Von Mangoldt Research Articles

Sum of the GL(3) Fourier coefficients over mixed powers

Let [Formula: see text] be the [Formula: see text]th Fourier coefficient of [Formula: see text] Hecke–Maass cusp form, denoted as [Formula: see text] or the triple divisor function, denoted as [Formula: see text]. Let [Formula: see text] be an integer. In this paper, we establish an asymptotic formula for the sum [Formula: see text] where [Formula: see text] is either von Mangoldt function or identity function, and [Formula: see text] is a binary quadratic polynomial. When [Formula: see text], then [Formula: see text] can be any bounded arithmetical function.

A smooth version of Landau’s explicit formula

In this paper, we present a smooth version of Landau’s explicit formula for the von Mangoldt arithmetical function. By assuming the validity of the Riemann hypothesis, we show that in order to determine whether a natural number [Formula: see text] is a prime number, it is sufficient to know the location of a number of nontrivial zeros of the Riemann zeta function of order [Formula: see text]. Next we use Heisenberg’s inequality to support the conjecture that this number of zeros cannot be essentially diminished.

On Ramanujan expansions and primes in arithmetic progressions

A celebrated theorem of Delange gives a sufficient condition for an arithmetic function to be the sum of the associated Ramanujan expansion with the coefficients provided by a previous result of Wintner. By applying the Delange theorem to the correlation of the von Mangoldt function with its incomplete form, we deduce an inequality involving the counting function of the prime numbers in arithmetic progressions. A remarkable aspect is that such an inequality is equivalent to the famous conjectural formula by Hardy and Littlewood for the twin primes.

A solution to the Erdős–Sárközy–Sós problem on asymptotic Sidon bases of order 3

A set $S\subset {\mathbb {N}}$ is a Sidon set if all pairwise sums $s_1+s_2$ (for $s_1, s_2\in S$ , $s_1\leqslant s_2$ ) are distinct. A set $S\subset {\mathbb {N}}$ is an asymptotic basis of order 3 if every sufficiently large integer $n$ can be written as the sum of three elements of $S$ . In 1993, Erdős, Sárközy and Sós asked whether there exists a set $S$ with both properties. We answer this question in the affirmative. Our proof relies on a deep result of Sawin on the $\mathbb {F}_q[t]$ -analogue of Montgomery's conjecture for convolutions of the von Mangoldt function.

On Universality of Some Beurling Zeta-Functions

Let P be the set of generalized prime numbers, and ζP(s), s=σ+it, denote the Beurling zeta-function associated with P. In the paper, we consider the approximation of analytic functions by using shifts ζP(s+iτ), τ∈R. We assume the classical axioms for the number of generalized integers and the mean of the generalized von Mangoldt function, the linear independence of the set {logp:p∈P}, and the existence of a bounded mean square for ζP(s). Under the above hypotheses, we obtain the universality of the function ζP(s). This means that the set of shifts ζP(s+iτ) approximating a given analytic function defined on a certain strip σ^<σ<1 has a positive lower density. This result opens a new chapter in the theory of Beurling zeta functions. Moreover, it supports the Linnik–Ibragimov conjecture on the universality of Dirichlet series. For the proof, a probabilistic approach is applied.

Sparse sets that satisfy the prime number theorem

For arbitrary real t>1 we examine the set {⌊x/nt⌋:n≤x}. Asymptotic formulas for the cardinality of this set and the number of primes in this set are given. The prime counting result uses an alternate Vaughan's decomposition for the von Mangoldt function, with triple exponential sums instead of double exponential sums. These sets are the sparsest known sets that satisfy the prime number theorem, in the sense that the number of primes is asymptotically given by the cardinality of the set divided by the natural logarithm of the cardinality of the set.

Quantitative bounds for Gowers uniformity of the Möbius and von Mangoldt functions

We establish quantitative bounds on the U^{k}[N] Gowers norms of the Möbius function \mu and the von Mangoldt function \Lambda for all k , with error terms of the shape O((\log\log N)^{-c}) . As a consequence, we obtain quantitative bounds for the number of solutions to any linear system of equations of finite complexity in the primes, with the same shape of error terms. We also obtain the first quantitative bounds on the size of sets containing no k -term arithmetic progressions with shifted prime difference.

