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Journal articles on the topic 'The Number of Primes at a Given Magnitude'

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Published: 3 June 2025
Last updated: 31 July 2025

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1

Broughan, Kevin A., and Rory J. Casey. "Harmonic sets and the harmonic prime number theorem." Bulletin of the Australian Mathematical Society 71, no. 1 (2005): 127–37. http://dx.doi.org/10.1017/s0004972700038089.

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We restrict primes and prime powers to sets . Let . Then the error in θH(x) has, unconditionally, the expected order of magnitude . However, if then ψH (x) = x log 2 + O (log x). Some reasons for and consequences of these sharp results are explored. A proof is given of the “harmonic prime number theorem”, πH (x)/π (x) → log 2.
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2

Siokis, Fotios M. "Exploring the Dynamic Behavior of Crude Oil Prices in Times of Crisis: Quantifying the Aftershock Sequence of the COVID-19 Pandemic." Mathematics 12, no. 17 (2024): 2743. http://dx.doi.org/10.3390/math12172743.

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Crude oil prices crashed and dropped into negative territory at the onset of the COVID-19 pandemic. This extreme event triggered a series of great-magnitude aftershocks. We seek to investigate the cascading dynamics and the characteristics of the series immediately following the oil market crash. Utilizing a robust method named the Omori law, we quantify the correlations of these events. This research presents empirical regularity concerning the number of times that the absolute value of the percentage change in the oil index exceeds a given threshold value. During the COVID-19 crisis, the Wes
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3

Xu, H., P. S. Heeger, and R. L. Fairchild. "Distinct roles for B7-1 and B7-2 determinants during priming of effector CD8+ Tc1 and regulatory CD4+ Th2 cells for contact hypersensitivity." Journal of Immunology 159, no. 9 (1997): 4217–26. http://dx.doi.org/10.4049/jimmunol.159.9.4217.

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Abstract Contact hypersensitivity (CHS) is a CD8+ T cell-mediated response to hapten sensitization and challenge of the epidermis. Both the IFN-gamma-producing effector CD8+ T cells and the IL-4/IL-10-producing CD4+ T (Th2) cells that restrict the magnitude and duration of the response are primed by hapten-presenting Langerhans cells (hpLC). hpLC isolated from hapten-sensitized animals expressed high levels of B7-2 and lower levels of B7-1, suggesting the availability of each molecule to provide costimulation during CD8+ and CD4+ T cell priming for CHS. Anti-B7-2 Ab given during hapten sensiti
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4

Spinks, Jean, Gang Chen, and Lara Donovan. "Does generic entry lower the prices paid for pharmaceuticals in Australia? A comparison before and after the introduction of the mandatory price-reduction policy." Australian Health Review 37, no. 5 (2013): 675. http://dx.doi.org/10.1071/ah13024.

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Objective We investigated the relationship between the number of generic medicines and pharmaceutical prices over time in Australia. Methods A dataset was utilised containing 76 items for 4 years (2003–2007) on the national subsidy scheme – the Pharmaceutical Benefits Scheme (PBS) – for which a generic brand is available. The PBS price was used as the dependent variable, and the number of generics available the key explanatory variable. The ordinary least-squares estimator was adopted for estimation. In the robustness analysis, an instrumental-variables method was used to account for potential
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5

BROUGHAN, KEVIN A., and QIZHI ZHOU. "FLAT PRIMES AND THIN PRIMES." Bulletin of the Australian Mathematical Society 82, no. 2 (2009): 282–92. http://dx.doi.org/10.1017/s0004972710000067.

