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Journal articles on the topic 'Von Mangoldt function'

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Published: 8 March 2025

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1

Bienvenu, Pierre-Yves. "Asymptotics for some polynomial patterns in the primes." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 149, no. 5 (January 17, 2019): 1241–90. http://dx.doi.org/10.1017/prm.2018.52.

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AbstractWe prove asymptotic formulae for sums of the form $$\sum\limits_{n\in {\open z}^d\cap K} {\prod\limits_{i = 1}^t {F_i} } (\psi _i(n)),$$where K is a convex body, each Fi is either the von Mangoldt function or the representation function of a quadratic form, and Ψ = (ψ1, …, ψt) is a system of linear forms of finite complexity. When all the functions Fi are equal to the von Mangoldt function, we recover a result of Green and Tao, while when they are all representation functions of quadratic forms, we recover a result of Matthiesen. Our formulae imply asymptotics for some polynomial patterns in the primes. For instance, they describe the asymptotic behaviour of the number of k-term arithmetic progressions of primes whose common difference is a sum of two squares.The paper combines ingredients from the work of Green and Tao on linear equations in primes and that of Matthiesen on linear correlations amongst integers represented by a quadratic form. To make the von Mangoldt function compatible with the representation function of a quadratic form, we provide a new pseudorandom majorant for both – an average of the known majorants for each of the functions – and prove that it has the required pseudorandomness properties.

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2

Kunik, Matthias, and Lutz G. Lucht. "Power series with the von Mangoldt function." Functiones et Approximatio Commentarii Mathematici 47, no. 1 (September 2012): 15–33. http://dx.doi.org/10.7169/facm/2012.47.1.2.

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3

EISNER, TANJA. "Nilsystems and ergodic averages along primes." Ergodic Theory and Dynamical Systems 40, no. 10 (April 11, 2019): 2769–77. http://dx.doi.org/10.1017/etds.2019.27.

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A celebrated result by Bourgain and Wierdl states that ergodic averages along primes converge almost everywhere for $L^{p}$-functions, $p>1$, with a polynomial version by Wierdl and Nair. Using an anti-correlation result for the von Mangoldt function due to Green and Tao, we observe everywhere convergence of such averages for nilsystems and continuous functions.

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4

Fujii, Akio. "Eigenvalues of the Laplace-Beltrami operator and the von-Mangoldt function." Proceedings of the Japan Academy, Series A, Mathematical Sciences 69, no. 5 (1993): 125–30. http://dx.doi.org/10.3792/pjaa.69.125.

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5

Hast, Daniel Rayor, and Vlad Matei. "Higher Moments of Arithmetic Functions in Short Intervals: A Geometric Perspective." International Mathematics Research Notices 2019, no. 21 (January 29, 2018): 6554–84. http://dx.doi.org/10.1093/imrn/rnx310.

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Abstract We study the geometry associated to the distribution of certain arithmetic functions, including the von Mangoldt function and the Möbius function, in short intervals of polynomials over a finite field $\mathbb{F}_{q}$. Using the Grothendieck–Lefschetz trace formula, we reinterpret each moment of these distributions as a point-counting problem on a highly singular complete intersection variety. We compute part of the ℓ-adic cohomology of these varieties, corresponding to an asymptotic bound on each moment for fixed degree n in the limit as $q \to \infty $. The results of this paper can be viewed as a geometric explanation for asymptotic results that can be proved using analytic number theory over function fields.

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6

BANKS, WILLIAM D., JOHN B. FRIEDLANDER, MOUBARIZ Z. GARAEV, and IGOR E. SHPARLINSKI. "EXPONENTIAL AND CHARACTER SUMS WITH MERSENNE NUMBERS." Journal of the Australian Mathematical Society 92, no. 1 (February 2012): 1–13. http://dx.doi.org/10.1017/s1446788712000109.

