n ≥ 0 and k ≥ 1 , PD ( 2 18 k − 1 ( 12 n + 1 ) ) ≡ 0 ( mod 27 ) . Furthermore, using some results due to Newman, we find some strange congruences modulo 27 for PD ( n ) . For example, we prove that for k ≥ 0 , PD ( 13 9 k ( 75 p + 2 ) ) ≡ 0 ( mod 27 ) and PD ( 2 × 13 9 k + 8 ) ≡ 0 ( mod 27 ) , where p is a prime and p ≡ 1 ( mod 12 ) . [ABSTRACT FROM AUTHOR]

Abstract: The main purpose of this paper is to prove a conjecture of the Euler numbers and its generalization by using the analytic methods. That is, for any prime and integer we proved , where are the Euler numbers and the Euler function. [Copyright &y& Elsevier]

Abstract: In this paper we consider the integers of the forms and , which are ever focused by F. Cohen, P. Erdős, J.L. Selfridge, W. Sierpiński, etc. We establish a general theorem. As corollaries, we prove that (i) there exists an infinite arithmetic progression of positive odd numbers for each term k of which and any nonnegative integer n, each of four integers , , and has at least two distinct odd prime factors; (ii) there exists an infinite arithmetic progression of positive odd numbers for each term k of which and any nonnegative integer n, each of ten integers , , , , , , , , and has at least two distinct odd prime factors; (iii) there exists an infinite arithmetic progression of positive odd numbers for each term k of which and any nonnegative integer n, each of ten integers , , , , , , , , and has at least two distinct odd prime factors. Furthermore, we pose several related open problems in the introduction and three conjectures in the last section. [Copyright &y& Elsevier]

Abstract: In this paper we obtain an improved asymptotic formula on the frequency of k-free numbers with a given difference. We also give a new upper bound of Barban–Davenport–Halberstam type for the k-free numbers in arithmetic progressions. [Copyright &y& Elsevier]

Abstract: This paper discusses tetrahedra with rational edges forming an arithmetic progression, focussing specifically on whether they can have rational volume or rational face areas. Several infinite families are found which have rational volume, a face can have rational area only if its edges are themselves in arithmetic progression, and a tetrahedron can have at most one such rational face area. [Copyright &y& Elsevier]