SUMS OF TWO SQUARES PAIR CORRELATION & DISTRIBUTION IN SHORT INTERVALS

SUMS OF TWO SQUARES PAIR CORRELATION & DISTRIBUTION IN SHORT INTERVALS YOTAM SMILANSKY Abstract. In this work we show that based on a conjecture for the air correlation of integers reresentable as sums of two squares, which was first suggested by Connors and Keating and reformulated here, the second moment of the distribution of the number of reresentable integers in short intervals is Poissonian, where short means of length comarable to the mean sacing between sums of two squares. In addition we resent a method for roducing such conjectures through calculations in rime owers modulo rings and describe how these conjectures, as well as the above stated result, may by generalized to other binary quadratic forms. While roducing these air correlation conjectures we arrive at a surrising result regarding Mertens formula for rimes in arithmetic rogressions, and in order to test the validity of the conjectures, we resent numerical comutations which suort our aroach.. Introduction Throughout this work n, k and h will denote ositive integers, denote rime numbers and for abbreviation reasons we use a b c) instead of a b mod c. In addition we say m n) = k if k n but k+ n... Background and motivation. When studying the distribution of a sequence of integers, for examle the sequence of rimes or of those reresentable as a sum of two squares, a natural first ste would be to understand the mean density of such integers. For rime numbers this was achieved by Hadamard and de la Vallee Poussin with their famous Prime Number Theorem, and for sums of squares by Landau [L]. In order to learn more about the distribution of such a set the next ste would be to look at the k-oint correlation, or in other words to find an exression for N N f n + d )... f n + d k ), N n= where f is the characteristic function of the set at hand and d,..., d k are distinct integers. These correlations give increasingly more recise data about the distribution, where the -oint correlation rovides the leading quantitative estimate of the fluctuations about the mean density of the sequence. Date: July, 0.

Regarding the sequence of rimes, Hardy and Littlewood gave [HL] the following k-tule conjecture for the number π d N) of ositive integers n N for which all of n + d,..., n + d k are rime, d = d,..., d k ) and d,..., d k distinct integers. The conjecture is:.) π d N) S d N log N) k as N rovided S d 0, where the singular series S d is S d = rimes k ν d )) ) k and ν d ) stands for the number of residue classes modulo occuied by d,..., d k. For k = this is exactly the Prime Number Theorem, and for k is has not been roved for any d... From a k-tule conjecture to distribution in short intervals. We will follow Gallagher s work [G] on rimes in order to obtain the moments of distribution of the number of integers reresentable as a sum of two squares in short intervals. Consider first the set of rimes and the rime number theorem, which states πn) N log N, N. This relation can be understood as the statement that the number of rimes in an interval n, n + α), averaged over n N, tends to the limit λ, when N and α tend to infinity in such a way that α λ log N with λ a ositive constant. Gallagher studies the distribution of values of πn + α) πn) for n N and α λ log N, and shows that assuming the rime k-tule conjecture of Hardy and Littlewood.), it suffices that.) S d H k d,..,d k H holds for all k N in order to rove that all the moments of the distribution tend to moments of Poisson distribution, and so the distribution tends to Poisson distribution with arameter λ as N. This means that the distribution of rimes in such intervals is similar to the distribution of a random set of integers with mean λ, and so even though clearly rimes are not distributed randomly, in the ersective of intervals such as those we deal with here they do. Gallagher has roved.) in [G], and a simler roof was resented by Kevin Ford [F]. We shall refer to this result as Gallagher s Lemma. Consider now the set of integers which are reresentable as sum of squares and Landau s theorem, which states that BN), the number of such integers

u to N, is given asymtotically by ) N N.3) BN) β + O, N log N log 3 4 N where β = ) is the Landau-Ramanujan constant. Similar to the rimes, this relation can be understood as the statement that the number of rimes in an interval n, n + α), averaged over n N, tends to the limit λ, when N and α tend to infinity in such a way that α λ β log N with λ a ositive constant. We wish to study the distribution of values of Bn + α) Bn) for n N and α λ β log N. In order to follow Gallagher s method we need first a conjecture analogues of Hardy and Littlewood s conjecture for sums of two squares, that is an asymtotic formula for the number B d N) of ositive integers n N for which all of n+d,..., n+d k can be reresented as a sum of two squares, d = d,..., d k ) and d,..., d k distinct integers. The conjecture, analogous to.), is that there exists a function T d, the singular series for our roblem, for which the limit.4) N B d N) T d ) k as N log N holds. If this is so, then the function T d deends only on the differences between the d,..., d k, in the sense that T d = T d+ where =,..., ). Assuming this conjecture, it is enough to show that the singular series T d has mean value β:.5) T d βh) k d,..,d k H for the moments to be Poisson with arameter λ..3. Main Result. Connors and Keating conjectured in [CK] that for k = and h = d d we have.6) T d,d = T h = W h) h mh)+) where m h) is the ower to which the rime is raised in the rime decomosition of h and { 4 m h) = 0 W h) = m h)+ 3 m m h)+ h) Our main result is that T d H h) T h = β H + oh ) d d H h H 3

