arXiv subjects
Publications and source records attributed to Timothy S. Trudgian.
We quantify the set of known exponent pairs $(k, \ell)$ and develop a framework to compute the optimal exponent pair for an arbitrary objective function. Applying this methodology, we make progress on several open problems, including bounds of the Riemann zeta-function $\zeta(s)$ in the critical strip, estimates of the moments of $\zeta(1/2 + it)$ and the generalised Dirichlet divisor problem.
Assuming the Riemann Hypothesis, it is known that there are infinitely many consecutive pairs of zeros of the Riemann zeta-function within 0.515396 times the average spacing. This is obtained using the method of Montgomery and Odlyzko. We prove that this method can never find infinitely many pairs of consecutive zeros within 0.5042 times the average spacing.
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 $|t| \geq \exp(1000)$.
Let $ρ(n)$ denote the maximal number of different primes that may occur in the order of a finite solvable group $G$, all elements of which have orders divisible by at most $n$ distinct primes. We show that $ρ(n)\leq 5n$ for all $n\geq 1$. As an application, we improve on a recent bound by Hung and Yang for arbitrary finite groups.
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.
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.
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.
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.
Let $L(s,\chi)$ be the Dirichlet $L$-function associated to a non-principal primitive Dirichlet character $\chi$ defined modulo $q$, where $q\ge 3$. We prove, under the assumption of the Generalised Riemann Hypothesis, the validity of estimates given by Lamzouri, Li, and Soundararajan on $\vert L(1,\chi) \vert$. As a corollary, we have that similar estimates hold for the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{-q})$, $q\ge 5$.
We consider sums of the form $\sum ϕ(γ)$, where $ϕ$ is a given function, and $γ$ ranges over the ordinates of nontrivial zeros of the Riemann zeta-function in a given interval. We show how the numerical estimation of such sums can be accelerated by a simple device, and give examples involving both convergent and divergent infinite sums.
We consider the sum $\sum 1/γ$, where $γ$ ranges over the ordinates of nontrivial zeros of the Riemann zeta-function in an interval $(0,T]$, and consider the behaviour of the sum as $T \to\infty$. We show that, after subtracting a smooth approximation $\frac{1}{4π} \log^2(T/2π),$ the sum tends to a limit $H \approx -0.0171594$ which can be expressed as an integral. We calculate $H$ to high accuracy, using a method which has error $O((\log T)/T^2)$. Our results improve on earlier results by Hassani and other authors.
We show that, on the Riemann hypothesis, $\limsup_{X\to\infty}I(X)/X^{2} \leq 0.8603$, where $I(X) = \int_X^{2X} (ψ(x)-x)^2\,dx.$ This proves (and improves on) a claim by Pintz from 1982. We also show unconditionally that $\frac{1}{5\,374}\leq I(X)/X^2 $ for sufficiently large $X$, and that the $I(X)/X^{2}$ has no limit as $X\rightarrow\infty$.
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.
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 $L(x)$.
This article considers the positive integers $N$ for which $ζ_{N}(s) = \sum_{n=1}^{N} n^{-s}$ has zeroes in the half-plane $\Re(s)>1$. Building on earlier results, we show that there are no zeroes for $1\leq N\leq 18$ and for $N=20, 21, 28$. For all other $N$ there are infinitely many zeroes.
We prove that the Riemann zeta-function $ζ(σ+ it)$ has no zeros in the region $σ\geq 1 - 1/(5.573412 \log|t|)$ for $|t|\geq 2$. This represents the largest known zero-free region within the critical strip for $3.06\cdot10^{10} < |t|<\exp(10151.5)$. Our improvements result from determining some favorable trigonometric polynomials having particular properties, and from analyzing the error term in the method of Kadiri. We also improve an upper bound in a question of Landau regarding nonnegative trigonometric polynomials.
Let $L(s, π\timesπ^\prime)$ be the Rankin--Selberg $L$-function attached to automorphic representations $π$ and $π^\prime$. Let $\tildeπ$ and $\tildeπ^\prime$ denote the contragredient representations associated to $π$ and $π^\prime$. Under the assumption of certain upper bounds for coefficients of the logarithmic derivatives of $L(s, π\times\tildeπ)$ and $L(s, π^\prime\times\tildeπ^\prime)$, we prove a log-free zero-density estimate for $L(s, π\timesπ^\prime)$ which generalises a result due to Fogels in the context of Dirichlet $L$-functions. We then employ this log-free estimate in studying the distribution of the Fourier coefficients of an automorphic representation $π$. As an application we examine the non-lacunarity of the Fourier coefficients $b_f(p)$ of a modular newform $f(z)=\sum_{n=1}^{\infty} b_f(n) e^{2πi n z}$ of weight $k$, level $N$, and character $χ$. More precisely for $f(z)$ and a prime $p$, set $j_f(p):=\max_{x;~x> p} J_{f} (p, x)$, where $J_{f} (p, x):=\#\{{\rm prime}~q;~a_π(q)=0~{\rm for~all~}p<q\leq x\}.$ We prove that $j_f(p)\ll_{f, θ} p^θ$ for some $0<θ<1$.