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Published: 13 July 2024

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1

Mignot, Teddy. "Points de hauteur bornée sur les hypersurfaces lisses de l'espace triprojectif." International Journal of Number Theory 11, no. 03 (March 31, 2015): 945–95. http://dx.doi.org/10.1142/s1793042115500529.

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Nous démontrons ici la conjecture de Batyrev et Manin pour le nombre de points de hauteur bornée de certaines hypersurfaces de l'espace triprojectif de tridegré (1, 1, 1). La constante intervenant dans le résultat final est celle conjecturée par Peyre. La méthode utilisée est inspirée de celle développée par Schindler pour traiter le cas des hypersurfaces des espaces biprojectifs. Celle-ci est essentiellement basée sur la méthode du cercle de Hardy–Littlewood. We prove the Batyrev–Manin conjecture for the number of points of bounded height on some smooth hypersurfaces of the triprojective space of tridegree (1, 1, 1). The constant appearing in the final result is the one conjectured by Peyre. The method used is the one developed by Schindler to study the case of hypersurfaces of biprojective spaces. It is essentially based on the Hardy–Littlewood method.

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2

Haglund, J., J. Morse, and M. Zabrocki. "A Compositional Shuffle Conjecture Specifying Touch Points of the Dyck Path." Canadian Journal of Mathematics 64, no. 4 (August 1, 2012): 822–44. http://dx.doi.org/10.4153/cjm-2011-078-4.

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Abstract We introduce a q, t-enumeration of Dyck paths that are forced to touch the main diagonal at specific points and forbidden to touch elsewhere and conjecture that it describes the action of the Macdonald theory ∇ operator applied to a Hall–Littlewood polynomial. Our conjecture refines several earlier conjectures concerning the space of diagonal harmonics including the “shuffle conjecture” (Duke J. Math. 126 (2005), pp. 195 − 232) for ∇ en[X]. We bring to light that certain generalized Hall–Littlewood polynomials indexed by compositions are the building blocks for the algebraic combinatorial theory of q, t-Catalan sequences, and we prove a number of identities involving these functions.

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3

CHAN, TSZ HO. "A NOTE ON PRIMES IN SHORT INTERVALS." International Journal of Number Theory 02, no. 01 (March 2006): 105–10. http://dx.doi.org/10.1142/s1793042106000437.

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Montgomery and Soundararajan obtained evidence for the Gaussian distribution of primes in short intervals assuming a quantitative Hardy–Littlewood conjecture. In this article, we show that their methods may be modified and an average form of the Hardy–Littlewood conjecture suffices.

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4

Xuan Jiang, Chun. "On the singular series in the Jiang prime k-tuple theorem." Physics & Astronomy International Journal 2, no. 6 (November 16, 2018): 514–17. http://dx.doi.org/10.15406/paij.2018.02.00134.

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Using Jiang function we prove Jiang prime k-tuple theorem. We find true singular series. Using the examples we prove the Hardy-Littlewood prime k-tuple conjecture with wrong singular series. Jiang prime k-tuple theorem will replace the Hardy-Littlewood prime k-tuple conjecture.

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5

Chindris, Calin, Harm Derksen, and Jerzy Weyman. "Counterexamples to Okounkov’s log-concavity conjecture." Compositio Mathematica 143, no. 6 (November 2007): 1545–57. http://dx.doi.org/10.1112/s0010437x07003090.

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6

de Mathan, Bernard. "Conjecture de Littlewood et récurrences linéaires." Journal de Théorie des Nombres de Bordeaux 15, no. 1 (2003): 249–66. http://dx.doi.org/10.5802/jtnb.401.

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7

Brüdern, Jörg. "On Waring's problem for cubes." Mathematical Proceedings of the Cambridge Philosophical Society 109, no. 2 (March 1991): 229–56. http://dx.doi.org/10.1017/s0305004100069711.