and so assuming this air correlation conjecture we show Gallagher s Lemma for sums of two squares and k = holds, or in other words we show that assuming the conjecture, the second moment of the distribution of values of Bn + α) Bn) for n N and α λ β log N is indeed Poisson..4. Mean Density and Pair correlation. We rovide a new aroach to the comutation of the air correlation function stated above, which goes through the mean density and air correlation of reresentable element in modulo rings of the form Z / k Z for rimes and k. The mean density is thus given by.7) Mn) = + ) which is the roduct of the densities in modulo rings mentioned above. We comare this exression with the leading term of the analytic result for the density of reresentable integers given by Landau.8) Ln) = β log n and roduce the recise ratio between the two Mn) lim n Ln) = π e γ where γ is Euler s constant, using a version of Mertens formula in geometric rogressions described in [LZ]. Next we derive.6) in similar methods to those used for the mean density.7). In Section 7 we resent numeric calculations to suort our conjecture..5. Generalization to other binary quadratic forms. Our methods allow us to exand our observation also to integers reresentable by other binary quadratic forms x + dy with d =, 3, 4, 7 in addition to d =, which are the sums of two squares. A surrising result is that the ratio between the roduct formulas M d n) we resent and the analytic results using variations on Landau s theorem L d n), for n, is in fact constant for the five different quadratic forms insected and is M d n).9) lim n L d n) = π e γ. We next roduce conjectures analogous to.6) and therefore to.4) with k = for integers reresentable by the forms at hand, and finally deduce that assuming our conjectures the second moment of the distributions in the aroriate short intervals is Poisson. Acknowledgments. This work is art of the author s M. Sc. thesis written under the suervision of Zeev Rudnick at Tel-Aviv University. Partially suorted by the Israel Science Foundation grant No. 083/0). 4

. Distribution In Small Intervals - Gallagher s Lemma In order to obtain the second moment for the distribution of reresentable integers in the intervals described above, we follow Gallagher s work for rimes, and we will show that T d = T d d = H h)t h = β H +oh ) d d H d d H h H where by the Connors and Keating conjecture T d,d = T h = W h) h mh)+) and W h) = We start by comuting { h H.. Dirichlet s Function. Set 4 m h)+ 3 ah) = T h = 4W h) m h) = 0 m m h)+ h) T h. h mh)+). Notice that ah) is multilicative: obviously a) = since is odd and has no rime factors, and for m, n) = we have amn) = am)an) because our function is comosed of roducts deending only on the rime factorizations. Comuting a k ) gives 4) a k ) = 3 k = k+ We can thus write Ds) = ah)h s = + h= k= 3 ) = + k ks k= 4) = Rs)P s)qs) 34) a k ) ) ks + s s 5. ) + k= k+ ks )

where Rs) = + s s+) 3 s s+) P s) = s ) Qs) = 4) s + s+) ) s+). s.. Comarison to Riemann s ζ function. Taking ζs) = s ), we will now show that Ds) ζs) is analytic for Res) > 0, thus Ds) is analytic in that region with a simle ole at s =. Ds) ζs) = + s 3 s+) s s+) s ) The first exression turns out to be + s s s+) s+) s ). Rs) s ) = s + s s s 3 s+) s+) s+) which is clearly analytic in the desired region. The second exression is Qs) s ) = s + s s+) s+) ) ) s+)). Notice that s + s s+) s+) ) s+) ) = + s+ ) s+ ) + s+ s+) = + s+ s s+ s+ + + s+ s+ s+ + ) s = + O s+ and so the roduct Qs) s ) = )) + O s+ converges in the desired region Res) > 0 in which it is analytic, imlying that Ds) ζs) is also analytic there. 6

