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Fujii obtained a formula for the average number of Goldbach representations with lower-order terms expressed as a sum over the zeros of the Riemann zeta function and a smaller error term. This assumed the Riemann Hypothesis. We obtain an unconditional version of this result and obtain applications conditional on various conjectures on zeros of the Riemann zeta function.
For a fixed integer h, the standard orthogonality relations for Ramanujan sums
$c_r(n)$
give an asymptotic formula for the shifted convolution
$\sum _{n\le N} c_q(n)c_r(n+h)$
. We prove a generalised formula for affine convolutions
$\sum _{n\le N} c_q(n)c_r(kn+h)$
. This allows us to study affine convolutions
$\sum _{n\le N} f(n)g(kn+h)$
of arithmetical functions
$f,g$
admitting a suitable Ramanujan–Fourier expansion. As an application, we give a heuristic justification of the Hardy–Littlewood conjectural asymptotic formula for counting Sophie Germain primes.
We establish the optimal order of Malliavin-type remainders in the asymptotic density approximation formula for Beurling generalized integers. Given
$\alpha \in (0,1]$
and
$c>0$
(with
$c\leq 1$
if
$\alpha =1$
), a generalized number system is constructed with Riemann prime counting function
$ \Pi (x)= \operatorname {\mathrm {Li}}(x)+ O(x\exp (-c \log ^{\alpha } x ) +\log _{2}x), $
and whose integer counting function satisfies the extremal oscillation estimate
$N(x)=\rho x + \Omega _{\pm }(x\exp (- c'(\log x\log _{2} x)^{\frac {\alpha }{\alpha +1}})$
for any
$c'>(c(\alpha +1))^{\frac {1}{\alpha +1}}$
, where
$\rho>0$
is its asymptotic density. In particular, this improves and extends upon the earlier work [Adv. Math. 370 (2020), Article 107240].
We prove that the Riemann hypothesis is equivalent to the condition
$\int _{2}^x\left (\pi (t)-\operatorname {\textrm {li}}(t)\right )\textrm {d}t<0$
for all
$x>2$
. Here,
$\pi (t)$
is the prime-counting function and
$\operatorname {\textrm {li}}(t)$
is the logarithmic integral. This makes explicit a claim of Pintz. Moreover, we prove an analogous result for the Chebyshev function
$\theta (t)$
and discuss the extent to which one can make related claims unconditionally.
Let
$g \geq 1$
be an integer and let
$A/\mathbb Q$
be an abelian variety that is isogenous over
$\mathbb Q$
to a product of g elliptic curves defined over
$\mathbb Q$
, pairwise non-isogenous over
$\overline {\mathbb Q}$
and each without complex multiplication. For an integer t and a positive real number x, denote by
$\pi _A(x, t)$
the number of primes
$p \leq x$
, of good reduction for A, for which the Frobenius trace
$a_{1, p}(A)$
associated to the reduction of A modulo p equals t. Assuming the Generalized Riemann Hypothesis for Dedekind zeta functions, we prove that
$\pi _A(x, 0) \ll _A x^{1 - \frac {1}{3 g+1 }}/(\operatorname {log} x)^{1 - \frac {2}{3 g+1}}$
and
$\pi _A(x, t) \ll _A x^{1 - \frac {1}{3 g + 2}}/(\operatorname {log} x)^{1 - \frac {2}{3 g + 2}}$
if
$t \neq 0$
. These bounds largely improve upon recent ones obtained for
$g = 2$
by Chen, Jones, and Serban, and may be viewed as generalizations to arbitrary g of the bounds obtained for
$g=1$
by Murty, Murty, and Saradha, combined with a refinement in the power of
$\operatorname {log} x$
by Zywina. Under the assumptions stated above, we also prove the existence of a density one set of primes p satisfying
$|a_{1, p}(A)|>p^{\frac {1}{3 g + 1} - \varepsilon }$
for any fixed
$\varepsilon>0$
.
We prove that most permutations of degree $n$ have some power which is a cycle of prime length approximately $\log n$. Explicitly, we show that for $n$ sufficiently large, the proportion of such elements is at least $1-5/\log \log n$ with the prime between $\log n$ and $(\log n)^{\log \log n}$. The proportion of even permutations with this property is at least $1-7/\log \log n$.
