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Artur Kawalec

Publications and source records attributed to Artur Kawalec.

15 recordsLinked to original sources
arXiv2026

On the series expansion of the secondary zeta function about $s=1$ and its coefficients

The secondary zeta function is defined as a generalized zeta series over the imaginary parts of non-trivial zeros assuming (RH). This function admits Laurent series expansion at the double pole at $s=1$. In this article, we derive a new formula for the expansion coefficients of the regular part, which is similar to the Stieltjes constants formula for the Riemann zeta function. We also numerically verify and compute the new formula to high precision for several test cases. Lastly, we also apply the Brent's (BPT) Theorem for improving convergence of the main formula.

math.NT↗
arXiv2026

Analytical continuation of Euler prime product for $\Re(s)>\tfrac{1}{2}$ assuming (RH)

We analytically continue the Euler prime product for $\Re(s)>\tfrac{1}{2}$ (except for its pole $s=1$) assuming (RH) by introducing a new factor to the Euler product. We also discuss how to recover the Mertens's 3rd Theorem at $s=1$ case, and how to apply the same technique to analytically continue other similar Euler products. In the last part, we also construct a simple script in Pari/GP to compute the Euler product and verify the calculations numerically.

math.GM↗
arXiv2026

On the series expansion of k-free Dirichlet series and its analytical continuation

In this article, we develop a k-free zeta Dirichlet series into a Laurent series with a simple pole, and prove a Stieltjes like formula for the expansion coefficients of the regular part. We also investigate another analytical continuation of these series and develop a formula for $ζ(\tfrac{1}{k})$ for positive integer $k\geq 2$ in terms of the k-free indicator function.

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arXiv2026

On the series expansion of the prime zeta function about $s=1$ and its coefficients

In this article, we derive a series expansion of the prime zeta function about the $s=1$ logarithmic singularity and prove general formula for its expansion coefficients, which is similar to the Stieltjes expansion coefficients for the Riemann zeta function. These results can also be viewed as a generalization of Mertens's Theorems to higher order. We also numerically verify and compute the presented formulas to high precision for several test cases.

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arXiv2024

On the series expansion of the secondary zeta function

In article, we explore the secondary zeta function $Z(s)$, which is defined as a generalized zeta type of series over imaginary parts of non-trivial zeros of the Riemann zeta function $ζ(s)$. This function has been analytically continued as a meromorphic function in $\mathbb{C}$ with one double pole and an infinity of simple poles. The secondary zeta function is of interest because it can naturally represent an analytical formula for non-trivial zeros of the Riemann zeta function that we will explore, and we show that the non-trivial zeros can be generated directly from primes by introducing a new form of an explicit formula written in terms of the prime zeta function. Additionally, we will also give several new series expansions for $Z(s)$ and numerically compute these coefficients to high precision, and also develop several new methods to analytically extend $Z(s)$ to larger domains and develop algorithms to compute them.

math.NT↗
arXiv2024

On the series expansion of a square-free zeta series

In this article, we develop a square-free zeta series associated with the Möbius function into a power series, and prove a Stieltjes like formula for these expansion coefficients. We also investigate another analytical continuation of these series and develop a formula for $ζ(\tfrac{1}{2})$ in terms of the Möbius function, and in the last part, we explore an alternating series version of these results.

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arXiv2022

The inverse Riemann zeta function

In this article, we develop a formula for an inverse Riemann zeta function such that for $w=ζ(s)$ we have $s=ζ^{-1}(w)$ for real and complex domains $s$ and $w$. The presented work is based on extending the analytical recurrence formulas for trivial and non-trivial zeros as to solve an equation $ζ(s)-w=0$ for a given $w$-domain using logarithmic differentiation and zeta recursive root extraction methods. We further explore formulas for trivial and non-trivial zeros of the Riemann zeta function in greater detail, and next, we also explore an expansion of the inverse zeta function by an attractor of its branch singularities, and develop some identities that emerge from them. In the last part, we extend the presented results as a general method for finding zeros and inverses of many other functions, such as the gamma function, the Bessel function of the first kind, or finite/infinite degree polynomials and rational functions, etc. We further compute all the presented formulas numerically to high precision and show that these formulas do indeed converge to the inverse of the Riemann zeta function and the related results. We also develop a fast algorithm to compute $ζ^{-1}(w)$ for complex $w$.