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A classical conjecture in the additive theory of numbers is that all sufficiently large natural numbers may be written as the sum of four positive cubes of integers. This is known as the Four Cubes Problem, and since the pioneering work of Hardy and Littlewood one expects a much more precise quantitative form of the conjecture to hold. Let v(n) be the number of representations of n in the proposed manner. Then the expected formula takes the shapewhere (n) is the singular series associated with four cubes as familiar in the Hardy–Littlewood theory.

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8

BUGEAUD, YANN, ALAN HAYNES, and SANJU VELANI. "METRIC CONSIDERATIONS CONCERNING THE MIXED LITTLEWOOD CONJECTURE." International Journal of Number Theory 07, no. 03 (May 2011): 593–609. http://dx.doi.org/10.1142/s1793042111004289.

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The main goal of this paper is to develop a metrical theory of Diophantine approximation within the framework of the de Mathan–Teulié Conjecture — also known as the "Mixed Littlewood Conjecture". Let p be a prime. A consequence of our main result is that, for almost every real number α, [Formula: see text]

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9

Krass, S. "On a conjecture of Littlewood in Diophantine approximations." Bulletin of the Australian Mathematical Society 32, no. 3 (December 1985): 379–87. http://dx.doi.org/10.1017/s0004972700002495.

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A conjecture of Littlewood States that for arbitrary , and any ε > 0 there exist m0 ≠ 0, m1,…,mn so that . In this paper we show this conjecture holds for all ξ̲ = (ξ1,…,ξn) such that 1, ξ1,…,ξn is a rational bass of a real algebraic number field of degree n+1.

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10

Klemes, Ivo. "A Note on Hardy's Inequality." Canadian Mathematical Bulletin 36, no. 4 (December 1, 1993): 442–48. http://dx.doi.org/10.4153/cmb-1993-059-4.

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11

Bugeaud, Yann. "Around the Littlewood conjecture in Diophantine approximation." Publications Mathématiques de Besançon, no. 1 (April 13, 2015): 5–18. http://dx.doi.org/10.5802/pmb.1.

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12

Harpaz, Yonatan, Alexei N. Skorobogatov, and Olivier Wittenberg. "The Hardy–Littlewood conjecture and rational points." Compositio Mathematica 150, no. 12 (September 10, 2014): 2095–111. http://dx.doi.org/10.1112/s0010437x14007568.

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AbstractSchinzel’s Hypothesis (H) was used by Colliot-Thélène and Sansuc, and later by Serre, Swinnerton-Dyer and others, to prove that the Brauer–Manin obstruction controls the Hasse principle and weak approximation on pencils of conics and similar varieties. We show that when the ground field is $\mathbb{Q}$ and the degenerate geometric fibres of the pencil are all defined over $\mathbb{Q}$, one can use this method to obtain unconditional results by replacing Hypothesis (H) with the finite complexity case of the generalised Hardy–Littlewood conjecture recently established by Green, Tao and Ziegler.

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13

Essouabri, Driss. "Preuve d'une conjecture de Hardy et Littlewood." Comptes Rendus de l'Académie des Sciences - Series I - Mathematics 328, no. 7 (April 1999): 557–62. http://dx.doi.org/10.1016/s0764-4442(99)80246-8.

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14

Fei, JinHua. "An application of the Hardy–Littlewood conjecture." Journal of Number Theory 168 (November 2016): 39–44. http://dx.doi.org/10.1016/j.jnt.2016.05.001.

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15

Haynes, Alan, and Henna Koivusalo. "A randomized version of the Littlewood Conjecture." Journal of Number Theory 178 (September 2017): 201–7. http://dx.doi.org/10.1016/j.jnt.2017.02.017.

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16

Vinogradov, A. I. "The Hardy-Littlewood conjecture. An algebraic approach." Journal of Mathematical Sciences 79, no. 5 (May 1996): 1273–76. http://dx.doi.org/10.1007/bf02366456.

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17

Haynes, Alan, Henna Koivusalo, and James Walton. "Perfectly ordered quasicrystals and the Littlewood conjecture." Transactions of the American Mathematical Society 370, no. 7 (February 8, 2018): 4975–92. http://dx.doi.org/10.1090/tran/7136.