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AbstractA number is called upper (lower) flat if its shift by +1 ( −1) is a power of 2 times a squarefree number. If the squarefree number is 1 or a single odd prime then the original number is called upper (lower) thin. Upper flat numbers which are primes arise in the study of multi-perfect numbers. Here we show that the lower or upper flat primes have asymptotic density relative to that of the full set of primes given by twice Artin’s constant, that more than 53% of the primes are both lower and upper flat, and that the series of reciprocals of the lower or the upper thin primes converges.
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6

YAMADA, TOMOYUKI, and SHINICHI MORISHITA. "COMPUTING HIGHLY SPECIFIC AND NOISE-TOLERANT OLIGOMERS EFFICIENTLY." Journal of Bioinformatics and Computational Biology 02, no. 01 (2004): 21–46. http://dx.doi.org/10.1142/s0219720004000454.

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The sequencing of the genomes of a variety of species and the growing databases containing expressed sequence tags (ESTs) and complementary DNAs (cDNAs) facilitate the design of highly specific oligomers for use as genomic markers, PCR primers, or DNA oligo microarrays. The first step in evaluating the specificity of short oligomers of about 20 units in length is to determine the frequencies at which the oligomers occur. However, for oligomers longer than about fifty units this is not efficient, as they usually have a frequency of only 1. A more suitable procedure is to consider the mismatch t
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7

BROUGHAN, KEVIN A. "ON SHIFTED PRIMES AND BALANCED PRIMES." International Journal of Number Theory 08, no. 08 (2012): 2017–33. http://dx.doi.org/10.1142/s179304211250114x.

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The asymptotic order of the number of primes, which are such that the shift by a fixed integer is a number supported by a given set of primes times a coprime squarefree number, is determined. The order is also determined when the shift and its negative have this same shape. In each case the order is dependent only on the set of primes and the squarefree core of the shift.
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8

Chenevier, Gaëtan. "On number fields with given ramification." Compositio Mathematica 143, no. 6 (2007): 1359–73. http://dx.doi.org/10.1112/s0010437x07003132.

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AbstractLet E be a CM number field and let S be a finite set of primes of E containing the primes dividing a given prime number l and another prime u split above the maximal totally real subfield of E. If ES denotes a maximal algebraic extension of E which is unramified outside S, we show that the natural maps $\mathrm {Gal}(\overline {E_u}/E_u) \longrightarrow \mathrm {Gal}(E_S/E)$ are injective. We discuss generalizations of this result.
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9

Maynard, James. "Dense clusters of primes in subsets." Compositio Mathematica 152, no. 7 (2016): 1517–54. http://dx.doi.org/10.1112/s0010437x16007296.

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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$p_{n+m}-p_{n}\leqslant {\it\epsilon}\log x$.
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10

Borrow, Ray, Nick Andrews, Helen Findlow, et al. "Kinetics of Antibody Persistence following Administration of a Combination Meningococcal Serogroup C and Haemophilus influenzae Type b Conjugate Vaccine in Healthy Infants in the United Kingdom Primed with a Monovalent Meningococcal Serogroup C Vaccine." Clinical and Vaccine Immunology 17, no. 1 (2009): 154–59. http://dx.doi.org/10.1128/cvi.00384-09.

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ABSTRACT The kinetics of antibody persistence following the administration of a combination meningococcal serogroup C and Haemophilus influenzae type b (Hib) conjugate vaccine (Menitorix) in the second year of life in children primed with two doses of one of three monovalent meningococcal serogroup C (MCC) vaccines was investigated. The study subjects were administered either Menitorix at 12 to 15 months of age, followed by the seven-valent pneumococcal conjugate vaccine (PCV7) and the measles, mumps, and rubella vaccine 4 to 6 weeks later, or all three vaccines concomitantly at 12 to 15 month
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11

Freiberg, Tristan. "Short intervals with a given number of primes." Journal of Number Theory 163 (June 2016): 159–71. http://dx.doi.org/10.1016/j.jnt.2015.11.009.

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12

Nakagoshi, Norikata. "On the class number of the lpth cyclotomic number field." Mathematical Proceedings of the Cambridge Philosophical Society 109, no. 2 (1991): 263–76. http://dx.doi.org/10.1017/s0305004100069735.