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AbstractWe give new bounds on sums of the form ∑ n≤NΛ(n)exp (2πiagn/m) and ∑ n≤NΛ(n)χ(gn+a), where Λ is the von Mangoldt function, m is a natural number, a and g are integers coprime to m, and χ is a multiplicative character modulo m. In particular, our results yield bounds on the sums ∑ p≤Nexp (2πiaMp/m) and ∑ p≤Nχ(Mp) with Mersenne numbers Mp=2p−1, where p is prime.

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7

Geštautas, Andrius, and Antanas Laurinčikas. "On Universality of Some Beurling Zeta-Functions." Axioms 13, no. 3 (February 23, 2024): 145. http://dx.doi.org/10.3390/axioms13030145.

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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.

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8

Berra-Montiel, Jasel, and Alberto Molgado. "Polymeric quantum mechanics and the zeros of the Riemann zeta function." International Journal of Geometric Methods in Modern Physics 15, no. 06 (May 8, 2018): 1850095. http://dx.doi.org/10.1142/s0219887818500950.

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We analyze the Berry–Keating model and the Sierra and Rodríguez-Laguna Hamiltonian within the polymeric quantization formalism. By using the polymer representation, we obtain for both models, the associated polymeric quantum Hamiltonians and the corresponding stationary wave functions. The self-adjointness condition provides a proper domain for the Hamiltonian operator and the energy spectrum, which turned out to be dependent on an introduced scale parameter. By performing a counting of semiclassical states, we prove that the polymer representation reproduces the smooth part of the Riemann–von Mangoldt formula, and also introduces a correction depending on the energy and the scale parameter. This may shed some light on the understanding of the fluctuation behavior of the zeros of the Riemann function from a purely quantum point of view.

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9

Pilatte, Cédric. "A solution to the Erdős–Sárközy–Sós problem on asymptotic Sidon bases of order 3." Compositio Mathematica 160, no. 6 (May 10, 2024): 1418–32. http://dx.doi.org/10.1112/s0010437x24007140.

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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.

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10

Jiang, Yujiao, and Guangshi Lü. "Exponential sums formed with the von Mangoldt function and Fourier coefficients of $${ GL}(m)$$ G L ( m ) automorphic forms." Monatshefte für Mathematik 184, no. 4 (May 27, 2017): 539–61. http://dx.doi.org/10.1007/s00605-017-1068-4.

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11

CASTRO, CARLOS. "ON THE RIEMANN HYPOTHESIS, AREA QUANTIZATION, DIRAC OPERATORS, MODULARITY, AND RENORMALIZATION GROUP." International Journal of Geometric Methods in Modern Physics 07, no. 01 (February 2010): 1–31. http://dx.doi.org/10.1142/s0219887810003938.

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Two methods to prove the Riemann Hypothesis are presented. One is based on the modular properties of Θ (theta) functions and the other on the Hilbert–Polya proposal to find an operator whose spectrum reproduces the ordinates ρn (imaginary parts) of the zeta zeros in the critical line: sn = ½ + iρn. A detailed analysis of a one-dimensional Dirac-like operator with a potential V(x) is given that reproduces the spectrum of energy levels En = ρn, when the boundary conditions ΨE (x = -∞) = ± ΨE (x = +∞) are imposed. Such potential V(x) is derived implicitly from the relation [Formula: see text], where the functional form of [Formula: see text] is given by the full-fledged Riemann–von Mangoldt counting function of the zeta zeros, including the fluctuating as well as the [Formula: see text] terms. The construction is also extended to self-adjoint Schroedinger operators. Crucial is the introduction of an energy-dependent cut-off function Λ(E). Finally, the natural quantization of the phase space areas (associated to nonperiodic crystal-like structures) in integer multiples of π follows from the Bohr–Sommerfeld quantization conditions of Quantum Mechanics. It allows to find a physical reasoning why the average density of the primes distribution for very large [Formula: see text] has a one-to-one correspondence with the asymptotic limit of the inverse average density of the zeta zeros in the critical line suggesting intriguing connections to the renormalization group program.

