Over the past three decades, fractional calculus has gained increasing importance and practical relevance in various fields of science and engineering. This article aims to develop enhanced ...

In this paper we bound character sums of the shape ∑ n≤N χ 1 f n χ 2 f n+l , ; where χ1 and χ2 are non-principal multiplicative characters modulo a prime p, f(x) is a real-valued, twice-differentiable ...

Harmonic functions, defined as twice continuously differentiable functions satisfying Laplace’s equation, have long been a subject of intense study in both pure and applied mathematics. Their ...

The original version of this story appeared in Quanta Magazine. Calculus is a powerful mathematical tool. But for hundreds of years after its invention in the 17th century, it stood on a shaky ...