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The first factor of the class number of a cyclotomic number field can be obtainable by the analytic class number formula and there are some tables which show the decompositions of the first factors into primes. But, using just the analytic formula, we cannot tell what kinds of primes will appear as the factors of the class number of a given cyclotomic number field, except for those of the genus number, or the irregular primes. It is significant to find in advance the prime factors, particularly those prime to the degree of the field. For instance, in the table of the first factors we can pick
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13

David, Chantal, and Ethan Smith. "Elliptic curves with a given number of points over finite fields." Compositio Mathematica 149, no. 2 (2012): 175–203. http://dx.doi.org/10.1112/s0010437x12000541.

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AbstractGiven an elliptic curve E and a positive integer N, we consider the problem of counting the number of primes p for which the reduction of E modulo p possesses exactly N points over 𝔽p. On average (over a family of elliptic curves), we show bounds that are significantly better than what is trivially obtained by the Hasse bound. Under some additional hypotheses, including a conjecture concerning the short-interval distribution of primes in arithmetic progressions, we obtain an asymptotic formula for the average.
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14

Chege, Gerald K., Enid G. Shephard, Ann Meyers, et al. "HIV-1 subtype C Pr55gag virus-like particle vaccine efficiently boosts baboons primed with a matched DNA vaccine." Journal of General Virology 89, no. 9 (2008): 2214–27. http://dx.doi.org/10.1099/vir.0.83501-0.

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A DNA vaccine expressing human immunodeficiency virus type 1 (HIV-1) southern African subtype C Gag (pTHGag) and a recombinant baculovirus Pr55gag virus-like particle prepared using a subtype C Pr55gag protein (Gag VLP) was tested in a prime–boost inoculation regimen in Chacma baboons. The response of five baboons to Gag peptides in a gamma interferon (IFN-γ) enzyme-linked immunospot (ELISPOT) assay after three pTHGag immunizations ranged from 100 to 515 spot-forming units (s.f.u.) per 106 peripheral blood mononuclear cells (PBMCs), whilst the response of two baboons to the Gag VLP vaccine ran
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15

Debaene, Korneel. "Explicit counting of ideals and a Brun–Titchmarsh inequality for the Chebotarev density theorem." International Journal of Number Theory 15, no. 05 (2019): 883–905. http://dx.doi.org/10.1142/s1793042119500477.

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We prove a bound on the number of primes with a given splitting behavior in a given field extension. This bound generalizes the Brun–Titchmarsh bound on the number of primes in an arithmetic progression. The proof is set up as an application of Selberg’s Sieve in number fields. The main new ingredient is an explicit counting result estimating the number of integral elements with certain properties up to multiplication by units. As a consequence of this result, we deduce an explicit estimate for the number of ideals of norm up to [Formula: see text].
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16

Roberts, David P. "Polynomials with prescribed bad primes." International Journal of Number Theory 11, no. 04 (2015): 1115–48. http://dx.doi.org/10.1142/s179304211550061x.

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We tabulate polynomials in ℚ[t] with a given factorization partition, bad reduction entirely within a given set of primes, and satisfying auxiliary conditions associated to 0, 1, and ∞. We explain how these polynomials are of particular interest because of their role in the construction of nonsolvable number fields of arbitrarily large degree and bounded ramification.
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17

Lehmer, Emma. "An indeterminate in number theory." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 46, no. 3 (1989): 469–72. http://dx.doi.org/10.1017/s1446788700030949.

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AbstractThis paper studies quintic residuacity of primes p of the form for which the expression for 4f modulo p given in the first volume of this journal becomes indeterminate, and replaces it by a much simpler expression.
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18

Zoeteman, M. "Uniformly counting primes with a given primitive root and in an arithmetic progression." International Journal of Number Theory 15, no. 10 (2019): 2115–34. http://dx.doi.org/10.1142/s1793042119501161.