Diophantine approximation with prime denominator in real quadratic function fields

In the thirties of the last century, I.M. Vinogradov proved that the inequality ||pα||≤p−1/5+ε has infinitely prime solutions p, where ||.|| denotes the distance to a nearest integer. This result has subsequently been improved by many authors. In particular, Vaughan (1978) replaced the exponent 1/5 by 1/4 using his celebrated identity for the von Mangoldt function and a refinement of Fourier analytic arguments. The current record is due to Matomäki (2009) who showed the infinitude of prime solutions of the inequality ||pα||≤p−1/3+ε. This exponent 1/3 is considered the limit of the current technology. Recently, in [3], the authors established an analogue of Matomäki's result on imaginary quadratic extensions of the function field k=Fq(T). In this paper, we consider the case of real quadratic extensions of k of class number 1, for which we prove a function field analogue of Vaughan's above-mentioned result (exponent θ=1/4). Our method uses versions of Vaughan's identity and the Dirichlet approximation theorem for function fields. The latter was established by Arijit Ganguly in the appendix to our previous paper [3] on the imaginary quadratic case. We also simplify arguments in the paper [1] on the same problem for real quadratic number fields by D. Mazumder and the first-named author.

A pretentious proof of Linnik's estimate for primes in arithmetic progressions

AbstractIn the present paper, the author adopts a pretentious approach and recovers an estimate obtained by Linnik for the sums of the von Mangoldt function Λ on arithmetic progressions. It is the analogue of an estimate that Linnik established in his attempt to prove his celebrated theorem concerning the size of the smallest prime number of an arithmetic progression. Our work builds on ideas coming from the pretentious large sieve of Granville, Harper, and Soundararajan and it also borrows insights from the treatment of Koukoulopoulos on multiplicative functions with small averages.

On the error term in the explicit formula of Riemann–von Mangoldt

We provide an explicit [Formula: see text] error term for the Riemann–von Mangoldt formula by making results of Wolke [On the explicit formula of Riemann–von Mangoldt, II, J. London Math. Soc. 2(3) (1983) 406–416] and Ramaré [Modified truncated Perron formulae, Ann. Math. Blaise Pascal 23(1) (2016) 109–128] explicit. We also include applications to primes between consecutive powers, the error term in the prime number theorem and an inequality of Ramanujan.

Double Dirichlet series associated with arithmetic functions II

This paper is a continuation of our previous work on double Dirichlet series associated with arithmetic functions such as the von Mangoldt function, the Möbius function, and so on. We consider the analytic behaviour around the non-positive integer points on singularity sets which are points of indeterminacy. In particular, we show a certain reciprocity law of their residues. Also on this occasion we correct some inaccuracies in our previous paper.

Primes between consecutive powers

This paper updates the explicit interval estimate for primes between consecutive powers. It is shown that there is least one prime between n155 and (n+1)155 for all n≥1. This result is in part obtained with a new explicit version of Goldston's 1983 estimate for the error in the truncated Riemann–von Mangoldt explicit formula.

Higher uniformity of arithmetic functions in short intervals I. All intervals

Abstract We study higher uniformity properties of the Möbius function $\mu $ , the von Mangoldt function $\Lambda $ , and the divisor functions $d_k$ on short intervals $(X,X+H]$ with $X^{\theta +\varepsilon } \leq H \leq X^{1-\varepsilon }$ for a fixed constant $0 \leq \theta < 1$ and any $\varepsilon>0$ . More precisely, letting $\Lambda ^\sharp $ and $d_k^\sharp $ be suitable approximants of $\Lambda $ and $d_k$ and $\mu ^\sharp = 0$ , we show for instance that, for any nilsequence $F(g(n)\Gamma )$ , we have $$\begin{align*}\sum_{X < n \leq X+H} (f(n)-f^\sharp(n)) F(g(n) \Gamma) \ll H \log^{-A} X \end{align*}$$ when $\theta = 5/8$ and $f \in \{\Lambda , \mu , d_k\}$ or $\theta = 1/3$ and $f = d_2$ . As a consequence, we show that the short interval Gowers norms $\|f-f^\sharp \|_{U^s(X,X+H]}$ are also asymptotically small for any fixed s for these choices of $f,\theta $ . As applications, we prove an asymptotic formula for the number of solutions to linear equations in primes in short intervals and show that multiple ergodic averages along primes in short intervals converge in $L^2$ . Our innovations include the use of multiparameter nilsequence equidistribution theorems to control type $II$ sums and an elementary decomposition of the neighborhood of a hyperbola into arithmetic progressions to control type $I_2$ sums.

Linnik's large sieve and the L1$L^{1}$ norm of exponential sums

AbstractThe elementary method of Balog and Ruzsa and the large sieve of Linnik are utilized to investigate the behaviour of the norm of an exponential sum over the primes. A new proof of a lower bound due to Vaughan for the norm of an exponential sum formed with the von Mangoldt function is furnished.