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18

Ferolito, Clarice. "An unresolved analogue of the Littlewood Conjecture." Involve, a Journal of Mathematics 3, no. 2 (August 11, 2010): 191–96. http://dx.doi.org/10.2140/involve.2010.3.191.

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19

BAIER, STEPHAN, and LIANGYI ZHAO. "ON PRIMES IN QUADRATIC PROGRESSIONS." International Journal of Number Theory 05, no. 06 (September 2009): 1017–35. http://dx.doi.org/10.1142/s1793042109002523.

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20

Bugeaud, Yann, Michael Drmota, and Bernard de Mathan. "On a mixed Littlewood conjecture in Diophantine approximation." Acta Arithmetica 128, no. 2 (2007): 107–24. http://dx.doi.org/10.4064/aa128-2-2.

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21

de Mathan, Bernard. "On a mixed Littlewood conjecture for quadratic numbers." Journal de Théorie des Nombres de Bordeaux 17, no. 1 (2005): 207–15. http://dx.doi.org/10.5802/jtnb.487.

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22

ADAMCZEWSKI, BORIS, and YANN BUGEAUD. "ON THE LITTLEWOOD CONJECTURE IN SIMULTANEOUS DIOPHANTINE APPROXIMATION." Journal of the London Mathematical Society 73, no. 02 (April 2006): 355–66. http://dx.doi.org/10.1112/s0024610706022617.

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23

Badziahin, Dzmitry. "Computation of the Infimum in the Littlewood Conjecture." Experimental Mathematics 25, no. 1 (October 12, 2015): 100–105. http://dx.doi.org/10.1080/10586458.2015.1031356.

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24

Bary-Soroker, Lior. "Hardy–Littlewood Tuple Conjecture Over Large Finite Fields." International Mathematics Research Notices 2014, no. 2 (November 16, 2012): 568–75. http://dx.doi.org/10.1093/imrn/rns249.

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25

Harrap, Stephen, and Alan Haynes. "The mixed Littlewood conjecture for pseudo-absolute values." Mathematische Annalen 357, no. 3 (March 26, 2013): 941–60. http://dx.doi.org/10.1007/s00208-013-0928-z.

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26

Giordano, George. "On the irregularity of the distribution of the sums of pairs of odd primes." International Journal of Mathematics and Mathematical Sciences 30, no. 6 (2002): 377–81. http://dx.doi.org/10.1155/s0161171202110325.

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LetP2(n)denote the number of ways of writingnas a sum of two odd primes. We support a conjecture of Hardy and Littlewood concerningP2(n)by showing that it holds in a certain average sense. Thereby showing the irregularity ofP2(n).

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27

Zhou, Mi. "The number of 8n+1 primes compare with 8n-1 primes." Advances in Engineering Technology Research 1, no. 1 (May 17, 2022): 189. http://dx.doi.org/10.56028/aetr.1.1.189.

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There is a phenomenon in mathematics: there is a phenomenon that before a natural number K, primes of the form 4n+1 do not appear more frequently than 4n-1 primes; Beyond k, between k and k+k', the above phenomenon is reversed, the frequency of 4n+1 primes is not less than 4n-1 primes; After exceeding k+k', between k+k 'and k+k'+k'', it is reversed again ...... The J.E. Littlewood proved the first stage of the phenomenon: primes of the form 4n+1 appear no more frequently than 4n-1 primes before a natural number k. In this paper used a more easy method and directly prove the phenomenon very shortly , provides a theoretical proof for this description.This method is more easy directly and elementary than Littlewood’,and It can help people understand this phenomenon better, and at the same time, it provides a good example for the optimization of number theory research methods and the use of some elementary methods to study mathematical problems. At the same time, there is a generalization conjecture: before a natural number K, which of 8n+1 and 8N-1 primes appear more frequently? The conjecture remains unsolved. Littlewood proved the occurrence frequency theorem of 4n+1 primes and 4N-1 primes, and this paper also gave the proof, the method is different from Littlewood, but he was the first; However, for 8n+1 primes compare with 8n-1 primes, we prove for the first time that the result is same as 4n+1 primes compare with 4n-1 primes.