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We study the number of primes with a given primitive root and in an arithmetic progression under the assumption of a suitable form of the generalized Riemann Hypothesis. Previous work of Lenstra, Moree and Stevenhagen has given asymptotics without an explicit error term, we provide an explicit error term by combining their work with the method of Hooley regarding Artin’s primitive root conjecture. We give an application to a Diophantine problem involving primes with a given primitive root.
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19

TRAN, DUC-SON. "THE PRIME-REPRESENTING FUNCTIONS." Mathematical Reports 26(76), no. 3-4 (2024): 251–54. https://doi.org/10.59277/mrar.2024.26.76.3.4.251.

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In this paper, we prove that there is a fixed number D such that the values ⌈Ddn ⌉ are primes for all positive integers n and real numbers d > 8/3. We also prove that there exists a value e such that the values ⌈Ben ⌉ are primes for all positive integers n and a given any real number B > 1, where ⌈.⌉ is the ceiling function.
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20

Radomskii, Artyom Olegovich. "Primes in tuples and Romanoff's theorem." Izvestiya: Mathematics 89, no. 1 (2025): 125–39. https://doi.org/10.4213/im9544e.

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21

LAISHRAM, SHANTA. "ON A CONJECTURE ON RAMANUJAN PRIMES." International Journal of Number Theory 06, no. 08 (2010): 1869–73. http://dx.doi.org/10.1142/s1793042110003848.

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For n ≥ 1, the nth Ramanujan prime is defined to be the smallest positive integer Rn with the property that if x ≥ Rn, then [Formula: see text] where π(ν) is the number of primes not exceeding ν for any ν > 0 and ν ∈ ℝ. In this paper, we prove a conjecture of Sondow on upper bound for Ramanujan primes. An explicit bound of Ramanujan primes is also given. The proof uses explicit bounds of prime π and θ functions due to Dusart.
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22

KANE, DANIEL M. "AN ASYMPTOTIC FOR THE NUMBER OF SOLUTIONS TO LINEAR EQUATIONS IN PRIME NUMBERS FROM SPECIFIED CHEBOTAREV CLASSES." International Journal of Number Theory 09, no. 04 (2013): 1073–111. http://dx.doi.org/10.1142/s1793042113500139.

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We extend results relating to Vinogradov's three primes theorem to provide asymptotic estimates for the number of solutions to a given linear equation in three or more prime numbers under the additional constraint that each of the primes involved satisfies specialized Chebotarev conditions. In particular, we show that such solutions can be expected to exist unless a solution would violate some local constraint.
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23

TANNER, NOAM. "STRINGS OF CONSECUTIVE PRIMES IN FUNCTION FIELDS." International Journal of Number Theory 05, no. 01 (2009): 81–88. http://dx.doi.org/10.1142/s1793042109001918.

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In a recent paper, Thorne [5] proved the existence of arbitrarily long strings of consecutive primes in arithmetic progressions in the polynomial ring 𝔽q[t]. Here we extend this result to show that given any k there exists a string of k consecutive primes of degree D in arithmetic progression for all sufficiently large D.
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24

MacDougall, Jim. "Mersenne composites and cyclotomic primes." Mathematical Gazette 87, no. 508 (2003): 71–75. http://dx.doi.org/10.1017/s0025557200172122.

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One of the long-standing problems of number theory, appealing to professional and recreational mathematicians alike, is the existence of Mersenne primes. These puzzling primes, for example 7, 31, 127 and 8191, are of the form 2P - 1, where p is itself a prime. The problem of their existence originated some 2400 years ago with the early Greek mathematicians' quest for the so-called perfect numbers, those like 6 and 28 which are the sum of their proper divisors. The connection was given in Euclid's Elements in 300 BC: if 2P - 1 is prime, then 2P-1(2P - 1) is a perfect number. Much later, Euler p
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25

Blomer, V., J. Brüdern, and R. Dietmann. "Sums of smooth squares." Compositio Mathematica 145, no. 6 (2009): 1401–41. http://dx.doi.org/10.1112/s0010437x09004254.

