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This article is devoted to the even entire functions, which are the exact solutions for the Laplace and diffusion equations. These functions are considered in the algebraic number field. We guess that the functions have purely real zeros in the entire complex plane. These are proposed as new connections with algebraic number theory and mathematical physics. [ABSTRACT FROM AUTHOR]

D. A. Goldston, J. Pintz and C. Y. Yıldırım [3] proved the existence of infinitely many prime pairs whose difference is arbitrarily small compared to the average gap, namely $$ \liminf_{n\to\infty} \frac{p_{n+1}-p_n}{\log p_n}=0. $$ In the present work we generalize their result to totally real number fields. We prove that if ω0 and ω1 run over distinct prime elements of the number field, then $$ \liminf_{\omega_0,\omega_1} \left|\frac{{\mathbf {N}}(\omega_1-\omega_0)}{{\mathbf {N}}\omega_0}\right|=0. $$ [ABSTRACT FROM AUTHOR]

Abstract: For an odd prime number p, we consider the p-primary part of the Brumer–Stark conjecture for a cyclic extension of number fields of degree 2p. We extend earlier work of Greither, Roblot, and Tangedal (2004) by proving the conjecture when the minus component of the p-primary part of the class group of K is not a cyclic Galois module. Consequently, we are able to prove the full Brumer–Stark conjecture for some new classes of number field extensions. [Copyright &y& Elsevier]

Abstract: Text: By a result of G. Freiman and A. Geroldinger [G. Freiman, A. Geroldinger, An addition theorem and its arithmetical application, J. Number Theory 85 (1) (2000) 59–73] it is known that the set of lengths of factorizations of an algebraic integer (in the ring of integers of an algebraic number field), or more generally of an element of a Krull monoid with finite class group, has a certain structure: it is an almost arithmetical multiprogression for whose difference and bound only finitely many values are possible, and these depend just on the class group. We establish a sort of converse to this result, showing that for each choice of finitely many differences and of a bound there exists some number field such that each almost arithmetical multiprogression with one of these difference and that bound is up to shift the set of lengths of an algebraic integer of that number field. Moreover, we give an explicit sufficient condition on the class group of the number field for this to happen. Video: For a video summary of this paper, please visit http://www.youtube.com/watch?v=c61xM-5D6Do. [Copyright &y& Elsevier]