arXiv subjects

Michael J. Mossinghoff

Publications and source records attributed to Michael J. Mossinghoff.

At least 19 recordsLinked to original sources

arXiv2026

The integer group determinants for $\mathbb{Z}_p^n$

We give new conditions on allowable powers of a prime $p$ dividing an integer group determinant for the group $\mathbb{Z}_p^n$, and show these conditions are sharp for $n\leq 5$. The values coprime to $p$ for these groups are already known; determining which multiples of allowable powers of $p$ occur is much more complicated. We show that all multiples of sufficiently large powers of $p$ occur as integer group determinants of $\mathbb{Z}_p^n$. We also provide a complete characterization for the group $\mathbb{Z}_3^3$, where particular arithmetic conditions are required for multiples of certain powers of $3$.

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arXiv2025

Mahler's problem and Turyn polynomials

Mahler's problem asks for the largest possible value of the Mahler measure, normalized by the $L_2$ norm, of a polynomial with $\pm1$ coefficients and large degree. We establish a new record value in this problem exceeding $0.95$ by analyzing certain Turyn polynomials, which are defined by cyclically shifting the coefficients of a Fekete polynomial by a prescribed amount. It was recently established that the distribution of values over the unit circle of Fekete polynomials of large degree is effectively modeled by a particular random point process. We extend this analysis to the Turyn polynomials, and determine expressions for the asymptotic normalized Mahler measure of these polynomials, as well as for their normalized $L_q$ norms. We also describe a number of calculations on the corresponding random processes, which indicate that the Turyn polynomials where the shift is approximately $1/4$ of the length have Mahler measure exceeding $95\%$ of their $L_2$ norm. Further, we show that these asymptotic values are not disturbed by a small change to make polynomials having entirely $\pm1$ coefficients, which establishes the result on Mahler's problem. We also estimate that the limiting value of the normalized $L_1$ norm of these polynomials exceeds $0.977$, in connection with a question of Newman.

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arXiv2024

A note on trigonometric polynomials for lower bounds of $ζ(s)$

Non-negative trigonometric polynomials satisfying certain properties are employed when studying a number of aspects of the Riemann zeta function. When establishing zero-free regions in the critical strip, the classical polynomial $3+4\cos(θ)+\cos(2θ)$ used by de la Vallée Poussin has since been replaced by more beneficial polynomials with larger degree. The classical polynomial was also employed by Titchmarsh to provide a lower bound on $|ζ(σ+it)|$ when $σ>1$. We show that this polynomial is optimal for this purpose.

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arXiv2023

Ideal solutions in the Prouhet-Tarry-Escott problem

For given positive integers $m$ and $n$ with $m<n$, the Prouhet-Tarry-Escott problem asks if there exist two disjoint multisets of integers of size $n$ having identical $k$th moments for $1\leq k\leq m$; in the ideal case one requires $m=n-1$, which is maximal. We describe some searches for ideal solutions to the Prouhet-Tarry-Escott problem, especially solutions possessing a particular symmetry, both over $\mathbb{Z}$ and over the ring of integers of several imaginary quadratic number fields. Over $\mathbb{Z}$, we significantly extend searches for symmetric ideal solutions at sizes $9$, $10$, $11$, and $12$, and we conduct extensive searches for the first time at larger sizes up to $16$. For the quadratic number field case, we find new ideal solutions of sizes $10$ and $12$ in the Gaussian integers, of size $9$ in $\mathbb{Z}[i\sqrt{2}]$, and of sizes $9$ and $12$ in the Eisenstein integers.

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arXiv2022

Explicit zero-free regions for the Riemann zeta-function

We prove that the Riemann zeta-function $ζ(σ+ it)$ has no zeros in the region $σ\geq 1 - 1/(55.241(\log|t|)^{2/3} (\log\log |t|)^{1/3})$ for $|t|\geq 3$. In addition, we improve the constant in the classical zero-free region, showing that the zeta-function has no zeros in the region $σ\geq 1 - 1/(5.558691\log|t|)$ for $|t|\geq 2$. We also provide new bounds that are useful for intermediate values of $|t|$. Combined, our results improve the largest known zero-free region within the critical strip for $3\cdot10^{12} \leq |t|\leq \exp(64.1)$ and $|t| \geq \exp(1000)$.

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arXiv2022

Prime power order circulant determinants

Newman showed that for primes $p\geq 5$ an integral circulant determinant of prime power order $p^t$ cannot take the value $p^{t+1}$ once $t\geq 2.$ We show that many other values are also excluded. In particular, we show that $p^{2t}$ is the smallest power of $p$ attained for any $t\geq 3$, $p\geq 3.$ We demonstrate the complexity involved by giving a complete description of the $25\times 25$ and $27\times 27$ integral circulant determinants. The former case involves a partition of the primes that are $1\bmod5$ into two sets, Tanner's \textit{perissads} and \textit{artiads}, which were later characterized by E. Lehmer.