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AbstractLet R(n,θ) denote the number of representations of the natural number n as the sum of four squares, each composed only with primes not exceeding nθ/2. When θ>e−1/3 a lower bound for R(n,θ) of the expected order of magnitude is established, and when θ>365/592, it is shown that R(n,θ)>0 holds for large n. A similar result is obtained for sums of three squares. An asymptotic formula is obtained for the related problem of representing an integer as the sum of two squares and two squares composed of small primes, as above, for any fixed θ>0. This last result is the key to bound
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26

Cho, Ilwoo, and Palle Jorgensen. "Primes in Intervals and Semicircular Elements Induced by p-Adic Number Fields Qp over Primes p." Mathematics 7, no. 2 (2019): 199. http://dx.doi.org/10.3390/math7020199.

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In this paper, we study free probability on (weighted-)semicircular elements in a certain Banach *-probability space ( LS , τ 0 ) induced by measurable functions on p-adic number fields Q p over primes p . In particular, we are interested in the cases where such free-probabilistic information is affected by primes in given closed intervals of the set R of real numbers by defining suitable “truncated” linear functionals on LS .
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27

Perucca, Antonella, and Pietro Sgobba. "Kummer Theory for Number Fields and the Reductions of Algebraic Numbers II." Uniform distribution theory 15, no. 1 (2020): 75–92. http://dx.doi.org/10.2478/udt-2020-0004.

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AbstractLet K be a number field, and let G be a finitely generated and torsion-free subgroup of K×. For almost all primes p of K, we consider the order of the cyclic group (G mod 𝔭), and ask whether this number lies in a given arithmetic progression. We prove that the density of primes for which the condition holds is, under some general assumptions, a computable rational number which is strictly positive. We have also discovered the following equidistribution property: if ℓe is a prime power and a is a multiple of ℓ (and a is a multiple of 4 if ℓ =2), then the density of primes 𝔭 of K such th
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28

Banerjee, Kumarjit, Satyendra Nath Mandal, and Sanjoy Kumar Das. "A Comparative Study of Different Techniques for Prime Testing in Implementation of RSA." American Journal of Advanced Computing 1, no. 1 (2020): 1–7. http://dx.doi.org/10.15864/ajac.1102.

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The RSA cryptosystem, invented by Ron Rivest, Adi Shamir and Len Adleman was first publicized in the August 1977 issue of Scientific American. The security level of this algorithm very much depends on two large prime numbers. The large primes have been taken by BigInteger in Java. An algorithm has been proposed to calculate the exact square root of the given number. Three methods have been used to check whether a given number is prime or not. In trial division approach, a number has to be divided from 2 to the half the square root of the number. The number will be not prime if it gives any fac
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29

Warren, Benjamin Lee. "Primes and Their Connection to Certain Polyhedral Number Sequences." Journal of Mathematics 2021 (August 9, 2021): 1–11. http://dx.doi.org/10.1155/2021/9956343.

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30

Kan, J. "On the upper bound for the number of primes or almost primes in a given integer sequence." Periodica Mathematica Hungarica 22, no. 1 (1991): 61–69. http://dx.doi.org/10.1007/bf02309110.

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31

Mousavi, Hamed. "Pointwise ergodic theorems for nonconventional bilinear polynomial averages along prime orbits." Project Repository Journal 21, no. 1 (2024): 140–44. http://dx.doi.org/10.54050/prj2122480.

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Pointwise ergodic theorems for nonconventional bilinear polynomial averages along prime orbits Following our proof of the first Goldbach result in number field setting after 60 years (Mitsui 1960s), We give new information on density ternary Goldbach problem after 10 years (Shao 2013), and also design a new strategy to prove density version of representing a number as sum of “certain” primes. We also apply Rosser Sieve arguments to prove represtation of integers into sum of primes coming from a zero density subset. From Ergodic theory side, we give the first pointwise bilinear theorem along th
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32

Woodford, Roger. "On Partitions into Powers of Primes and Their Difference Functions." Canadian Journal of Mathematics 61, no. 2 (2009): 465–80. http://dx.doi.org/10.4153/cjm-2009-024-6.

