We consider the polynomial p(x)=(x+1)(x-2-i)(x-3-i)(x-1-i) over the complex numbers. We have deg p′(x) = deg p(x)−1 = 4-1 = 3. The Fundamental Theorem of Algebra asserts that the same reduction occurs for the total number of zeros of the polynomials. When we consider the fractional derivatives the following questions arise:… Continue reading…
The video below shows the α-th Grünwald-Letnikov fractional derivative of the Riemann Zeta Function ζ(s) for α between 0 and 10 on -20 ≤ R(s) ≤ 20 and −3 ≤ I(s) ≤ 27. The hue represents the argument with red representing the positive real direction and cyan the negative real… Continue reading…
Explore the sieve of Eratosthenes. Click on a number to have all its multiples marked by changing the field color to red and crossing them out. Numbers that you have clicked appear on green background. When there are no white fields left, the numbers in green fields are prime numbers…. Continue reading…
One identifies the space of binary quadratic forms with the upper half plane $$\mathfrak{h}$$. The set of well-rounded forms corresponds to an infinite tree (red). The vertices of this tree correspond to perfect forms, and dualizing gives the Voronoi tessellation of h by hyperbolic triangles (blue). This structure can be… Continue reading…
The Hermite constant for the ring of integers in real quadratic fields $$\mathbb{Q}(\sqrt{d})$$ for square-free positive integers $$d$$ are plotted above. The color of each point signifies the class number of the field. This is part of a current project of D. Yasaki into scaled trace forms over real quadratic… Continue reading…
Let $$p$$ be an odd prime number. The graph shows the subgroup lattice of the group $$E_1$$, which is the unique non-abelian group of order $$p^3$$ of exponent $$p$$. $$C_p$$ denotes the cyclic group with $$p$$ elements. The subfield lattice of the unique extension of $$\mathbb{Q}_p$$ with Galois group $$E_1$$… Continue reading…
^ Level 19 Name Index con len c2 c3 > Cusps Gal Supergroups Subgroups 19A14 285 1 285 5 6 1915 61 91 1815 19A2 ^ Level 21 Name Index con len c2 c3 > Cusps Gal Supergroups Subgroups 21A14 252 2 63 8 0 2112 67 127 21B4 21D5 21C6 ^ Level 25 Name Index… Continue reading…
The mathematics journal Involve is dedicated to showcasing and encouraging high quality mathematical research involving students (at all levels). The editorial board consists of mathematical scientists each of whom is personally committed to nurturing student participation in research. Filip Saidak is one of the editors of Involve.
The absolute value of the Riemann zeta function $$\zeta(\sigma+it)$$ for $$0 \le \sigma \le 8$$ and $$0.1 \le t \le 60$$ over the complex plane. The white dots in the valleys are the non trivial zeros of the function. The Riemann hypothesis claims that the real part of all these… Continue reading…
The plot shows the distribution of the zeros of the 38th derivative of the Riemann zeta function on the complex plane with the zero free regions for $$\zeta^{(38)}$$ as described in New Zero-Free Regions for the Derivatives of the Riemann Zeta Function by Thomas Binder, Sebastian Pauli, and Filip Saidak.