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arXiv2022

Small polygons with large area

A polygon is \textit{small} if it has unit diameter. The maximal area of a small polygon with a fixed number of sides $n$ is not known when $n$ is even and $n\geq14$. We determine an improved lower bound for the maximal area of a small $n$-gon for this case. The improvement affects the $1/n^3$ term of an asymptotic expansion; prior advances affected less significant terms. This bound cannot be improved by more than $O(1/n^3)$. For $n=6$, $8$, $10$, and $12$, the polygon we construct has maximal area.

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arXiv2022

Wolstenholme and Vandiver primes

A prime $p$ is a Wolstenholme prime if $\binom{2p}{p}\equiv2$ mod $p^4$, or, equivalently, if $p$ divides the numerator of the Bernoulli number $B_{p-3}$; a Vandiver prime $p$ is one that divides the Euler number $E_{p-3}$. Only two Wolstenholme primes and eight Vandiver primes are known. We increase the search range in the first case by a factor of $10$, and show that no additional Wolstenholme primes exist up to $10^{11}$, and in the second case by a factor of $20$, proving that no additional Vandiver primes occur up to this same bound. To facilitate this, we develop a number of new congruences for Bernoulli and Euler numbers mod $p$ that are favorable for computation, and we implement some highly parallel searches using GPUs.

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arXiv2021

Fake Mu's

Let $f(n)$ denote a multiplicative function with range $\{-1,0,1\}$, and let $F(x) = \sum_{n\leq x} f(n)$. Then $F(x)/\sqrt{x} = a\sqrt{x} + b + E(x)$, where $a$ and $b$ are constants and $E(x)$ is an error term that either tends to $0$ in the limit, or is expected to oscillate about $0$ in a roughly balanced manner. We say $F(x)$ has persistent bias $b$ (at the scale of $\sqrt{x}$) in the first case, and apparent bias $b$ in the latter. For example, if $f(n)=μ(n)$, the Möbius function, then $F(x) = \sum_{n\leq x} μ(n)$ has $b=0$ so exhibits no persistent or apparent bias, while if $f(n)=λ(n)$, the Liouville function, then $F(x) = \sum_{n\leq x} λ(n)$ has apparent bias $b=1/ζ(1/2)$. We study the bias when $f(p^k)$ is independent of the prime $p$, and call such functions fake $μ's$. We investigate the conditions required for such a function to exhibit a persistent or apparent bias, determine the functions in this family with maximal and minimal bias of each type, and characterize the functions with no bias of either type. For such a function $F(x)$ with apparent bias $b$, we also show that $F(x)/\sqrt{x}-a\sqrt{x}-b$ changes sign infinitely often.

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arXiv2021

The Integer group determinants for the Heisenberg group of order $p^3$

We establish a congruence satisfied by the integer group determinants for the non-abelian Heisenberg group of order $p^3$. We characterize all determinant values coprime to $p$, give sharp divisibility conditions for multiples of $p$, and determine all values when $p=3$. We also provide new sharp conditions on the power of $p$ dividing the group determinants for $\mathbb Z_p^2$. For a finite group, the integer group determinants can be understood as corresponding to Lind's generalization of the Mahler measure. We speculate on the Lind-Mahler measure for the discrete Heisenberg group and for two other infinite non-abelian groups arising from symmetries of the plane and 3-space.

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arXiv2021

Explicit lower bounds on $|L(1, χ)|$

Let $χ$ denote a primitive, non-quadratic Dirichlet character with conductor $q$, and let $L(s, χ)$ denote its associated Dirichlet $L$-function. We show that $|L(1, χ)| \geq 1/(9.12255 \log(q/π))$ for sufficiently large $q$, and that $|L(1, χ)| \geq 1/(9.69030 \log(q/π))$ for all $q\geq2$, improving some results of Louboutin. The improvements stem principally from the construction, via simulated annealing, of some real trigonometric polynomials having particularly favorable properties.

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arXiv2020

Oscillations in weighted arithmetic sums

We examine oscillations in a number of sums of arithmetic functions involving $Ω(n)$, the total number of prime factors of $n$, and $ω(n)$, the number of distinct prime factors of $n$. In particular, we examine oscillations in $S_α(x) = \sum_{n\leq x} (-1)^{n - Ω(n)}/n^α$ and in $H_α(x) = \sum_{n\leq x} (-1)^{ω(n)}/n^α$ for $α\in[0,1]$, and in $W(x)=\sum_{n\leq x} (-2)^{Ω(n)}$. We show for example that each of the inequalities $S_0(x)<0$, $S_0(x)>3.3\sqrt{x}$, $S_1(x)>0$, and $S_1(x)\sqrt{x}<-3.3$ is true infinitely often, disproving some hypotheses of Sun.