We prove that neither a prime nor an l-almost prime number theorem holds in the class of regular Toeplitz subshifts. But when a quantitative strengthening of the regularity with respect to the periodic structure involving Euler’s totient function is assumed, then the two theorems hold.
Let $K=\mathbb{Q}(\unicode[STIX]{x1D714})$ with $\unicode[STIX]{x1D714}$ the root of a degree $n$ monic irreducible polynomial $f\in \mathbb{Z}[X]$. We show that the degree $n$ polynomial $N(\sum _{i=1}^{n-k}x_{i}\unicode[STIX]{x1D714}^{i-1})$ in $n-k$ variables takes the expected asymptotic number of prime values if $n\geqslant 4k$. In the special case $K=\mathbb{Q}(\sqrt[n]{\unicode[STIX]{x1D703}})$, we show that $N(\sum _{i=1}^{n-k}x_{i}\sqrt[n]{\unicode[STIX]{x1D703}^{i-1}})$ takes infinitely many prime values, provided $n\geqslant 22k/7$.
Our proof relies on using suitable ‘Type I’ and ‘Type II’ estimates in Harman’s sieve, which are established in a similar overall manner to the previous work of Friedlander and Iwaniec on prime values of $X^{2}+Y^{4}$ and of Heath-Brown on $X^{3}+2Y^{3}$. Our proof ultimately relies on employing explicit elementary estimates from the geometry of numbers and algebraic geometry to control the number of highly skewed lattices appearing in our final estimates.
We show that for all large enough x the interval [x, x + x1/2 log1.39x] contains numbers with a prime factor p > x18/19. Our work builds on the previous works of Heath–Brown and Jia (1998) and Jia and Liu (2000) concerning the same problem for the longer intervals [x, x + x1/2 + ϵ]. We also incorporate some ideas from Harman’s book Prime-detecting sieves (2007). The main new ingredient that we use is the iterative argument of Matomäki and Radziwiłł (2016) for bounding Dirichlet polynomial mean values, which is applied to obtain Type II information. This allows us to take shorter intervals than in the above-mentioned previous works. We have also had to develop ideas to avoid losing any powers of log x when applying Harman’s sieve method.
In this note we examine Littlewood’s proof of the prime number theorem. We show that this can be extended to provide an equivalence between the prime number theorem and the nonvanishing of Riemann’s zeta-function on the one-line. Our approach goes through the theory of almost periodic functions and is self-contained.
We prove that for every $m\geq 0$ there exists an $\unicode[STIX]{x1D700}=\unicode[STIX]{x1D700}(m)>0$ such that if $0<\unicode[STIX]{x1D706}<\unicode[STIX]{x1D700}$ and $x$ is sufficiently large in terms of $m$ and $\unicode[STIX]{x1D706}$, then
The value of $\unicode[STIX]{x1D700}(m)$ and the dependence of the implicit constant on $\unicode[STIX]{x1D706}$ and $m$ may be made explicit. This is an improvement of the author’s previous result. Moreover, we will show that a careful investigation of the proof, apart from some slight changes, can lead to analogous estimates when allowing the parameters $m$ and $\unicode[STIX]{x1D706}$ to vary as functions of $x$ or replacing the set $\mathbb{P}$ of all primes by primes belonging to certain specific subsets.
We prove that for every sufficiently large integer $n$, the polynomial $1+x+x^{2}/11+x^{3}/111+\cdots +x^{n}/111\ldots 1$ is irreducible over the rationals, where the coefficient of $x^{k}$ for $1\leqslant k\leqslant n$ is the reciprocal of the decimal number consisting of $k$ digits which are each $1$. Similar results following from the same techniques are discussed.