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arXiv2022

Analytical recurrence formulas for non-trivial zeros of the Riemann zeta function

In this article, we develop four types of analytical recurrence formulas for non-trivial zeros of the Riemann zeta function on critical line assuming (RH). Thus, all non-trivial zeros up to the $n$th order must be known in order to generate the $n$th+1 non-trivial zero. All the presented formulas are based on certain closed-form representations of the secondary zeta function family, which are already available in the literature. We also present a formula to generate the non-trivial zeros directly from primes. Thus all primes can be converted into an individual non-trivial zero, and we also give a set of formulas to convert all non-trivial zeros into an individual prime. We also extend the presented results to other Dirichlet-L functions, and in particular, we develop an analytical recurrence formula for non-trivial zeros of the Dirichlet beta function. Throughout this article, we also numerically compute these formulas to high precision for various test cases and review the computed results.

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arXiv2021

The recurrence formulas for primes and non-trivial zeros of the Riemann zeta function

In this article, we explore the Riemann zeta function with a perspective on primes and non-trivial zeros. We develop the Golomb's recurrence formula for the $n$th+1 prime, and assuming (RH), we propose an analytical recurrence formula for the $n$th+1 non-trivial zero of the Riemann zeta function. Thus all non-trivial zeros up the $n$th order must be known to generate the $n$th+1 non-trivial zero. We also explore a variation of the recurrence formulas for primes based on the prime zeta function, which will be a basis for the development of the recurrence formulas for non-trivial zeros based on the secondary zeta function. In the last part, we review the presented formulas and outline the duality between primes and non-trivial zeros. The proposed formula implies that all primes can be converted into an individual non-trivial zero (assuming RH), and conversely, all non-trivial zeros can be converted into an individual prime (not assuming RH). Also, throughout this article, we summarize numerical computation and verify the presented results to high precision.

math.GM↗
arXiv2020

Nested formulas for cosine and inverse cosine functions based on Viète's formula for $π$

In this article, we develop nested representations for cosine and inverse cosine functions, which is a generalization of Viète's formula for $π$. We explore a natural inverse relationship between these representations and develop numerical algorithms to compute them. Throughout this article, we perform numerical computation for various test cases, and demonstrate that these nested formulas are valid for complex arguments and a $k$th branch. We further extend the presented results to hyperbolic cosine and logarithm functions, and using additional trigonometric identities, we explore the sine and tangent functions and their inverses.

math.GM↗
arXiv2020

On the complex magnitude of Dirichlet beta function

In this article, we derive an expression for the complex magnitude of the Dirichlet beta function $β(s)$ represented as a Euler prime product and compare with similar results for the Riemann zeta function. We also obtain formulas for $β(s)$ valid for an even and odd $k$th positive integer argument and present a set of generated formulas for $β(k)$ up to $11$th order, including Catalan's constant and compute these formulas numerically. Additionally, we derive a second expression for the complex magnitude of $β(s)$ valid in the critical strip from which we obtain a formula for the Euler-Mascheroni constant expressed in terms of zeros of the Dirichlet beta function on the critical line. Finally, we investigate the asymptotic behavior of the Euler prime product on the critical line.

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arXiv2019

Asymptotic formulas for harmonic series in terms of a non-trivial zero on the critical line

In this article, we develop two types of asymptotic formulas for harmonic series in terms of single non-trivial zeros of the Riemann zeta function on the critical line. The series is obtained by evaluating the complex magnitude of an alternating and non-alternating series representation of the Riemann zeta function. Consequently, if the asymptotic limit of the harmonic series is known, then we obtain the Euler-Mascheroni constant with $\log(k)$. We further numerically compute these series for different non-trivial zeros. We also investigate a recursive formula for non-trivial zeros.

math.NT↗
arXiv2019

Prime product formulas for the Riemann zeta function and related identities

In this article, we derive a Euler prime product formula for the magnitude of the Riemann zeta function $ζ(s)$ valid for $\Re(s)>1$, as well as similar formulas for $ζ(s)$ valid for an even and odd $k$th positive integer argument. We shall further give a set of generated formulas for $ζ(k)$ up to $11$th order, including Apéry's constant, and also construct formulas for $ζ(3/2)$. We'll also validate these formulas numerically.

math.GM↗
arXiv2016

The $n$th+1 Prime Number Limit Formulas

A new derivation of Golomb's limit formula for generating the $n$th$+1$ prime number is presented. The limit formula is derived by extracting $p_{n+1}$ from Euler's prime product representation of the Riemann zeta function $ζ(s)$ in the limit as $s$ approaches infinity. Also, new variations of these limit formulas are explored, such as the logarithm and a half-prime formulas for the $p_{n+1}$.

math.GM↗