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We show that, on the Riemann hypothesis, $\limsup_{X\to\infty}I(X)/X^{2} \leq 0.8603$, where $I(X) = \int_X^{2X} (\psi(x)-x)^2\,dx.$ 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 $X\rightarrow\infty$., Comment: 23 pages

We examine oscillations in a number of sums of arithmetic functions involving [Formula: see text], the total number of prime factors of [Formula: see text], and [Formula: see text], the number of distinct prime factors of [Formula: see text]. In particular, we examine oscillations in [Formula: see text] and in [Formula: see text] for [Formula: see text], and in [Formula: see text]. We show for example that each of the inequalities [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text] is true infinitely often, disproving some hypotheses of Sun.

We show that the natural density of positive integers n for which σ ( 2 n + 1 ) ≥ σ ( 2 n ) is between 0.053 and 0.055.

We provide an explicit estimate on the least primitive root mod p 2 . We show, in particular, that every prime p has a primitive root mod p 2 that is less than p 0.99 .

We give a method for producing explicit bounds on g ( p ) g(p) , the least primitive root modulo p p . Using our method we show that \[ g ( p ) > 2 r 2 r ω ( p − 1 ) p 1 4 + 1 4 r g(p)>2r\,2^{r\omega (p-1)}\,p^{\frac {1}{4}+\frac {1}{4r}} \] for p > 10 56 p>10^{56} where r ≥ 2 r\geq 2 is an integer parameter. This result beats existing bounds that rely on explicit versions of the Burgess inequality. Our main result allows one to derive bounds of differing shapes for various ranges of p p . For example, our method also allows us to show that g ( p ) > p 5 / 8 g(p)>p^{5/8} for all p ≥ 10 22 p\geq 10^{22} and g ( p ) > p 1 / 2 g(p)>p^{1/2} for p ≥ 10 56 p\geq 10^{56} .

Chebyshev presented a conjecture after observing the apparent bias towards primes congruent to [Formula: see text]. His conjecture is equivalent to a version of the Generalized Riemann Hypothesis. Fujii strengthened this conjecture; we strengthen it still further using detailed computations of zeroes of Dirichlet [Formula: see text]-functions.

With $\Fq$ the finite field of $q$ elements, we investigate the following question. If $\gamma$ generates $\Fqn$ over $\Fq$ and $\beta$ is a non-zero element of $\Fqn$, is there always an $a \in \Fq$ such that $\beta(\gamma + a)$ is a primitive element? We resolve this case when $n=3$, thereby proving a conjecture by Cohen. We also improve substantially on what is known when $n=4$., Comment: To appear in Math. Comp

We give explicit bounds on sums of $$d(n)^{2}$$ and $${d}_{4}(n)$$ , where d(n) is the number of divisors of n and $${d}_{4}(n)$$ is the number of ways of writing n as a product of four positive integers. In doing so, we make a slight improvement on the upper bound for class numbers of quartic number fields.

Let V(T) denote the number of sign changes in $$\psi (x) - x$$ for $$x\in [1, T]$$ . We show that $$\liminf _{T\rightarrow \infty } V(T)/\log T\ge \gamma _{1}/\pi + 1.867\cdot 10^{-30}$$ , where $$\gamma _{1} = 14.13\ldots $$ denotes the ordinate of the lowest-lying non-trivial zero of the Riemann zeta-function. This improves on a long-standing result by Kaczorowski.

Let $g^{\square}(p)$ denote the least square-free primitive root modulo $p$. We show that $g^{\square}(p)< p^{0.96}$ for all $p$., 7 pages

We consider Diophantine quintuples $\{a, b, c, d, e\}$. These are sets of distinct positive integers, the product of any two elements of which is one less than a perfect square. It is conjectured that there are no Diophantine quintuples; we improve on current estimates to show that there are at most $1.9\cdot 10^{29}$ Diophantine quintuples., Comment: 16 pages

