Search

Using a new smoothing argument, we establish explicit versions of the prime number theorem in short intervals and consider the limiting behaviour of this result., Comment: 13 pages, any comments or suggestions are warmly welcomed! (Extra lit review and minor typographical updates in update.)

Assuming the Riemann Hypothesis, we combine powerful theoretical methods, computational techniques, and the latest explicit knowledge on the zeros of the Riemann zeta-function to prove $$|\psi(x) - x| \leq \frac{\sqrt{x}\log{x}(\log{x} - \log\log{x})}{8\pi}$$ for all $x\geq 401\,211\,790$, where $\psi(x)$ is the Chebyshev $\psi$-function. These bounds for the error term in the prime number theorem refine classical results from Schoenfeld. We also demonstrate how our results can be used to refine bounds for other prime counting functions, which have broad applications throughout the literature., Comment: 28 pages, 2 tables, any feedback welcomed! (Latest version implements valuable feedback and suggestions which resulted in more impressive outcomes.)

A 1976 result from Norton may be used to give an asymptotic (but not explicit) description of the constant in Mertens' second theorem for primes in arithmetic progressions. Assuming the Generalized Riemann Hypothesis, we give an effective description of Norton's observation., Comment: 13 pages, comments welcome! To appear in the Proceedings of the Integers Conference 2023

Using an explicit version of Selberg's upper sieve, we obtain explicit upper bounds for the number of $n\leq x$ such that a non-empty set of irreducible polynomials $F_i(n)$ with integer coefficients are simultaneously prime; this set can contain as many polynomials as desired. To demonstrate, we present computations for some irreducible polynomials and obtain an explicit upper bound for the number of Sophie Germain primes up to $x$, which have practical applications in cryptography., Comment: 18 pages, one table, comments welcome!

We improve upon the traditional error term in the truncated Perron formula for the logarithm of an $L$-function. All our constants are explicit., Comment: 11 pages

We update Sunley's explicit estimate for the ideal-counting function, which is the number of integral ideals of bounded norm in a number field., Comment: 26 pages, 6 tables, comments welcome! Updates explained in the article

Under GRH, we establish a version of Duke's short-sum theorem for entire Artin $L$-functions. This yields corresponding bounds for residues of Dedekind zeta functions. All numerical constants in this work are explicit., Comment: 20 pages, to appear in the Journal of Number Theorey

We obtain an effective analytic formula, with explicit constants, for the number of distinct irreducible factors of a polynomial $f \in \mathbb{Z}[x]$. We use an explicit version of Mertens' theorem for number fields to estimate a related sum over rational primes. For a given $f \in \mathbb{Z}[x]$, our result yields a finite list of primes that certifies the number of distinct irreducible factors of $f$., Comment: 15 pages

We obtain unconditional, effective number-field analogues of the three Mertens' theorems, all with explicit constants and valid for $x\geq 2$. Our error terms are explicitly bounded in terms of the degree and discriminant of the number field. To this end, we provide unconditional bounds, with explicit constants, for the residue of the corresponding Dedekind zeta function at $s=1$., Comment: 19 pages

Assuming the Generalized Riemann Hypothesis we obtain uniform, effective number-field analogues of Mertens' theorems., Comment: 8 Pages