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12

Matomäki, Kaisa, Maksym Radziwiłł, and Terence Tao. "Correlations of the von Mangoldt and higher divisor functions I. Long shift ranges." Proceedings of the London Mathematical Society 118, no. 2 (July 31, 2018): 284–350. http://dx.doi.org/10.1112/plms.12181.

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13

Matomäki, Kaisa, Maksym Radziwiłł, and Terence Tao. "Correlations of the von Mangoldt and higher divisor functions II: divisor correlations in short ranges." Mathematische Annalen 374, no. 1-2 (January 21, 2019): 793–840. http://dx.doi.org/10.1007/s00208-018-01801-4.

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14

Satman, Abdurrahman. "An Analytical Study of Interference in Composite Reservoirs." Society of Petroleum Engineers Journal 25, no. 02 (April 1, 1985): 281–90. http://dx.doi.org/10.2118/10902-pa.

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Satman, Abdurrahman; SPE; Technical U. of Istanbul Abstract This paper discusses the interference test in composite reservoirs. The composite model considers all important parameters of interest: the hydraulic diffusivity, the mobility ratio, the distance to the radial discontinuity, the distance between wells, the wellbore storage, and skin effect at the active well. Type curves expressed as a function of proper combinations of these parameters are presented. Introduction Interference tests are widely used to estimate the reservoir properties. An interference test is a multiwell test that requires at least one active well, either a producer or injector, and at least one observation well. During the test, pressure effects caused by the active well are measured at the shut-in observation wells. Basic techniques for analyzing interference tests in uniform systems are discussed in Ref. 1. Usually, type-curve matching is the preferred technique for analyzing the pressure data from the test. Early interference test studies assumed that the storage capacity of the active well and the skin region around the sandface have a negligible effect on the observation well response. Recently, investigators have focused on wellbore storage and skin effects. Tongpenyai and Raghavan presented a new solution for analyzing the pressure response at the presented a new solution for analyzing the pressure response at the observation well, which took into account the effects of wellbore storage and skin at both the active and the observation wells. They produced type curves expressed as a function of exp(2S) products, the ( / ) ratios, and ( / ) to correlate the pressure response at the observation well. Composite systems are encountered in a wide variety of reservoir situations. In a composite system, there is a circular inner region with fluid and rock properties different from those in the outer region. Such a system can occur in hydrocarbon reservoirs and geothermal reservoirs. The injection of fluids during EOR processes can cause the development of fluid banks around the injection wells. This would be true in the case of a in-situ combustion or a steamflood. In a geothermal reservoir, pressure reduction in the vicinity of the well may cause the phase boundaries. A producing well completed in the center of a circular hot zone surrounded by producing well completed in the center of a circular hot zone surrounded by a concentric cooler water region is also a composite system. During the early to late 1960's, there was great interest in the composite reservoir flow problem. Hurst discussed the "sands in series" problem. He presented the formulas to describe the pressure behavior of problem. He presented the formulas to describe the pressure behavior of the unsteady-state flow phenomenon for fluid movement through two sands in series in a radial configuration, with each sand of different permeability. Mortada studied the interference pressure drop for oil fields located in a nonuniform extensive aquifer comprising two regions of different properties. He presented an expression for the interference pressure drop properties. He presented an expression for the interference pressure drop in an oil field resulting from a constant rate of water influx in another oil field. Loucks and Guerrero presented a qualitative discussion of pressure drop characteristics in composite reservoirs. Ramey and Rowan and pressure drop characteristics in composite reservoirs. Ramey and Rowan and Clegg developed approximate solutions. Refs. 11 through 13 also discuss composite reservoir systems and present either analytical or numerical solutions. Composite system model solutions have been used to determine some critical parameters during the application of EOR processes. The formation of a fluid bank around the injection well makes the reservoir a composite system. Van Poollen and Kazemi discussed how to determine the mean distance to the radial discontinuity in an in-situ combustion project. Refs. 16 and 17 discuss the effect of radial discontinuity in interpretation of pressure falloff tests in reservoirs with fluid banks. Sosa et al. examined the effect of relative permeability and mobility ratio on falloff behavior in reservoirs with water banks. The presence of different temperature zones in nonisothermal reservoirs may resemble permeability boundaries during well testing. Mangold et al. presented a numerical study of a thermal discontinuity in well test analysis. Their results indicated that nonisothermal influence could be detected and accounted for by tests of sufficient duration with suitably placed observation wells. Horne et al. indicated the possibility of determining compressibility and permeability contrasts across the phase boundaries in geothermal reservoirs. The most recent study of well test analysis in composite reservoirs was by Eggenschwiler, Satman et al. Their studies presented a very general composite system model. The problem was solved analytically by using the Laplace transformation with numerical inversion. The solution concerned the transient flow of a slightly compressible fluid in a porous medium during injection or falloff for a single well confined in concentric regions of differing mobilities and hydraulic diffusivities. The system assumed both wellbore storage and a skin effect. Their results indicated that a pseudosteady-state pressure response exists in the transition region between the inner region and outer region semilog straight lines. This response is drawn on a Cartesian vs. plot, the slope of which is used to estimate the bulk volume of the inner region. SPEJ p. 281