Let As) be an analytic function in Res) > 0 defined by Ds) = As)ζs). Since Res s= ζs) =, in order to comute Res s= Ds) we can simly comute A) and so Res s= Ds) = A) = = β..3. Some Hel From Harmonic Analysis. Theorem. Let F s) = n= bn)n s be a Dirichlet series with ositive real coefficients and absolutely convergent for Res) >. Suose that F s) can be extended to a meromorhic function in the region Res) having no oles excet for a simle ole at s = with residue R 0. Then bn) = Rx + ox) as x. n x This is a version of the Wiener-Ikehara Theorem, for a roof see [M]. Our series ah) meets the condition of the theorem, and so ah) = β H ) + oh ) = β H + oh). h H Since ah) = T h we have h H The next ste is to calculate AH ) = fh) = h so using summation by arts h H T h = β H + oh). h H h H ht h = fh )AH ) = β H + oh ) ˆ = β H + oh ) β = β H ht h. Set T h = β H + oh) ˆ H At)f t)dt ) β t + ot) dt ˆ H H H ) t {t})dt + oh ) + H + ) {H } + oh ) = β H + oh ) 7

and therefore d,d H T d h H H h) T h = β H + oh ) which is effectively Gallagher s Lemma for sums of two squares and k =. 3. Sums of Squares in Modulo Rings Following Keating-Connors we attemt to roduce a -tule conjecture using essentially heuristic methods and Landau s theorem. The first ste would be to comute an exression for the air correlation of reresentable integers: lim # {n N : n, n + h are both reresentable}. N N To roceed, consider the consequences of the following lemma: Lemma. An integer is reresentable if and only if it is reresentable in Z/ k Z for every rime and integer k N. Equied with this lemma we shall examine Z / k Z for all rimes and k N, and determine which are the reresentable elements in these modulo rings. This will allow us to give an exression for the the density of reresentable elements, and then of reresentable airs. 3.. Reresentable elements in modulo rings. Definition 3. Say a Z / k Z has a non trivial reresentation as a sum of two squares if there exists x, y Z / k Z such that a x + y k) with x, ) =. Definition 4. Say a Z / k Z has a comletely non trivial reresentation as a sum of two squares if there exists x, y Z / k Z such that a x + y k) with x, ) = and y, ) =. Definition 5. Say an element a Z / k Z lifts to an element b Z / k+l Z if b a k). The following roositions give the information needed in order to rove Lemma, to sort the reresentable elements in the modulo rings and to comute their densities: Proosition 6. Let be an odd rime, k N and a Z / k Z. a) If a has a non trivial reresentation and a, ) =, then all lifts of a in Z / k+ Z have non trivial reresentations as well. b) If a has a comletely non trivial reresentation and a, ) >, then all lifts of a in Z / k+ Z have comletely non trivial reresentations as well. 8

Proof. For k, take a Z / k+ Z such that a x + y k), x, ) =, so a is lifted from an element with a non trivial reresentation in Z / k Z. Since x Z / k Z) the element x = x x +y a x is well defined and gives x + y = a + x + y a) 4x. We assume x + y a 0 k) and so x + y a) 0 k). Since k we have x + y a) 0 k+), meaning x + y a k+). In case a) we assume a, ) = and so and at least one of x, y must be co-rime, as required. In case b) we assume a, ) = and we have y, ) =, and so we must have x, ) = as well. Proosition 7. Let be an odd rime and a Z /Z, a, ) =, so a has a non trivial reresentation. Proof. We claim that for any nonzero element in Z /Z there are x, y so that x a y ). If u v ) then u ±v ), therefore x and a y each take + different values. Since there are only different elements in Z /Z there is a coule that solves the congruence and so a is reresentable. Since a 0 ) it is imossible that x, ) > and y, ) >, and so we can choose a reresentation with x, ) = as required. Proosition 8. Let be a rime such that 4), so 0 Z /Z has a comletely non trivial reresentation. Proof. Since 4), is a quadratic residue modulo so there exists a nonzero element α such that α ) and so + α + ) 0 ) as required. Proosition 9. Let be rime such that 3 4). Elements 0 a Z / k Z, a, ) > are reresentable if and only if m a) is even. Proof. If m a) is even, we can write a = d l, d, ) =. The element d is reresentable as a lift of a nonzero element in Z /Z by Proositions 6 and 7, while l is a square itself and therefore obviously reresentable. Since the roduct of two reresentable elements is also reresentable, we have the required result. Now assume a Z / k Z is reresentable, m a) = l + and k > l + : a = d l+ x + y k), d, ) = = x + y = d l+ + m k = d + m k l ) l+, m N. Since k l > 0 and d, ) = we have d + m k l, ) =, and so has an odd multilicity in the rime decomosition of x + y, which is a contradiction. 9