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28

GOLDSTON, D. A., and A. H. LEDOAN. "JUMPING CHAMPIONS AND GAPS BETWEEN CONSECUTIVE PRIMES." International Journal of Number Theory 07, no. 06 (September 2011): 1413–21. http://dx.doi.org/10.1142/s179304211100471x.

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The most common difference that occurs among the consecutive primes less than or equal to x is called a jumping champion. Occasionally there are ties. Therefore there can be more than one jumping champion for a given x. In 1999 Odlyzko, Rubinstein and Wolf provided heuristic and empirical evidence in support of the conjecture that the numbers greater than 1 that are jumping champions are 4 and the primorials 2, 6, 30, 210, 2310,…. As a step toward proving this conjecture they introduced a second weaker conjecture that any fixed prime p divides all sufficiently large jumping champions. In this paper we extend a method of Erdős and Straus from 1980 to prove that the second conjecture follows directly from the prime pair conjecture of Hardy and Littlewood.

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29

Badziahin, D. "On $t$-adic Littlewood conjecture for certain infinite products." Proceedings of the American Mathematical Society 149, no. 11 (August 6, 2021): 4527–40. http://dx.doi.org/10.1090/proc/15475.

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30

Ayadi, Khalil. "A note on the Littlewood conjecture in positive characteristic." Quaestiones Mathematicae 43, no. 1 (January 15, 2019): 107–16. http://dx.doi.org/10.2989/16073606.2018.1539047.

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31

Krenedits, S. "On mockenhoupt’s conjecture in the Hardy-Littlewood majorant problem." Journal of Contemporary Mathematical Analysis 48, no. 3 (May 2013): 91–109. http://dx.doi.org/10.3103/s1068362313030011.

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32

Chun-Xuan, Jiang. "The Hardy-Littlewood Prime K-Tuple Conjecture Is False." Journal of Middle East and North Africa Sciences 2, no. 7 (2016): 5–10. http://dx.doi.org/10.12816/0032684.

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33

Táfula, Christian. "An elementary heuristic for Hardy–Littlewood extended Goldbach’s conjecture." São Paulo Journal of Mathematical Sciences 14, no. 1 (August 27, 2019): 391–405. http://dx.doi.org/10.1007/s40863-019-00146-3.

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34

Vlamos, Panayiotis. "Properties of the functionf(x)=x/π(x)." International Journal of Mathematics and Mathematical Sciences 28, no. 5 (2001): 307–11. http://dx.doi.org/10.1155/s0161171201005725.

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We obtain the asymptotic estimations for∑k=2nf(k)and∑k=2n1/f(k), wheref(k)=k/π(k),k≥2. We study the expression2f(x+y)−f(x)−f(y)for integersx,y≥2and as an application we make several remarks in connection with the conjecture of Hardy and Littlewood.

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35

Hong, Shao Fang, and Wei Cao. "Notes on the Borwein-Choi conjecture of Littlewood cyclotomic polynomials." Acta Mathematica Sinica, English Series 25, no. 1 (November 17, 2008): 65–76. http://dx.doi.org/10.1007/s10114-008-6444-5.

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36

Technau, Niclas, and Agamemnon Zafeiropoulos. "The Discrepancy of (nkx)k=1∞ With Respect to Certain Probability Measures." Quarterly Journal of Mathematics 71, no. 2 (March 12, 2020): 573–97. http://dx.doi.org/10.1093/qmathj/haz058.

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Abstract Let $(n_k)_{k=1}^{\infty }$ be a lacunary sequence of integers. We show that if $\mu$ is a probability measure on $[0,1)$ such that $|\widehat{\mu }(t)|\leq c|t|^{-\eta }$, then for $\mu$-almost all $x$, the discrepancy $D_N(n_kx)$ satisfies $$\begin{equation*}\frac{1}{4} \leq \limsup_{N\to\infty}\frac{N D_N(n_kx)}{\sqrt{N\log\log N}} \leq C\end{equation*}$$for some constant $C>0$. This proves a conjecture of Haynes, Jensen and Kristensen and allows an improvement on their previous result relevant to an inhomogeneous version of the Littlewood conjecture.