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15

Languasco, Alessandro, and Alessandro Zaccagnini. "A Cesàro average of Goldbach numbers." Forum Mathematicum 27, no. 4 (January 1, 2015). http://dx.doi.org/10.1515/forum-2012-0100.

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16

Lü, Xiaodong, and Xinyue Xu. "On fractional sum of the von Mangoldt function." Colloquium Mathematicum, 2024. https://doi.org/10.4064/cm9418-10-2024.

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17

Révész, Szilárd Gy. "A Riemann–von Mangoldt-Type Formula for the Distribution of Beurling Primes." Mathematica Pannonica, November 8, 2021. http://dx.doi.org/10.1556/314.2021.00019.

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In this paper we work out a Riemann–von Mangoldt type formula for the summatory function := , where is an arithmetical semigroup (a Beurling generalized system of integers) and is the corresponding von Mangoldt function attaining with a prime element and zero otherwise. On the way towards this formula, we prove explicit estimates on the Beurling zeta function , belonging to , to the number of zeroes of in various regions, in particular within the critical strip where the analytic continuation exists, and to the magnitude of the logarithmic derivative of , under the sole additional assumption that Knopfmacher’s Axiom A is satisfied. We also construct a technically useful broken line contour to which the technic of integral transformation can be well applied. The whole work serves as a first step towards a further study of the distribution of zeros of the Beurling zeta function, providing appropriate zero density and zero clustering estimates, to be presented in the continuation of this paper.

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18

Kiuchi, Isao, and Wataru Takeda. "On Sums of Sums Involving the Von Mangoldt Function." Results in Mathematics 79, no. 7 (September 15, 2024). http://dx.doi.org/10.1007/s00025-024-02276-3.

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19

Cantarini, Marco, Alessandro Gambini, and Alessandro Zaccagnini. "Cesàro averages for Goldbach representations with summands in arithmetic progressions." International Journal of Number Theory, July 6, 2021, 1–15. http://dx.doi.org/10.1142/s1793042121500937.

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Let [Formula: see text] be the von Mangoldt function, let [Formula: see text] be an integer and let [Formula: see text] be the counting function for the Goldbach numbers with summands in arithmetic progression modulo a common integer [Formula: see text]. We prove an asymptotic formula for the weighted average, with Cesàro weight of order [Formula: see text], with [Formula: see text], of this function. Our result is uniform in a suitable range for [Formula: see text].

