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On the Optimal Weight Function in the Goldston–Pintz–Yıldırım Method for Finding Small Gaps Between Consecutive Primes

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On the Optimal Weight Function in the Goldston–Pintz–Yıldırım Method for Finding Small Gaps Between Consecutive Primes i i János Pintz, András Biró, Kálmán Győry, Gergely Harcos, Miklós Simonovits, József Szabados (Eds.): Number Theory, Analysis, and Combinatorics — 2013/11/19 — 12:08 — page 75 i i Bálint Farkas, János Pintz, and Szilárd Révész On the optimal weight function in the Goldston–Pintz–Yıldırım method for finding small gaps between consecutive primes Abstract: We work out the optimization problem, initiated by K. Soundararajan, for the choice of the underlying polynomial used in the construction of the weight func­ tion in the Goldston–Pintz–Yıldırım method for finding small gaps between primes. First we reformulate to a maximization problem on 2[0, 1] for a self-adjoint oper­ ator , the norm of which is then the maximal eigenvalue of . To find eigenfunc­ tions and eigenvalues, we derive a differential equation which can be explicitly solved. The aimed maximal value is () = 4/(1/3 + ),achievedbythe−1st integral of 1−/2 1/3 nd −2 (1√),where1 ∼ is the first positive root of the −2 Bessel func­ tion −2. As this naturally gives rise to a number of technical problems in the appli­ cation of the GPY method, we also construct a polynomial which is a simpler func­ tion yet it furnishes an approximately optimal extremal quantity, 4/( + 1/3) with some other constant . In the forthcoming paper of J. Pintz [8] it is indeed shown how this quasi-optimal choice of the polynomial in the weight finally can exploit the GPY method to its theoretical limits. Keywords: Prime distribution, gaps between primes, Goldston–Pintz–Yıldırım method, self-adjoint Fredholm type operator, Bessel differential equation, Bessel functions of the first and second kind Classification: Primary 11N05, 47A75. Secondary 47A53, 49J05, 49K15, 49N10 || Bálint Farkas: Bergische Universität Wuppertal, Faculty of Mathematics and Natural Science, Gaußs­ traße 20, 42119 Wuppertal, Germany, E-mail: [email protected] János Pintz: Hungarian Academy of Sciences, Alfréd Rényi Institute of Mathematics, 1364 Budapest, P.O. Box 127, Hungary, E-mail: [email protected], [email protected] Szilárd Révész: Kuwait University, Department of Mathematics, P.O. Box 5969 Safat, 13060 Kuwait, E-mail: [email protected] 1 Introduction 1.1 The extremal problem as given by Soundararajan In his work [11] Soundararajan presents and analyzes the proof of Goldston–Pintz– Yıldırım yielding small gaps between primes. Among others he raises and answers one of the most important problems of the field: Is it possible to modify the weight function Bereitgestellt von | De Gruyter / TCS Angemeldet Heruntergeladen am | 16.10.19 13:23 i i i i i i János Pintz, András Biró, Kálmán Győry, Gergely Harcos, Miklós Simonovits, József Szabados (Eds.): Number Theory, Analysis, and Combinatorics — 2013/11/19 — 12:08 — page 76 i i 76 | Bálint Farkas, János Pintz, and Szilárd Révész () in such a way that the method would lead to infinitely many bounded gaps be­ tween consecutive primes. If we consider the weight functions in full generality, that is all functions (), then this leads essentially to a tautology. For example, defining () =1 if both and +2are primes, and otherwise setting () =0, the summatory function of () describes the number of twin primes up to . Thus we cannot hope for an asymptotic evaluation of the summatory function. We briefly describe the feasible H choices of the weight function . Let us take an admissible -tuple ={ℎ1,...,ℎ} H meaning that there is no prime with the property that the elements ℎ of cover all log(/) residue classes mod .LetH() ==1 ∏ ( + ℎ) and let us define = ()( log ) with a nice function , for example a polynomial, with the additional property 1 =1, which is equivalent to (1) =. 1 Afterwards we reduce our choice of () to those of type () =∑≤,|H() and try to evaluate the summatory function of () and that of ()( +ℎ),whereℎ is an arbitrary number with ℎ<logand is the characteris­ tic function of the primes. (In case of bounded gaps between primes it is sufficient to considerthecasewhenℎ=ℎ, =1,2,...,.) Soundararajan explains how the optimal weight function (),hence,should be chosen to obtain best result: see formula (8) in [11]. In order to get this optimum, he log(/) also explains the choice := ()( log ) where is some suitably nice function, like a polynomial or at least a sufficiently many times (at least times) differentiable, smooth function on [0, 1] (or at least on [0, 1)), vanishing at leastin the order at 0,and satisfying the normalization (1) =. 1 Then, according to the analysis by Soundarara­ jan, the optimal choice for () and is equivalent to looking for the maximal possi­ ble value of (12) of [11], i.e., to determining 1 1 −2 −1 2 2 () := sup ( ∫ ((−1)(1 − )) )/( ∫ (()(1 − )) ), (1) ( − 2)! ( − 1)! 