Using Proositions 6, 7 and 8 we establish that all elements in Z / k Z for 4) are reresentable, while Proositions 6, 7 and 9 establish that the non reresentable elements in Z / k Z for 3 4) are those with odd m. Let us now examine the elements in Z / k Z: Proosition 0. An element a Z / k Z such that a + 4n k) for some n, is reresentable as a sum of squares. Proof. For k =, the only element in question is, and +0 4), and for k = 3 we also have 5 + 8). Take k 3, a Z / k+ Z, a 4). By our assumtion there are x, y in Z / k Z such that a x + y k) and we can assume x, ) = since a 4). If x + y a k+) we are done. The only other otion is that x + y a + k k+). In this case we have x + k ) + y x + k x + k + y x + k + y a + k + k a k+). since k x k k+) and k 0 k+) for k 3. Proosition. An element a Z / k Z of the form a = j + 4n), 0 j k is reresentable, and these are all the reresentable elements. Proof. Using the multilicativity of reresentable integers it is enough to show that j is reresentable. For j = l take l ) + 0 = l, and for j = l + take l ) + l ) = l+. There are no other reresentable elements since all reresentable integers in N must take the form j + 4n), which is not changed modulo k. The roof of Lemma is now clear: Say a = x + y, so obviously a x + y k). Conversely assume a is not reresentable, then for some 3 4) m a) is odd, so a is not reresentable in Z / k Z for k m a). 3.. Mean density of reresentable elements in modulo rings. We now wish to calculate the densities of reresentable elements in Z / k Z for all rimes, k. The following roositions rovide a method for deriving these limits, and resent ideas which can be useful also for calculating correlations of higher degrees. Proosition. The mean density of reresentable elements in Z / k Z, 4) for k is. Proof. This is immediate from the fact that all elements in Z / k Z for all k are reresentable. Proosition 3. For 3 4), the density of reresentable lifts of 0 Z/ l+ Z in Z / k Z tends to + as k. The density of reresentable lifts of 0 Z / l Z in Z / k Z tends to as k. + 0

Proof. Consider first the lifts of 0 Z /Z in Z / Z given by 0,,,..., ). Obviously the multilicity of in the nonzero lifts is odd, and therefore they are not reresentable in Z / Z, and so are all of their lifts in Z / k Z for k. Consider now 0 Z / Z and it s lifts 0,,,..., ) in Z / 3 Z. Here the multilicity of in the nonzero lifts is even, and therefore they are reresentable in Z / Z, and so are all of their lifts in Z / k Z for k 3. So in Z/ Z we have a single reresentable lift of 0 Z /Z, and in Z / 3 Z we have + ) such lifts. The next ste is similar to the first, where for 0 Z / 3 Z the only reresentable lift is 0 Z / 4 Z, and each of the nonzero lifts in Z / 3 Z has reresentable lifts in Z / 4 Z. So there are exactly + ) reresentable lifts of 0 Z /Z in Z / 4 Z, and similarly + ) + ) such lifts in Z/ 5 Z, + ) + 3 ) lifts in Z / 6 Z and so on. The total number of lifts of 0 Z /Z in Z / k Z is k, and so the density of reresentable lifts is given by k k + ) + ) k 3) n=0 k 4) n=0 n = k + ) k + for odd k, + n = k + ) k + + for even k. In both cases it is clear that the density tends to + as k. Note that the exact same results holds for reresentable lifts of 0 Z / l+ Z in Z / k Z as k. For the density of reresentable lifts of 0 Z / l Z one follows the exact same stes with a single shift, meaning that the above exression is multilied by and thus the density of reresentable lifts of 0 Z / l Z in Z / k Z tends to + as k. Proosition 4. The mean density of reresentable elements in Z / k Z, 3 4) for k is + ). Proof. It is enough to look at elements a Z /Z. There are elements such that a, ) =, and by Proosition 9 these are all reresentable and so are all their lifts. The density of reresentable lifts of a = 0 Z /Z in Z / k Z tends to as k. Combining the above we arrive at the desired + exression + + = + = + ). Proosition 5. An element a Z / k Z of the form j +4n), 0 j k is lifted into two elements in Z / k+ Z both can be written in a similar form. Only one of the two lifts of 0 and k can be written in that form for all Z/ l Z, l k +.

