We establish a connection between the conjectural two-over-two ratios formula for the Riemann zeta-function and a conjecture concerning correlations of a certain arithmetic function. Specifically, we prove that the ratios conjecture and the arithmetic correlations conjecture imply the same result. This casts a new light on the underpinnings of the ratios conjecture, which previously had been mo...

The Geng hierarchy is derived with the aid of Lenard recursion sequences. Based on Lax matrix, a hyperelliptic curve arithmetic genus n+1 introduced, from which meromorphic function ϕ ...

Since the discovery of fuzzy sets, the arithmetic operations of fuzzy numbers (Zadeh [7, 8]) which may be viewed as a generalization of interval arithmetic (Moore [5]) have emerged as an important area of research within the theory of fuzzy sets (Mizumoto and Tanaka [4], Dubois and Prade [2]). The arithmetic operations of fuzzy numbers have been performed either by extension principle [7, 8] or...

In the present paper, we propose a new definition of discrete parabolas, the so-called arithmetic discrete parabolas. We base our approach on a non-constant thickness function and characterized the 0connected and 1-connected parabolas in terms of thickness function. This results extend the well-known characterization of the κ-connectedness of arithmetic discrete lines, depending on the norm ‖ ·...

The Veneziano amplitude for the tree-level scattering of four tachyonic scalar of open string theory has an arithmetic analogue in terms of the p-adic gamma function. We propose a quantum extension of this amplitude using the q-extended p-adic gamma function given by Koblitz. This provides a one parameter deformation of the arithmetic Veneziano amplitude. We also comment on the dificulty in gen...

In this paper we give new estimates for integrals involving some arithmetic functions defined over prime numbers. The main focus here is on the counting function $\pi(x)$ and Chebyshev $\vartheta$-function. Some of these depend correctness Riemann hypothesis nontrivial zeros zeta $\zeta(s)$.

This paper presents a new characterization of provably recursive functions of first-order arithmetic. We consider functions defined by sets of equations. The equations can be arbitrary, not necessarily defining primitive recursive, or even total, functions. The main result states that a function is provably recursive iff its totality is provable (using natural deduction) from the defining set o...

Abstract We show that the conjecture of [27] for special value at $s=1$ zeta function an arithmetic surface is equivalent to Birch–Swinnerton–Dyer Jacobian generic fibre.