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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 more...

Let L ( s , χ ) be the Dirichlet L-function associated to a non-principal primitive Dirichlet character χ defined mod q , where q ≥ 3 . We prove, under the assumption of the Generalised Riemann Hypothesis, the validity of estimates given by Lamzouri, Li, and Soundararajan on | L ( 1 , χ ) | . As a corollary, we have that similar estimates hold for the class number of the imaginary quadratic field Q ( − q ) , q ≥ 5 . more...

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., 26 pages; to appear in Ramanujan J more...

We consider sums of the form $\sum \phi(\gamma)$, where $\phi$ is a given function, and $\gamma$ 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. more...

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. more...

We make explicit a theorem of Pintz concerning the error term in the prime number theorem. This gives an improved version of the prime number theorem with error term roughly square-root of that which was previously known. We apply this to a long-standing problem concerning an inequality studied by Ramanujan. more...

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 prove that for any prime power $q\notin\{3,4,5\}$, the cubic extension $\mathbb{F}_{q^3}$ of the finite field $\mathbb{F}_q$ contains a primitive element $\xi$ such that $\xi+\xi^{-1}$ is also primitive, and $\textrm{Tr}_{\mathbb{F}_{q^3}/\mathbb{F}_q}(\xi)=a$ for any prescribed $a\in\mathbb{F}_q$. This completes the proof of a conjecture of Gupta, Sharma, and Cohen concerning the analogous problem over an extension of arbitrary degree $n\ge3$., Comment: 4 pages more...

We consider $\omega(n)$ and $\Omega(n)$, 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 $L(x)=\sum_{n\leq x} (-1)^{\Omega(n)}$ and $H(x)=\sum_{n\leq x} (-1)^{\omega(n)}$ 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 $\omega(n)$ and $\Omega(n)$ 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) more...

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} . more...

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. more...

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 more...

Brun’s constant is \(B=\sum _{p \in P_{2}} p^{-1} + (p+2)^{-1}\), where the summation is over all twin primes. We improve the unconditional bounds on Brun’s constant to \(1.840503< B < 2.288490\), which are about 13% tighter. more...

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. more...

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. more...

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

Let $R_k(x)$ denote the error incurred by approximating the number of $k$-free integers less than $x$ by $x/\zeta(k)$. It is well known that $R_k(x)=\Omega(x^{\frac{1}{2k}})$, and widely conjectured that $R_k(x)=O(x^{\frac{1}{2k}+\epsilon})$. 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 more...

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 more...

We prove that the Riemann zeta-function ζ ( σ + i t ) has no zeros in the region σ ≥ 1 − 1 / ( 5.573412 log ⁡ | t | ) for | t | ≥ 2 . This represents the largest known zero-free region within the critical strip for 3.06 ⋅ 10 10 | t | exp ⁡ ( 10 151.5 ) . 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. more...

Let θ(x) = ∑p≤xlogp. We show that θ(x) < x for 2 < x < 1.39·1017. We also show that there is an x < exp(727.951332668) for which θ(x) > x.

We examine the problem of writing every sufficiently large even number as the sum of two primes and at most K powers of 2. We outline an approach that only just falls short of improving the current bounds on K. Finally, we improve the estimates in other Waring–Goldbach problems. more...

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 more...

We prove that for all $q>61$, every non-zero element in the finite field $\mathbb{F}_{q}$ can be written as a linear combination of two primitive roots of $\mathbb{F}_{q}$. This resolves a conjecture posed by Cohen and Mullen., 8 pages; to appear in Mathematics of Computation more...

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. more...

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 $409 3.67\times 10^{71}$. more...

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.

This article considers linear relations between the non-trivial zeroes of the Riemann zeta-function. The main application is an alternative disproof to Mertens’ conjecture by showing that limsupx!1 M(x)x 1=2 1:6383, and liminfx!1 M(x)x 1=2 1:6383 more...

This paper contains new explicit upper bounds for the number of zeroes of Dirichlet L-functions and Dedekind zeta-functions in rectangles.

Text This paper concerns the function S ( T ) , where π S ( T ) is the argument of the Riemann zeta-function along the critical line. The main result is that | S ( T ) | ⩽ 0.112 log T + 0.278 log log T + 2.510 , which holds for all T ⩾ e . Video For a video summary of this paper, please click here or visit http://youtu.be/FldP0idE0aI . more...

It is known that $\zeta (1+ it)\ll \mathop{(\log t)}\nolimits ^{2/ 3} $ when $t\gg 1$. This paper provides a new explicit estimate $\vert \zeta (1+ it)\vert \leq \frac{3}{4} \log t$, for $t\geq 3$. This gives the best upper bound on $\vert \zeta (1+ it)\vert $ for $t\leq 1{0}^{2\cdot 1{0}^{5} } $. more...

For nearly a century, mathematicians have explored connections between the Liouville function and the Riemann hypothesis. We describe a number of connections regarding oscillations in sums involving the Liouville function, including L0(x) = ∑ n ≤ x λ(n), first studied by Polya, and L1(x) = ∑ n ≤ x λ(n)∕n, explored by Turan. We establish new lower bounds on the size of the oscillations in such sums. In particular, we prove that \(L_{0}(x)> \sqrt{x}\) infinitely often, show that \(L_{1}(x) more...

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 more...

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 more...

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 more...

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

We investigate the behaviour of the function $L_{\alpha }(x) = \sum _{n\leq x}\lambda (n)/n^{\alpha }$, where $\lambda (n)$ is the Liouville function and $\alpha $ is a real parameter. The case where $\alpha =0$ was investigated by Pólya; the case $\alpha =1$, by Turán. The question of the existence of sign changes in both of these cases is related to the Riemann hypothesis. Using both analytic and computational methods, we investigate similar problems for the more general family $L_{\alpha }(x)$, where $0\leq \alpha \leq 1$, and their relationship to the Riemann hypothesis and other properties of the zeros of the Riemann zeta function. The case where $\alpha =1/2$is of particular interest. more...

This paper concerns the function S(T), the argument of the Rie- mann zeta-function along the critical line. The main result is that |S(T)| � 0.111logT + 0.275 log logT + 2.450, which holds for all Te. more...

This article improves the estimate of the size of the definite integral of S ( t ) S(t) , the argument of the Riemann zeta-function. The primary application of this improvement is Turing’s Method for the Riemann zeta-function. Analogous improvements are given for the arguments of Dirichlet L L -functions and of Dedekind zeta-functions. more...

The conjecture in question concerns the function ϕ n related to the distribution of the zeroes of the Riemann zeta-function, γ n , over the Gram points g n . It is the purpose of this article to show that for any α > 0 the sum ∑ n = 1 K ϕ n K α → 0 , and this was conjectured, on numerical evidence, by Shanks (1961) [7] to be true for α = 1 2 . more...

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 more...

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 more...