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arXiv2020

The size of oscillations in the Goldbach conjecture

Let $R(n) = \sum_{a+b=n} Λ(a)Λ(b)$, where $Λ(\cdot)$ is the von Mangoldt function. The function $R(n)$ is often studied in connection with Goldbach's conjecture. On the Riemann hypothesis (RH) it is known that $\sum_{n\leq x} R(n) = x^2/2 - 4x^{3/2} G(x) + O(x^{1+ε})$, where $G(x)=\Re \sum_{γ>0} \frac{x^{iγ}}{(\frac{1}{2} + iγ)(\frac{3}{2} + iγ)}$ and the sum is over the ordinates of the nontrivial zeros of the Riemann zeta function in the upper half-plane. We prove (on RH) that each of the inequalities $G(x) < -0.02093$ and $G(x)> 0.02092$ hold infinitely often, and establish improved bounds under an assumption of linearly independence for zeros of the zeta function. We also show that the bounds we obtain are very close to optimal.

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arXiv2020

A tale of two omegas

We consider $ω(n)$ and $Ω(n)$, which respectively count the number of distinct and total prime factors of $n$. We survey a number of similarities and differences between these two functions, and study the summatory functions $L(x)=\sum_{n\leq x} (-1)^{Ω(n)}$ and $H(x)=\sum_{n\leq x} (-1)^{ω(n)}$ in particular. Questions about oscillations in both of these functions are connected to the Riemann hypothesis and other questions concerning the Riemann zeta function. We show that even though $ω(n)$ and $Ω(n)$ have the same parity approximately 73.5\% of the time, these summatory functions exhibit quite different behaviors: $L(x)$ is biased toward negative values, while $H(x)$ is unbiased. We also prove that $H(x)>1.7\sqrt{x}$ for infinitely many integers $x$, and $H(x)<-1.7\sqrt{x}$ infinitely often as well. These statements complement results on oscillations for $L(x)$.

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arXiv2020

The distribution of $k$-free numbers

Let $R_k(x)$ denote the error incurred by approximating the number of $k$-free integers less than $x$ by $x/ζ(k)$. It is well known that $R_k(x)=Ω(x^{\frac{1}{2k}})$, and widely conjectured that $R_k(x)=O(x^{\frac{1}{2k}+ε})$. By establishing weak linear independence of some subsets of zeros of the Riemann zeta function, we establish an effective proof of the lower bound, with significantly larger bounds on the constant compared to those obtained in prior work. For example, we show that $R_k(x)/x^{1/2k} > 3$ infinitely often and that $R_k(x)/x^{1/2k} < -3$ infinitely often, for $k=2$, $3$, $4$, and $5$. We also investigate $R_2(x)$ and $R_3(x)$ in detail and establish that our bounds far exceed the oscillations exhibited by these functions over a long range: for $0<x\leq10^{18}$ we show that $|R_2(x)| < 1.12543x^{1/4}$ and $|R_3(x)| < 1.27417x^{1/6}$. We also present some empirical results regarding gaps between square-free numbers and between cube-free numbers.

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arXiv2018

The Lind-Lehmer Constant for $\mathbb Z_2^r \times \mathbb Z_{4}^s$

We show that the minimal positive logarithmic Lind-Mahler measure for a group of the form $G=\mathbb Z_2^r\times\mathbb Z_4^s$ with $|G|\geq 4$ is $\frac{1}{|G|} \log (|G|-1).$ We also show that for $G=\mathbb Z_2 \times \mathbb Z_{2^n}$ with $n\geq 3$ this value is $\frac{1}{|G|} \log 9.$ Previously the minimal measure was only known for $2$-groups of the form $\mathbb Z_2^k$ or $\mathbb Z_{2^k}.$

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arXiv2018

A generalization of the Goresky-Klapper conjecture, Part II

Suppose that $f(x)=Ax^k$ mod $p$ is a permutation of the least residues mod $p$. With the exception of the maps $f(x)=Ax$ and $Ax^{(p+1)/2}$ mod $p$ we show that for fixed $n\geq 2$ the image of each residue class mod $n$ contains elements from every residue classe mod $n$, once $p$ is sufficiently large. If $f(x)=Ax$ mod $p$, then for each $p$ and $n$ there will be exactly $(1+o(1))\frac{6}{π^2}n^2$ readily describable values of $A$ for which the image of some residue class mod $n$ misses at least one residue class mod $n,$ even when $p$ is large relative to $n$. A similar situation holds for $f(x)=Ax^{(p+1)/2}$ mod $p$.

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arXiv2018

A generalization of the Goresky-Klapper conjecture, Part I

For a fixed integer $n\geq 2,$ we show that a permutation of the least residues mod $p$ of the form $f(x)=Ax^k$ mod $p$ cannot map a residue class mod $n$ to just one residue class mod $n$ once $p$ is sufficiently large, other than the maps $f(x)=\pm x$ mod $p$ when $n$ is even and $f(x)=\pm x$ or $\pm x^{(p+1)/2}$ mod $p$ when $n$ is odd.

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