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20

Kuperberg, Vivian, and Matilde Lalín. "Sums of divisor functions and von Mangoldt convolutions in 𝔽_q[T] leading to symplectic distributions." Forum Mathematicum, March 26, 2022. http://dx.doi.org/10.1515/forum-2021-0171.

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Abstract In [J. P. Keating, B. Rodgers, E. Roditty-Gershon and Z. Rudnick, Sums of divisor functions in 𝔽 q ⁢ [ t ] \mathbb{F}_{q}[t] and matrix integrals, Math. Z. 288 2018, 1–2, 167–198], the authors established relationships of the mean-square of sums of the divisor function d k ⁢ ( f ) {d_{k}(f)} over short intervals and over arithmetic progressions for the function field 𝔽 q ⁢ [ T ] {\mathbb{F}_{q}[T]} to certain integrals over the ensemble of unitary matrices. We consider similar problems leading to distributions over the ensemble of symplectic matrices. We also consider analogous questions involving convolutions of the von Mangoldt function.

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21

Laporta, Maurizio. "On Ramanujan expansions and primes in arithmetic progressions." Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, August 23, 2024. http://dx.doi.org/10.1007/s12188-024-00282-4.

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AbstractA 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.

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22

Lichtman, Jared Duker. "Averages of the Möbius Function on Shifted Primes." Quarterly Journal of Mathematics, November 29, 2021. http://dx.doi.org/10.1093/qmath/haab054.

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Abstract It is a folklore conjecture that the Möbius function exhibits cancellation on shifted primes; that is, $\sum_{p{\,\leqslant} X}\mu(p+h) \ = \ o(\pi(X))$ as $X\to\infty$ for any fixed shift h > 0. This appears in print at least since Hildebrand in 1989. We prove the conjecture on average for shifts $h{\,\leqslant} H$, provided $\log H/\log\log X\to\infty$. We also obtain results for shifts of prime k-tuples, and for higher correlations of Möbius with von Mangoldt and divisor functions. Our argument combines sieve methods with a refinement of Matomäki, Radziwiłł and Tao’s work on an averaged form of Chowla’s conjecture.

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23

ROBLES, NICOLAS, and ARINDAM ROY. "UNEXPECTED AVERAGE VALUES OF GENERALIZED VON MANGOLDT FUNCTIONS IN RESIDUE CLASSES." Journal of the Australian Mathematical Society, July 17, 2020, 1–18. http://dx.doi.org/10.1017/s1446788719000715.

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In order to study integers with few prime factors, the average of $\unicode[STIX]{x1D6EC}_{k}=\unicode[STIX]{x1D707}\ast \log ^{k}$ has been a central object of research. One of the more important cases, $k=2$ , was considered by Selberg [‘An elementary proof of the prime-number theorem’, Ann. of Math. (2)50 (1949), 305–313]. For $k\geq 2$ , it was studied by Bombieri [‘The asymptotic sieve’, Rend. Accad. Naz. XL (5)1(2) (1975/76), 243–269; (1977)] and later by Friedlander and Iwaniec [‘On Bombieri’s asymptotic sieve’, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4)5(4) (1978), 719–756], as an application of the asymptotic sieve. Let $\unicode[STIX]{x1D6EC}_{j,k}:=\unicode[STIX]{x1D707}_{j}\ast \log ^{k}$ , where $\unicode[STIX]{x1D707}_{j}$ denotes the Liouville function for $(j+1)$ -free integers, and $0$ otherwise. In this paper we evaluate the average value of $\unicode[STIX]{x1D6EC}_{j,k}$ in a residue class $n\equiv a\text{ mod }q$ , $(a,q)=1$ , uniformly on $q$ . When $j\geq 2$ , we find that the average value in a residue class differs by a constant factor from the expected value. Moreover, an explicit formula of Weil type for $\unicode[STIX]{x1D6EC}_{k}(n)$ involving the zeros of the Riemann zeta function is derived for an arbitrary compactly supported ${\mathcal{C}}^{2}$ function.