We prove that for all $q>211$, there always exists a primitive root $g$ in the finite field $\mathbb{F}_{q}$ such that $Q(g)$ is also a primitive root, where $Q(x)= ax^2 + bx + c$ is a quadratic polynomial with $a, b, c\in \mathbb{F}_{q}$ such that $b^{2} - 4ac \neq 0$., 12 pages; code available at https://arxiv.org/src/1803.01435v2/anc/

We consider Dirichlet $L$-functions $L(s, \chi)$ where $\chi$ is a real, non-principal character modulo $q$. Using Pintz's refinement of Page's theorem, we prove that for $q\geq 3$ the function $L(s, \chi)$ has at most one real zero $\beta$ with $1- 1.011/\log q < \beta < 1$., Comment: 8 pages

This article proves the bound | ζ ( 1 2 + i t ) | ≤ 0.732 t 1 6 log ⁡ t for t ≥ 2 , which improves on a result by Cheng and Graham. We also show that | ζ ( 1 2 + i t ) | ≤ 0.732 | 4.678 + i t | 1 6 log ⁡ | 4.678 + i t | for all t.

Grosswald's conjecture is that $g(p)$, the least primitive root modulo $p$, satisfies $g(p) \leq \sqrt{p} - 2$ for all $p>409$. We make progress towards this conjecture by proving that $g(p) \leq \sqrt{p} -2$ for all $409 3.67\times 10^{71}$.

We demonstrate that there are infinitely many integers that cannot be expressed as the sum of two squares of integers and up to two nonnegative integer powers of 2.

A Lehmer number modulo a prime $p$ is an integer $a$ with $1 \leq a \leq p-1$ whose inverse $\bar{a}$ within the same range has opposite parity. Lehmer numbers that are also primitive roots have been discussed by Wang and Wang in an endeavour to count the number of ways $1$ can be expressed as the sum of two primitive roots that are also Lehmer numbers (an extension of a question of S. Golomb). In this paper we give an explicit estimate for the number of Lehmer primitive roots modulo $p$ and prove that, for all primes $p \neq 2,3,7$, Lehmer primitive roots exist. We also make explicit the known expression for the number of Lehmer numbers modulo $p$ and improve the Wang--Wang estimate for the number of solutions to the Golomb--Lehmer primitive root problem., Comment: 11 pages

We examine linear sums of primitive roots and their inverses in finite fields. In particular, we refine a result by Li and Han, and show that every $p> 13$ has a pair of primitive roots $a$ and $b$ such that $a+ b$ and $a^{-1} + b^{-1}$ are also primitive roots mod $p$., 19 pages; to appear in Finite Fields Appl

In this paper we examine Grosswald's conjecture on $g(p)$, the least primitive root modulo $p$. Assuming the Generalized Riemann Hypothesis (GRH), and building on previous work by Cohen, Oliveira e Silva and Trudgian, we resolve Grosswald's conjecture by showing that $g(p)< \sqrt{p} - 2$ for all $p>409$. Our method also shows that under GRH we have $\hat{g}(p)< \sqrt{p}-2$ for all $p>2791$, where $\hat{g}(p)$ is the least prime primitive root modulo $p$., 10 pages

We consider Diophantine quintuples \(\{a, b, c, d, e\}\), sets of integers with \(a

Turing's method uses explicit bounds on $|\int_{t_{1}}^{t_{2}} S(t)\, dt|$, where $\pi S(t)$ is the argument of the Riemann zeta-function. This article improves the bound on $|\int_{t_{1}}^{t_{2}} S(t)\, dt|$ given in an earlier paper by the author., Comment: 7 pages; to appear in Rocky Mountain J. Math; references updated

This article considers the positive integers $N$ for which $��_{N}(s) = \sum_{n=1}^{N} n^{-s}$ 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., 5 Pages - Final Version will appear in LMS JCM

We consider Diophantine quintuples $\{a, b, c, d, e\}$. These are sets of distinct positive integers, the product of any two elements of which is one less than a perfect square. It is conjectured that there are no Diophantine quintuples; we improve on current estimates to show that there are at most $1.18\cdot 10^{27}$ Diophantine quintuples., Comment: 15 pages

Research of the authors is partially supported by NSERC. The second author research is also partially supported by ARC.