In this paper, the scaling-law vector calculus, which is connected between the vector calculus and the scaling law in fractal geometry, is addressed based on the Leibniz derivative and Stieltjes integral for the first time. The scaling-law Gauss-Ostrogradsky-like, Stokes-like, and Green-like theorems, and Green-like identities are considered in sense of the scaling-law vector calculus. The scaling-law Navier-Stokes-like equations are obtained in detail. The obtained result is as a potentially mathematical tool proposed to develop an important way of approaching this challenge for the scaling-law flows.

The main target of this article is to prove the products, behaviors and simple zeros for the classes of the entire functions associated with the Weierstrass-Hadamard product and the Taylor series.

In the present paper we propose a new approach for the generalized Riemann hypothesis in theoretical framework of the Dirichlet $L$-series. The Dirichlet’s lambda function is used as the testing function to prove the generalized Riemann hypothesis. The obtained results can be also applied to consider the other classes of the Dirichlet $L$-series.

In this paper, we investigate some new inequalities based on the Riesz-type fractional integral operator for synchronous and bounded functions.

In this paper, we investigate some Tur\’{an}-type inequalities for the hypergeometric superhyperbolic sine and cosine functions associated with Kummer confluent hypergeometric series of first type.

This paper is devoted to the study of Tur\’{an} type inequalities for some well-known special functions such as supersine and supercosine which are derived by using a new form of the Cauchy-Bunyakovsky-Schwarz inequality.

The element model of the viscoelastic constitutive model has its significant advantage in describing the three stages of rheological curve. Through the medium of the k-Hilfer-Prabhakar fractional derivative, our primary intension in this paper is to establish a viscoelastic constitutive model coupling with the nonlinear time varying elastic element E(t). The k-Hilfer-Prabhakar fractional derivative is characterized by the recovery between the modified model and the known classical models of linear viscoelasticity. According to the theory of the Laplace transform and Boltzmann superposition principle, the strain of the proposed model is obtained. In summing up it may be stated that the analysis of parameter identification indicates the validity and rationality of the modified model.