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37

Sándor, József. "On certain inequalities for the prime counting function – Part III." Notes on Number Theory and Discrete Mathematics 29, no. 3 (July 3, 2023): 454–61. http://dx.doi.org/10.7546/nntdm.2023.29.3.454-461.

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As a continuation of [10] and [11], we offer some new inequalities for the prime counting function $\pi (x).$ Particularly, a multiplicative analogue of the Hardy–Littlewood conjecture is provided. Improvements of the converse of Landau's inequality are given. Some results on $\pi (p_n^2)$ are offered, $p_n$ denoting the $n$-th prime number. Results on $\pi (\pi (x))$ are also considered.

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38

Graham, William, and Markus Hunziker. "Multiplication of Polynomials on Hermitian Symmetric spaces and Littlewood–Richardson Coefficients." Canadian Journal of Mathematics 61, no. 2 (April 1, 2009): 351–72. http://dx.doi.org/10.4153/cjm-2009-018-2.

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Abstract. Let K be a complex reductive algebraic group and V a representation of K. Let S denote the ring of polynomials on V. Assume that the action of K on S is multiplicity-free. If ƛ denotes the isomorphism class of an irreducible representation of K, let ρƛ : K → GL(Vƛ) denote the corresponding irreducible representation and Sƛ the ƛ-isotypic component of S. Write Sƛ ・ Sμ for the subspace of S spanned by products of Sƛ and Sμ. If Vν occurs as an irreducible constituent of Vƛ ⊗ Vμ, is it true that Sν ⊆ Sƛ ・ Sμ? In this paper, the authors investigate this question for representations arising in the context of Hermitian symmetric pairs. It is shown that the answer is yes in some cases and, using an earlier result of Ruitenburg, that in the remaining classical cases, the answer is yes provided that a conjecture of Stanley on the multiplication of Jack polynomials is true. It is also shown how the conjecture connects multiplication in the ring S to the usual Littlewood–Richardson rule.

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39

Venkatesh, Akshay. "The work of Einsiedler, Katok and Lindenstrauss on the Littlewood conjecture." Bulletin of the American Mathematical Society 45, no. 01 (October 29, 2007): 117–35. http://dx.doi.org/10.1090/s0273-0979-07-01194-9.

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40

Bengoechea, Paloma, and Evgeniy Zorin. "On the Mixed Littlewood Conjecture and continued fractions in quadratic fields." Journal of Number Theory 162 (May 2016): 1–10. http://dx.doi.org/10.1016/j.jnt.2015.10.007.

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41

Gopalakrishna Gadiyar, H., and Ramanathan Padma. "Ramanujan-Fourier series and the conjecture D of Hardy and Littlewood." Czechoslovak Mathematical Journal 64, no. 1 (March 2014): 251–67. http://dx.doi.org/10.1007/s10587-014-0098-5.

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42

Cho, Nak, Virendra Kumar, and Ji Park. "The Coefficients of Powers of Bazilević Functions." Mathematics 6, no. 11 (November 18, 2018): 263. http://dx.doi.org/10.3390/math6110263.

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In the present work, a sharp bound on the modulus of the initial coefficients for powers of strongly Bazilević functions is obtained. As an application of these results, certain conditions are investigated under which the Littlewood-Paley conjecture holds for strongly Bazilević functions for large values of the parameters involved therein. Further, sharp estimate on the generalized Fekete-Szegö functional is also derived. Relevant connections of our results with the existing ones are also made.

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43

GREEN, BEN, TERENCE TAO, and TAMAR ZIEGLER. "AN INVERSE THEOREM FOR THE GOWERS U4-NORM." Glasgow Mathematical Journal 53, no. 1 (August 25, 2010): 1–50. http://dx.doi.org/10.1017/s0017089510000546.