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Abstract.In this paper, we extend the approach first outlined by Hardy and Ramanujan for calculating the asymptotic formulae for the number of partitions into r-th powers of primes, pP(r) (n), to include their difference functions. In doing so, we rectify an oversight of said authors, namely that the first difference function is perforce positive for all values of n, and include the magnitude of the error term.
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33

INTERLANDO, J. CARMELO, JOSÉ OTHON DANTAS LOPES, and TRAJANO PIRES DA NÓBREGA NETO. "THE DISCRIMINANT OF ABELIAN NUMBER FIELDS." Journal of Algebra and Its Applications 05, no. 01 (2006): 35–41. http://dx.doi.org/10.1142/s0219498806001636.

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A formula for computing the discriminant of any Abelian number field K is given. It is presented as a function of the conductor m of K and of the degrees of the fields K ∩ ℚ(ζpα) over ℚ, where p runs through the set of primes that divide m, and pα is the greatest power of p that divides m.
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34

Allakov, Ismail, and Nargiza Saydullayevna Muzropova. "The solution of some equation in primes." Chebyshevskii Sbornik 25, no. 4 (2025): 5–26. https://doi.org/10.22405/2226-8383-2024-25-4-5-26.

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The work proves that, under some additional conditions, every sufficiently large integer can be represented as the sum of two prime numbers and the square of a third prime number. A lower bound for the quantity of representation of a given integer in the specified form has also been proven.
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35

Kourbatov, Alexei, and Marek Wolf. "Predicting Maximal Gaps in Sets of Primes." Mathematics 7, no. 5 (2019): 400. http://dx.doi.org/10.3390/math7050400.

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Let q > r ≥ 1 be coprime integers. Let P c = P c ( q , r , H ) be an increasing sequence of primes p satisfying two conditions: (i) p ≡ r (mod q) and (ii) p starts a prime k-tuple with a given pattern H. Let π c ( x ) be the number of primes in P c not exceeding x. We heuristically derive formulas predicting the growth trend of the maximal gap G c ( x ) = max p ′ ≤ x ( p ′ − p ) between successive primes p , p ′ ∈ P c. Extensive computations for primes up to 10 14 show that a simple trend formula G c ( x ) ∼ x π c ( x ) · ( log π c ( x ) + O k ( 1 ) ) works well for maximal gaps between ini
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36

He, Zilong. "Inequalities for inert primes and their applications." International Journal of Number Theory 16, no. 08 (2020): 1819–32. http://dx.doi.org/10.1142/s1793042120500943.

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For any given non-square integer [Formula: see text], we prove Euclid’s type inequalities for the sequence [Formula: see text] of all primes satisfying the Kronecker symbol [Formula: see text], [Formula: see text] and give a new criterion on a ternary quadratic form to be irregular as an application, which simplifies Dickson and Jones’s argument in the classification of regular ternary quadratic forms to some extent.
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37

Ramakrishnan, B., and Brundaban Sahu. "On the Fourier expansions of Jacobi forms of half-integral weight." International Journal of Mathematics and Mathematical Sciences 2006 (2006): 1–11. http://dx.doi.org/10.1155/ijmms/2006/14726.

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Using the relationship between Jacobi forms of half-integral weight and vector valued modular forms, we obtain the number of components which determine the given Jacobi form of indexp,p2orpq, wherepandqare odd primes.
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38

CHAN, TSZ HO. "HIGHER MOMENTS OF PRIMES IN SHORT INTERVALS II." International Journal of Number Theory 01, no. 02 (2005): 207–14. http://dx.doi.org/10.1142/s1793042105000169.