0 0 where the set of functions , to be taken into account in the supremum, can be the set of certain polynomials as before, or more generally a family of functions subject to some conditions. Soundararajan [11] shows that the question whether we are able to find in this way infinitely many bounded gaps between primes is equivalent to the problem whether there exists any natural number with () > .4/ Then he mentions that the opposite inequality () < 4/ holds for all and therefore the method cannot yield infinitely many bounded prime gaps. (In an earlier unpublished note [10] he gives the short proof of this fact; we will reproduce this in Subsection 2.3. His considerations also lead easily to the stronger inequality () < 4/( + log),cf.Subsection2.3).Althoughhis work answered negatively the above mentioned central problem, it gave some hints but did not answer the question: What is the best weight function that can be chosen, and what size of gaps are implied by it? In their work [5] Goldston, Pintz and Yıldırım showed that if one takes ()+ℓ = ,where and ℓ areallowedtotendtoinfinitywith the size of the primes considered, then with several essential modifications of the Bereitgestellt von | De Gruyter / TCS Angemeldet Heruntergeladen am | 16.10.19 13:23 i i i i i i János Pintz, András Biró, Kálmán Győry, Gergely Harcos, Miklós Simonovits, József Szabados (Eds.): Number Theory, Analysis, and Combinatorics — 2013/11/19 — 12:08 — page 77 i i On the weight function in GPY for small gaps between primes | 77 original method one can reach infinitely many prime gaps of size essentially √log . (To have an idea of the difficulties it is enough to mention that the rather condensed proof of the result needs about 40 additional pages beyond the original one, presented with many details and explanations in [4]. However, a shortened, simplified and more condensed version [6] needs only 5 pages). In this case ℓ=√ and the value of the fraction (1) is 4/( + √) for the given choice of ()+ℓ = . Beyond the mentioned important fact that and ℓ are unbounded in [5], the scheme of the proof is similar but not the same as in the simplified version of Soundararajan [11]. However, a careful analysis suggests that in order to find the limits of the method it is necessary (but as discussed a little later, not necessarily sufficient) to find the size of () as tends to infinity together with the function which yields a maximum (if it exists) in the supre­ mum, or at least a function which yields a value “enough close” to the supremum. 1.2 Conditions and normalizations Before proceeding, let us discuss right here the issue of conditions and normalizations in the formulation of this maximization problem. First, it is clear that () remains un­ changed, if we add any constant to (−1). Thus the extremal problem becomes un­ bounded under addition of a free constant, hence at least some conditions must cer­ tainly control this divergence. In the number theory construction of Goldston–Pintz–Yıldırım, (by now gener­ ally abbreviated as the “GPY method”) the natural restriction is that must be a poly­ nomial divisible by – or, if we try to generalize the method, then a -times con­ tinuously differentiable function with () vanishing at 0 for = 0,...,−1.Thatis () (0) +1 () = ! +( ). The reason for that is the fact that the whole idea hinges upon the use of the generalized Möbius inversion, more precisely of the function, which must be zero for numbers having at least prime factors – always satisfied by the numbers represented by the product form (+ℎ1)⋅⋅⋅(+ℎ) in the construction. So for any meaningful weight function we need to use weights not containing any smaller power than . In other words, we should assume here having a zero of order , i.e., (0) = (0) = ⋅ ⋅ ⋅ = (−1)(0) = 0,while()(0) can be arbitrary. The analysis of Soundararajan exposed the question, whether a linear combina­ tion of monomials, i.e., a polynomial, or perhaps some more sophisticated choice of a weight function, may perhaps improve even upon this. We can say that the theoretical limit of the GPY method is the result, obtainable in principle by a choice of the weight function maximizing the extremal quantity (1). Yet it is to be noted that the techni­ calities of GPY are far more substantial than to simply “substituting any ”initwould automatically lead to a result – it is not even that clear, what result would follow from a given weight function. Therefore, to test the limits of the GPY method, we should break our approach into two parts. First, we look for the optimization of the weight , in the sense of (1), and second, we extend the GPY method using that weight function. Bereitgestellt von | De Gruyter / TCS Angemeldet Heruntergeladen am | 16.10.19 13:23 i i i i i i János Pintz, András Biró, Kálmán Győry, Gergely Harcos, Miklós Simonovits, József Szabados (Eds.): Number Theory, Analysis, and Combinatorics — 2013/11/19 — 12:08 — page 78 i i 78 | Bálint Farkas, János Pintz, and Szilárd Révész This paper is concerned with this first question, and the second part of this program is carried out in [8].
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