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AbstractWe prove the so-called inverse conjecture for the Gowers Us+1-norm in the case s = 3 (the cases s < 3 being established in previous literature). That is, we show that if f : [N] → ℂ is a function with |f(n)| ≤ 1 for all n and ‖f‖U4 ≥ δ then there is a bounded complexity 3-step nilsequence F(g(n)Γ) which correlates with f. The approach seems to generalise so as to prove the inverse conjecture for s ≥ 4 as well, and a longer paper will follow concerning this.By combining the main result of the present paper with several previous results of the first two authors one obtains the generalised Hardy–Littlewood prime-tuples conjecture for any linear system of complexity at most 3. In particular, we have an asymptotic for the number of 5-term arithmetic progressions p1 < p2 < p3 < p4 < p5 ≤ N of primes.

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44

Sababheh, Mohammad. "Hardy Inequalities on the Real Line." Canadian Mathematical Bulletin 54, no. 1 (March 1, 2011): 159–71. http://dx.doi.org/10.4153/cmb-2010-091-8.

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AbstractWe prove that some inequalities, which are considered to be generalizations of Hardy's inequality on the circle, can be modified and proved to be true for functions integrable on the real line. In fact we would like to show that some constructions that were used to prove the Littlewood conjecture can be used similarly to produce real Hardy-type inequalities. This discussion will lead to many questions concerning the relationship between Hardy-type inequalities on the circle and those on the real line.

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45

Badziahin, Dmitry, Yann Bugeaud, Manfred Einsiedler, and Dmitry Kleinbock. "On the complexity of a putative counterexample to the -adic Littlewood conjecture." Compositio Mathematica 151, no. 9 (May 19, 2015): 1647–62. http://dx.doi.org/10.1112/s0010437x15007393.

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Let $\Vert \cdot \Vert$ denote the distance to the nearest integer and, for a prime number $p$, let $|\cdot |_{p}$ denote the $p$-adic absolute value. Over a decade ago, de Mathan and Teulié [Problèmes diophantiens simultanés, Monatsh. Math. 143 (2004), 229–245] asked whether $\inf _{q\geqslant 1}$$q\cdot \Vert q{\it\alpha}\Vert \cdot |q|_{p}=0$ holds for every badly approximable real number ${\it\alpha}$ and every prime number $p$. Among other results, we establish that, if the complexity of the sequence of partial quotients of a real number ${\it\alpha}$ grows too rapidly or too slowly, then their conjecture is true for the pair $({\it\alpha},p)$ with $p$ an arbitrary prime.

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46

LIU, WENCAI. "SOME REFINED RESULTS ON THE MIXED LITTLEWOOD CONJECTURE FOR PSEUDO-ABSOLUTE VALUES." Journal of the Australian Mathematical Society 107, no. 1 (August 22, 2018): 91–109. http://dx.doi.org/10.1017/s1446788718000198.

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In this paper, we study the mixed Littlewood conjecture with pseudo-absolute values. For any pseudo-absolute-value sequence ${\mathcal{D}}$, we obtain a sharp criterion such that for almost every $\unicode[STIX]{x1D6FC}$ the inequality $$\begin{eqnarray}|n|_{{\mathcal{D}}}|n\unicode[STIX]{x1D6FC}-p|\leq \unicode[STIX]{x1D713}(n)\end{eqnarray}$$ has infinitely many coprime solutions $(n,p)\in \mathbb{N}\times \mathbb{Z}$ for a certain one-parameter family of $\unicode[STIX]{x1D713}$. Also, under a minor condition on pseudo-absolute-value sequences ${\mathcal{D}}_{1},{\mathcal{D}}_{2},\ldots ,{\mathcal{D}}_{k}$, we obtain a sharp criterion on a general sequence $\unicode[STIX]{x1D713}(n)$ such that for almost every $\unicode[STIX]{x1D6FC}$ the inequality $$\begin{eqnarray}|n|_{{\mathcal{D}}_{1}}|n|_{{\mathcal{D}}_{2}}\cdots |n|_{{\mathcal{D}}_{k}}|n\unicode[STIX]{x1D6FC}-p|\leq \unicode[STIX]{x1D713}(n)\end{eqnarray}$$ has infinitely many coprime solutions $(n,p)\in \mathbb{N}\times \mathbb{Z}$.