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Given good knowledge on the even moments, we derive asymptotic formulas for λth moments of primes in short intervals and prove "equivalence" result on odd moments. We also provide numerical evidence in support of these results.
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39

BARADI, RAJU, SRINIVAS BOGA, SRINIVAS BOGA, NARESH THOTI, and RAMAKRISHNA GATTADI. "ON THE COUNTING FUNCTION OF SEMIPRIMES." Turkish Journal of Computer and Mathematics Education (TURCOMAT) 9, no. 3 (2018): 1282–90. http://dx.doi.org/10.61841/turcomat.v9i3.14471.

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A semiprime is a natural number which can be written as the product of two primes. The asymptotic behaviour of the function π2(x), the number of semiprimes less than or equal to x, is studied. Using a combinatorial argument, asymptotic series of π2(x) is determined, with all the terms explicitly given. An algorithm for the calculation of the constants involved in the asymptotic series is presented and the constants are computed to 20 significant digits. The errors of the partial sums of the asymptotic series are investigated. A generalization of this approach to products of k primes, for k ≥ 3
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40

Goldston, D. A., and C. Y. Yildirim. "Primes in Short Segments of Arithmetic Progressions." Canadian Journal of Mathematics 50, no. 3 (1998): 563–80. http://dx.doi.org/10.4153/cjm-1998-031-9.

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AbstractConsider the variance for the number of primes that are both in the interval [y,y + h] for y ∈ [x,2x] and in an arithmetic progression of modulus q. We study the total variance obtained by adding these variances over all the reduced residue classes modulo q. Assuming a strong form of the twin prime conjecture and the Riemann Hypothesis one can obtain an asymptotic formula for the total variance in the range when 1 ≤ h/q ≤ x1/2-∈ , for any ∈ > 0. We show that one can still obtain some weaker asymptotic results assuming the Generalized Riemann Hypothesis (GRH) in place of the twin pri
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41

Broughan, Kevin A. "Adic Topologies for the Rational Integers." Canadian Journal of Mathematics 55, no. 4 (2003): 711–23. http://dx.doi.org/10.4153/cjm-2003-030-3.

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AbstractA topology on ℤ, which gives a nice proof that the set of prime integers is infinite, is characterised and examined. It is found to be homeomorphic to ℚ, with a compact completion homeomorphic to the Cantor set. It has a natural place in a family of topologies on ℤ, which includes the p-adics, and one in which the set of rational primes ℙ is dense. Examples from number theory are given, including the primes and squares, Fermat numbers, Fibonacci numbers and k-free numbers.
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42

THORNER, JESSE. "A variant of the Bombieri–Vinogradov theorem in short intervals and some questions of Serre." Mathematical Proceedings of the Cambridge Philosophical Society 161, no. 1 (2016): 53–63. http://dx.doi.org/10.1017/s0305004116000050.

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AbstractWe generalise the classical Bombieri–Vinogradov theorem for short intervals to a non-abelian setting. This leads to variants of the prime number theorem for short intervals where the primes lie in arithmetic progressions that are “twisted” by a splitting condition in a Galois extension of number fields. Using this result in conjunction with the recent work of Maynard, we prove that rational primes with a given splitting condition in a Galois extensionL/$\mathbb{Q}$exhibit bounded gaps in short intervals. We explore several arithmetic applications related to questions of Serre regarding
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43

Aiazzi, Bruno, Stefano Baronti, Leonardo Santurri, and Massimo Selva. "An Investigation on the Prime and Twin Prime Number Functions by Periodical Binary Sequences and Symmetrical Runs in a Modified Sieve Procedure." Symmetry 11, no. 6 (2019): 775. http://dx.doi.org/10.3390/sym11060775.

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In this work, the Sieve of Eratosthenes procedure (in the following named Sieve procedure) is approached by a novel point of view, which is able to give a justification of the Prime Number Theorem (P.N.T.). Moreover, an extension of this procedure to the case of twin primes is formulated. The proposed investigation, which is named Limited INtervals into PEriodical Sequences (LINPES) relies on a set of binary periodical sequences that are evaluated in limited intervals of the prime characteristic function. These sequences are built by considering the ensemble of deleted (that is, 0) and undelet
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44

Changa, M. E. "On the number of primes yielding square-free sums with given numbers." Russian Mathematical Surveys 58, no. 3 (2003): 613–14. http://dx.doi.org/10.1070/rm2003v058n03abeh000637.