Denote by $\mathbb{P}$ the set of all prime numbers and by $P(n)$ the largest prime factor of positive integer $n\geq 1$ with the convention $P(1)=1$. In this paper, we prove that, for each $\unicode[STIX]{x1D702}\in (\frac{32}{17},2.1426\cdots \,)$, there is a constant $c(\unicode[STIX]{x1D702})>1$ such that, for every fixed nonzero integer $a\in \mathbb{Z}^{\ast }$, the set
$$\begin{eqnarray}\{p\in \mathbb{P}:p=P(q-a)\text{ for some prime }q\text{ with }p^{\unicode[STIX]{x1D702}}<q\leq c(\unicode[STIX]{x1D702})p^{\unicode[STIX]{x1D702}}\}\end{eqnarray}$$
has relative asymptotic density one in $\mathbb{P}$. This improves a similar result due to Banks and Shparlinski [‘On values taken by the largest prime factor of shifted primes’, J. Aust. Math. Soc.82 (2015), 133–147], Theorem 1.1, which requires $\unicode[STIX]{x1D702}\in (\frac{32}{17},2.0606\cdots \,)$ in place of $\unicode[STIX]{x1D702}\in (\frac{32}{17},2.1426\cdots \,)$.
Let $h(n)$ denote the largest product of distinct primes whose sum does not exceed $n$. The main result of this paper is that the property for all $n\geq 1$, we have $\log h(n)<\sqrt{\text{li}^{-1}(n)}$ (where $\text{li}^{-1}$ denotes the inverse function of the logarithmic integral) is equivalent to the Riemann hypothesis.
It has been known since Vinogradov that, for irrational $\unicode[STIX]{x1D6FC}$, the sequence of fractional parts $\{\unicode[STIX]{x1D6FC}p\}$ is equidistributed in $\mathbb{R}/\mathbb{Z}$ as $p$ ranges over primes. There is a natural second-order equidistribution property, a pair correlation of such fractional parts, which has recently received renewed interest, in particular regarding its relation to additive combinatorics. In this paper we show that the primes do not enjoy this stronger equidistribution property.
Let $b$ be an integer larger than 1. We give an asymptotic formula for the exponential sum
$$\begin{eqnarray}\mathop{\sum }_{\substack{ p\leqslant x \\ g(p)=k}}\exp \big(2\text{i}\unicode[STIX]{x1D70B}\unicode[STIX]{x1D6FD}p\big),\end{eqnarray}$$
where the summation runs over prime numbers $p$ and where $\unicode[STIX]{x1D6FD}\in \mathbb{R}$, $k\in \mathbb{Z}$, and $g:\mathbb{N}\rightarrow \mathbb{Z}$ is a strongly $b$-additive function such that $\operatorname{pgcd}(g(1),\ldots ,g(b-1))=1$.
We prove a generalization of the author’s work to show that any subset of the primes which is ‘well distributed’ in arithmetic progressions contains many primes which are close together. Moreover, our bounds hold with some uniformity in the parameters. As applications, we show there are infinitely many intervals of length $(\log x)^{{\it\epsilon}}$ containing $\gg _{{\it\epsilon}}\log \log x$ primes, and show lower bounds of the correct order of magnitude for the number of strings of $m$ congruent primes with $p_{n+m}-p_{n}\leqslant {\it\epsilon}\log x$.
Let $P(n)$ denote the largest prime factor of an integer $n\geq 2$. In this paper, we study the distribution of the sequence $\{f(P(n)):n\geq 1\}$ over the set of congruence classes modulo an integer $b\geq 2$, where $f$ is a strongly $q$-additive integer-valued function (that is, $f(aq^{j}+b)=f(a)+f(b),$ with $(a,b,j)\in \mathbb{N}^{3}$, $0\leq b<q^{j}$). We also show that the sequence $\{{\it\alpha}P(n):n\geq 1,f(P(n))\equiv a\;(\text{mod}~b)\}$ is uniformly distributed modulo 1 if and only if ${\it\alpha}\in \mathbb{R}\!\setminus \!\mathbb{Q}$.
An integer $d$ is called a jumping champion for a given $x$ if $d$ is the most common gap between consecutive primes up to $x$. Occasionally, several gaps are equally common. Hence, there can be more than one jumping champion for the same $x$. In 1999, Odlyzko et al provided convincing heuristics and empirical evidence for the truth of the hypothesis that the jumping champions greater than 1 are 4 and the primorials $2,6,30,210,2310,\ldots \,$. In this paper, we prove that an appropriate form of the Hardy–Littlewood prime $k$-tuple conjecture for prime pairs and prime triples implies that all sufficiently large jumping champions are primorials and that all sufficiently large primorials are jumping champions over a long range of $x$.