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47

Pellegrino, Daniel, and Eduardo V. Teixeira. "Towards sharp Bohnenblust–Hille constants." Communications in Contemporary Mathematics 20, no. 03 (February 21, 2018): 1750029. http://dx.doi.org/10.1142/s0219199717500298.

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We investigate the optimality problem associated with the best constants in a class of Bohnenblust–Hille-type inequalities for [Formula: see text]-linear forms. While germinal estimates indicated an exponential growth, in this work we provide strong evidences to the conjecture that the sharp constants in the classical Bohnenblust–Hille inequality are universally bounded, irrespectively of the value of [Formula: see text]; hereafter referred as the Universality Conjecture. In our approach, we introduce the notions of entropy and complexity, designed to measure, to some extent, the complexity of such optimization problems. We show that the notion of entropy is critically connected to the Universality Conjecture; for instance, that if the entropy grows at most exponentially with respect to [Formula: see text], then the optimal constants of the [Formula: see text]-linear Bohnenblust–Hille inequality for real scalars are indeed bounded universally with respect to [Formula: see text]. It is likely that indeed the entropy grows as [Formula: see text], and in this scenario, we show that the optimal constants are precisely [Formula: see text]. In the bilinear case, [Formula: see text], we show that any extremum of the Littlewood’s [Formula: see text] inequality has entropy [Formula: see text] and complexity [Formula: see text], and thus we are able to classify all extrema of the problem. We also prove that, for any mixed [Formula: see text]-Littlewood inequality, the entropy do grow exponentially and the sharp constants for such a class of inequalities are precisely [Formula: see text]. In addition to the notions of entropy and complexity, the approach we develop in this work makes decisive use of a family of strongly non-symmetric [Formula: see text]-linear forms, which has further consequences to the theory, as we explain herein.

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48

Gross, Robert, and John H. Smith. "A Generalization of a Conjecture of Hardy and Littlewood to Algebraic Number Fields." Rocky Mountain Journal of Mathematics 30, no. 1 (March 2000): 195–215. http://dx.doi.org/10.1216/rmjm/1022008986.

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49

FOX, JACOB, MATTHEW KWAN, and LISA SAUERMANN. "Combinatorial anti-concentration inequalities, with applications." Mathematical Proceedings of the Cambridge Philosophical Society 171, no. 2 (June 30, 2021): 227–48. http://dx.doi.org/10.1017/s0305004120000183.

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AbstractWe prove several different anti-concentration inequalities for functions of independent Bernoulli-distributed random variables. First, motivated by a conjecture of Alon, Hefetz, Krivelevich and Tyomkyn, we prove some “Poisson-type” anti-concentration theorems that give bounds of the form 1/e + o(1) for the point probabilities of certain polynomials. Second, we prove an anti-concentration inequality for polynomials with nonnegative coefficients which extends the classical Erdős–Littlewood–Offord theorem and improves a theorem of Meka, Nguyen and Vu for polynomials of this type. As an application, we prove some new anti-concentration bounds for subgraph counts in random graphs.

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Heath-Brown, D. R. "Christopher Hooley. 7 August 1928—13 December 2018." Biographical Memoirs of Fellows of the Royal Society 69 (September 9, 2020): 225–46. http://dx.doi.org/10.1098/rsbm.2020.0027.

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Christopher Hooley was one of the leading analytic number theorists of his day, world-wide. His early work on Artin’s conjecture for primitive roots remains the definitive investigation in the area. His greatest contribution, however, was the introduction of exponential sums into every corner of analytic number theory, bringing the power of Deligne’s ‘Riemann hypothesis’ for varieties over finite fields to bear throughout the subject. For many he was a figure who bridged the classical period of Hardy and Littlewood with the modern era. This biographical sketch describes how he succeeded in applying the latest tools to famous old problems.

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