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45

Acedo, Luis. "On the General Divergent Arithmetic Sums over the Primes and the Symmetries of Riemann’s Zeta Function." Symmetry 16, no. 8 (2024): 970. http://dx.doi.org/10.3390/sym16080970.

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In this paper, we address the problem of the divergent sums of general arithmetic functions over the set of primes. In classical analytic number theory, the sum of the logarithm of the prime numbers plays a crucial role. We consider the sums of powers of the logarithm of primes and its connection with Riemann’s zeta function (z.f.). This connection is achieved through the second Chebyshev function of order n, which can be estimated by exploiting the symmetry properties of Riemann’s zeta function. Finally, a heuristic approach to evaluating more general sums is also given.
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46

Cha, Byungchul. "Chebyshev’s bias in function fields." Compositio Mathematica 144, no. 6 (2008): 1351–74. http://dx.doi.org/10.1112/s0010437x08003631.

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AbstractWe study a function field analog of Chebyshev’s bias. Our results, as well as their proofs, are similar to those of Rubinstein and Sarnak in the case of the rational number field. Following Rubinstein and Sarnak, we introduce the grand simplicity hypothesis (GSH), a certain hypothesis on the inverse zeros of Dirichlet L-series of a polynomial ring over a finite field. Under this hypothesis, we investigate how primes, that is, irreducible monic polynomials in a polynomial ring over a finite field, are distributed in a given set of residue classes modulo a fixed monic polynomial. In part
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47

Lourenco, Stella, and Vladislav Ayzenberg. "Number subliminally primes area judgments: Novel evidence for a general magnitude system in human adults." Journal of Vision 16, no. 12 (2016): 1279. http://dx.doi.org/10.1167/16.12.1279.

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48

De Koninck, Jean-Marie, and Imre Kátai. "The distribution of additive functions in short intervals on the set of shifted primes having a fixed number of prime factors." Annales Universitatis Scientiarum Budapestinensis de Rolando Eötvös Nominatae. Sectio computatorica, no. 38 (2012): 57–70. https://doi.org/10.71352/ac.38.057.

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Given a strongly additive function f, we establish short interval estimates for f on the set of shifted primes. We also consider similar sums, but running on sets of integers m + 1, where each integer m has a fixed number of prime factors.
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49

Li, Meng. "Diophantine Approximation with Prime Variables and Mixed Powers." Journal of Innovation and Development 10, no. 2 (2025): 63–70. https://doi.org/10.54097/dpg8r633.

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Let , , , , be non-zero real numbers, not all negative, and let be irrational and algebraic. Let be a well-spaced sequence and .We prove that for any given ,the number of satisfying and for which has no solution in primes , , , , does not exceed .
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50

Oktaviani, Dinni Rahma, Muhammad Habiburrohman, and Fiki Syaban Nugroho. "ALTERNATIVE PROOF OF THE INFINITUDE PRIMES AND PRIME PROPERTIES." BAREKENG: Jurnal Ilmu Matematika dan Terapan 17, no. 1 (2023): 0475–80. http://dx.doi.org/10.30598/barekengvol17iss1pp0475-0480.

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Prime numbers is one of kind number that have many uses, one of which is cryptography. The uniqueness of prime numbers in their divisors and distributions causes prime numbers to be widely used in digital security systems. In number theory, one of famous theorem is Euclid theorem. Euclid theorem says about infinitely of prime numbers. Many alternative proof has been given by mathematician to find new theory or approximation of prime properties. The construction of proof give new idea about properties of prime number. So, in this study, we will give an alternative proof of Euclid theorem and in
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