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24

Coppola, Giovanni. "An elementary property of correlations." Hardy-Ramanujan Journal, January 23, 2019. http://dx.doi.org/10.46298/hrj.2019.5108.

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International audience We study the "shift-Ramanujan expansion" to obtain a formulae for the shifted convolution sum $C_{f,g} (N,a)$ of general functions f, g satisfying Ramanujan Conjecture; here, the shift-Ramanujan expansion is with respect to a shift factor a > 0. Assuming Delange Hypothesis for the correlation, we get the "Ramanujan exact explicit formula", a kind of finite shift-Ramanujan expansion. A noteworthy case is when f = g = Λ, the von Mangoldt function; so $C_{\Lamda, \Lambda} (N, 2k)$, for natural k, corresponds to 2k-twin primes; under the assumption of Delange Hypothesis, we easily obtain the proof of Hardy-Littlewood Conjecture for this case.

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25

TAO, TERENCE, and TAMAR ZIEGLER. "POLYNOMIAL PATTERNS IN THE PRIMES." Forum of Mathematics, Pi 6 (2018). http://dx.doi.org/10.1017/fmp.2017.3.

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Let $P_{1},\ldots ,P_{k}:\mathbb{Z}\rightarrow \mathbb{Z}$ be polynomials of degree at most $d$ for some $d\geqslant 1$, with the degree $d$ coefficients all distinct, and admissible in the sense that for every prime $p$, there exists integers $n,m$ such that $n+P_{1}(m),\ldots ,n+P_{k}(m)$ are all not divisible by $p$. We show that there exist infinitely many natural numbers $n,m$ such that $n+P_{1}(m),\ldots ,n+P_{k}(m)$ are simultaneously prime, generalizing a previous result of the authors, which was restricted to the special case $P_{1}(0)=\cdots =P_{k}(0)=0$ (though it allowed for the top degree coefficients to coincide). Furthermore, we obtain an asymptotic for the number of such prime pairs $n,m$ with $n\leqslant N$ and $m\leqslant M$ with $M$ slightly less than $N^{1/d}$. This asymptotic is already new in general in the homogeneous case $P_{1}(0)=\cdots =P_{k}(0)=0$. Our arguments rely on four ingredients. The first is a (slightly modified) generalized von Neumann theorem of the authors, reducing matters to controlling certain averaged local Gowers norms of (suitable normalizations of) the von Mangoldt function. The second is a more recent concatenation theorem of the authors, controlling these averaged local Gowers norms by global Gowers norms. The third ingredient is the work of Green and the authors on linear equations in primes, allowing one to compute these global Gowers norms for the normalized von Mangoldt functions. Finally, we use the Conlon–Fox–Zhao densification approach to the transference principle to combine the preceding three ingredients together. In the special case $P_{1}(0)=\cdots =P_{k}(0)=0$, our methods also give infinitely many $n,m$ with $n+P_{1}(m),\ldots ,n+P_{k}(m)$ in a specified set primes of positive relative density $\unicode[STIX]{x1D6FF}$, with $m$ bounded by $\log ^{L}n$ for some $L$ independent of the density $\unicode[STIX]{x1D6FF}$. This improves slightly on a result from our previous paper, in which $L$ was allowed to depend on $\unicode[STIX]{x1D6FF}$.

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26

Nakamura, Takashi. "On Lerch’s formula and zeros of the quadrilateral zeta function." Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, February 10, 2025. https://doi.org/10.1007/s12188-025-00286-8.