Generalized von Mangoldt surfaces of revolution and asymmetric two-spheres of revolution with simple cut locus structure

It is known that if the Gaussian curvature function along each meridian on a surface of revolution $(\mathbb{R}^{2}, dr^{2} + m(r)^{2} d\theta^{2})$ is decreasing, then the cut locus of each point of $\theta^{-1} (0)$ is empty or a subarc of the opposite meridian $\theta^{-1} (\pi)$. Such a surface is called a von Mangoldt's surface of revolution. A surface of revolution $(\mathbb{R}^{2}, dr^{2} + m(r)^{2} d\theta^{2})$ is called a generalized von Mangoldt surface of revolution if the cut locus of each point of $\theta^{-1} (0)$ is empty or a subarc of the opposite meridian $\theta^{-1} (\pi)$. For example, the surface of revolution $(\mathbb{R}^{2}, dr^{2} + m_0(r)^{2} d\theta^{2})$, where $m_{0}(x) = x/(1 + x^{2})$, has the same cut locus structure as above and the cut locus of each point in $r^{-1}((0, \infty))$ is nonempty. Note that the Gaussian curvature function is not decreasing along a meridian for this surface. In this article, we give sufficient conditions for a surface of revolution $(\mathbb{R}^{2}, dr^{2} + m(r)^{2} d\theta^{2})$ to be a generalized von Mangoldt surface of revolution. Moreover, we prove that for any surface of revolution with finite total curvature $c$, there exists a generalized von Mangoldt surface of revolution with the same total curvature $c$ such that the Gaussian curvature function along a meridian is not monotone on $[a, \infty)$ for any $a > 0$.

Spectral exponential sums on hyperbolic surfaces

We study an exponential sum over Laplacian eigenvalues of Maaß forms on $$\varGamma \backslash {\mathbb {H}}$$ , where $$\varGamma $$ is a congruence subgroup of $${{\,\mathrm{SL}\,}}_{2}({\mathbb {Z}})$$ and $${\mathbb {H}}$$ is the upper half-plane. The goal is to establish an asymptotic formula that expresses the spectral exponential sum in terms of an oscillatory main term, the von Mangoldt function and the Selberg zeta function.

Riemann-type functional equations: Julia line and counting formulae

We study Riemann-type functional equations with respect to value-distribution theory and derive implications for their solutions. In particular, for a fixed complex number a≠0 and a function from the Selberg class L, we prove a Riemann–von Mangoldt formula for the number of a-points of the Δ-factor of the functional equation of L and an analog of Landau’s formula over these points. From the last formula we derive that the ordinates of these a-points are uniformly distributed modulo one. Lastly, we show the existence of the mean-value of the values of L(s) taken at these points.

On arithmetic series involving the fractional part function

We present new relationships between the work of H. Davenport and A. I. Popov. A new general formula involving the Von Mangoldt function is presented, as well as a criteria for the Riemann Hypothesis.

On certain sums of number theory

We study sums of the shape [Formula: see text] where [Formula: see text] is either the von Mangoldt function or the Dirichlet–Piltz divisor functions. We improve previous estimates when [Formula: see text] and [Formula: see text], and provide new results when [Formula: see text] with [Formula: see text], breaking the [Formula: see text]-barrier in each case. The functions [Formula: see text], [Formula: see text] and [Formula: see text] are also investigated.

Improving and maximal inequalities for primes in progressions

Assume that \(y < N\) are integers, and that \((b,y) =1\). Define an average along the primes in a progression of spacing y, given by integer \((b,y)=1\). $$\begin{aligned} A_{N,y,b} := \frac{\phi (y)}{N} \sum _{\begin{subarray}{c} n <N\\ n\equiv b\mod y \end{subarray}} \Lambda (n) f(x-n) \end{aligned}$$Above, \(\Lambda\) is the von Mangoldt function and \(\phi\) is the totient function. We establish improving and maximal inequalities for these averages. These bounds are uniform in the choice of progression. For instance, for \(1< r < \infty\) there is an integer \(N _{y, r}\) so that for \((b,y)=1\), we have $$\begin{aligned} \Vert \sup _{N>N _{y,r}} |A_{N,y,b} f |\Vert _{r}\ll \Vert f\Vert _{r}. \end{aligned}$$The implied constant is only a function of r. The uniformity over progressions imposes several novel elements on the proof.