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Abstract Let $$0 < a \le 1/2$$ 0 < a ≤ 1 / 2 and define the quadrilateral zeta function by $$2Q(s,a):= \zeta (s,a) + \zeta (s,1-a) + \mathrm{{Li}}_s (e^{2\pi ia}) + \mathrm{{Li}}_s(e^{2\pi i(1-a)})$$ 2 Q ( s , a ) : = ζ ( s , a ) + ζ ( s , 1 - a ) + Li s ( e 2 π i a ) + Li s ( e 2 π i ( 1 - a ) ) , where $$\zeta (s,a)$$ ζ ( s , a ) is the Hurwitz zeta function and $$\mathrm{{Li}}_s (e^{2\pi ia})$$ Li s ( e 2 π i a ) is the periodic zeta function. In the present paper, we show that there exists a unique real number $$a_0 \in (0,1/2)$$ a 0 ∈ ( 0 , 1 / 2 ) such that all real zeros of Q(s, a) are simple and are located only at the negative even integers just like $$\zeta (s)$$ ζ ( s ) if and only if $$a_0 < a \le 1/2$$ a 0 < a ≤ 1 / 2 . Moreover, we prove that Q(s, a) has infinitely many complex zeros in the region of absolute convergence and the critical strip when $$a \in {\mathbb {Q}} \cap (0,1/2) \setminus \{1/6, 1/4, 1/3\}$$ a ∈ Q ∩ ( 0 , 1 / 2 ) \ { 1 / 6 , 1 / 4 , 1 / 3 } . The Lerch formula, Hadamard product formula, Riemann-von Mangoldt formula for Q(s, a) are also shown.

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27

Matomäki, Kaisa, Xuancheng Shao, Terence Tao, and Joni Teräväinen. "Higher uniformity of arithmetic functions in short intervals I. All intervals." Forum of Mathematics, Pi 11 (2023). http://dx.doi.org/10.1017/fmp.2023.28.

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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.

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28

Révész, Szilárd Gy. "Oscillation of the Remainder Term in the Prime Number Theorem of Beurling, “Caused by a Given ζ-Zero”." International Mathematics Research Notices, October 5, 2022. http://dx.doi.org/10.1093/imrn/rnac274.

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Abstract:

Abstract Continuing previous studies of the Beurling zeta function, here, we prove two results, generalizing long existing knowledge regarding the classical case of the Riemann zeta function and some of its generalizations. First, we address the question of Littlewood, who asked for explicit oscillation results provided a zeta-zero is known. We prove that given a zero $\rho _0$ of the Beurling zeta function $\zeta _{{\mathcal {P}}}$ for a given number system generated by the primes ${\mathcal {P}}$, the corresponding error term $\Delta (x):=\psi _{{\mathcal {P}}}(x)-x$, where $\psi _{{\mathcal {P}}}(x)$ is the von Mangoldt summatory function shows oscillation in any large enough interval, as large as $\frac {\pi /2-\varepsilon }{|\rho _0|}x^{\Re \rho _0}$. The somewhat mysterious appearance of the constant $\pi /2$ is explained in the study. Finally, we prove as the next main result of the paper the following: given $\varepsilon>0$, there exists a Beurling number system with primes ${\mathcal {P}}$, such that $|\Delta (x)| \le \frac {\pi /2+\varepsilon }{|\rho _0|}x^{\Re \rho _0}$. In this 2nd part, a nontrivial construction of a low norm sine polynomial is coupled by the application of the wonderful recent prime random approximation result of Broucke and Vindas, who sharpened the breakthrough probabilistic construction due to Diamond, Montgomery, and Vorhauer.

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29

Ferrari, Matteo. "On a basic mean value theorem with explicit exponents." International Journal of Number Theory, August 31, 2021, 1–18. http://dx.doi.org/10.1142/s1793042122500361.

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Abstract:

We follow a paper by Sedunova regarding Vaughan’s basic mean value Theorem to improve and complete a more general demonstration for a suitable class of arithmetic functions as started by Cojocaru and Murty. As an application we derive a basic mean value theorem for the von Mangoldt generalized functions.

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30

Tao, Terence, and Joni Teräväinen. "Quantitative bounds for Gowers uniformity of the Möbius and von Mangoldt functions." Journal of the European Mathematical Society, December 12, 2023, 1–64. http://dx.doi.org/10.4171/jems/1404.

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