Proof. The first lift of j + 4n) is itself and so trivially can be written in that form. The second is j + 4n) + k = j + 4n + k j ) = j + 4m) since k j. If a 0 k) it can be written as a = l c with c odd and l k, so in the cases that c 3 4) we will not be able to write a in the required form for all Z / l Z. If a k k) it can be written as a = k c with c odd, and again we will not be able to write a in the required form for all Z / l Z. Proosition 6. The mean density of reresentable elements in Z / k Z, for k is. Proof. Again we look at elements in Z /Z. By Proosition 5 the density of reresentable lifts of 0, Z /Z in Z / k Z for k is, and so the density of reresentable elements in Z / k Z for k is =. We wish to calculate the mean density of reresentable integers, so following our aroach we take the roduct of all the above densities for n: 3.) Mn) = + ). On the other hand, the leading term in Landau s analytic exression for the mean density of reresentable integers is Ln) = β log n. The events that an integer is reresentable in modulo rings of different rimes show some deendency, a deendency which gives rise to a term yn) which must be taken into consideration. This term should give In the next section we show that Ln) Mn) yn). Mn) lim yn) = lim n n Ln) converges to some real number y. Unfortunately we do not have a Landau exression for the density of reresentable airs. Using the ratio y between our roduct of densities to Landau s density we attemt to give a conjecture for airs given by a roduct of densities in modulo rings.

4. Ratio Between the Product of Densities and Landau s Result Our exression Mn) = + ) which stands for the roduct of the densities of reresentable elements in modulo rings, can be calculated using a generalization of Mertesns famous formula for arithmetic rogressions. Using these methods one can calculate the term given by y in the revious section. Mertens original formula states that ) ) = e γ log n + O log n where the roduct is over all rimes less than n and γ denotes Euler s constant. For co-rime integers a, q, Languasco and Zaccagnini show [LZ] a generalization of Mertens formula 4.) lim log n/ϕq) n a q) [ ) = e γ where ϕ is Eulers totient function, and a; a, q) is given by { ϕq), a q) a; a, q) =, otherwise ) α;a,q)] /ϕq) Theorem 7. The ratio between Landau s leading term and the roduct of densities in modulo rings converges and is given by Mn) y = lim n Ln) = π e γ. Proof. Plugging a = 3, q = 4 in Mertens formula for rimes in arithmetic rogression we have and since ) e γ/ log n ) ) ) 4) + ) = 3 /

we arrive at + ) e γ/ log n We are interested in the ratio lim n Mn) Ln) = lim n ) + ) Mn) β/ log n with β the Landau-Ramanujan constant given by β = ) / and so lim n Mn) Ln) = e γ/ + ) / 4) 4) ) /. The two roducts are exactly L) which is calculated in [SW], where Ls) is the Dirichlet series for the non rincial character modulo 4. Therefore Mn) π y = lim n Ln) = e γ/ 4 = π e γ. This is quite an elegant result, which can be easily generalized using similar tools as will be done in section 6. 5. Reresentable Pairs In Modulo Rings And Their Densities We are now in a osition to look at the distribution of reresentable airs n, n + h in Z / k Z for an odd rime. We first notice that since all elements in Z / k Z for 4) are reresentable, for every h all coules n, n + h are reresentable. This is not the case for rimes 3 4). Proosition 8. The density of reresentable airs a, a + h) in Z / k Z, k, for rimes 3 4) is mh)+) +. Proof. First say m h) = 0 and let us look at airs a, a + h) in Z /Z. There are airs such that both elements are nonzero and therefore reresentable, and two additional airs such that one of the elements is 0. By Proositions 6 and 7 all nonzero elements lift into reresentable elements and therefore all lifts of the nonzero airs are reresentable. Therefore by Proosition 3 the density of airs with a zero element which are lifted into reresentable coules is, and so the density of reresentable airs is + + + 4 = +. ).

Next say m h) =, so there are airs with nonzero elements and a single air of two zeros in Z /Z. This does not rovide enough data and so we look at these airs in Z / Z. There are ) airs a, a + h) in Z / Z which are lifts of the nonzero airs, and airs which are lifted from the zero air, each one consisting of at least one nonzero lift of zero which, as we have seen, is not reresentable and so these airs are not reresentable. The density of reresentable airs is therefore ) = +. For m h) =, again we have nonzero airs with ) reresentable lifts into Z / Z. The zero air s lift remains a zero air and so we look at airs in Z / 3 Z. There are now ) airs in Z / Z lifted from the nonzero airs, and in addition airs which are nonzero in Z / 3 Z but are lifts of the zero air. These lifts are reresentable, and the two remaining lifts of the zero air consist as before of a nonzero element and the zero element, and so similarly to the case of m h) = 0 the density of reresentable airs is ) + ) + + 3 = 3 +. Following these ideas for even m h) yields ) and for odd m h) mh) n= ) ) n + + + mh)+ = mh)+ + mh) n= ) n mh)+ = mh)+ +. Proosition 9. The density of reresentable airs a, a + h) in Z / k Z, k, is { 4 m h) = 0 W h) = m h)+ 3 m m h)+ h). Proof. We now take into consideration the distribution of airs a, a + h) with h = i in Z / k Z. For odd rimes the distribution is similar to that of a, a + ) since, ) = i, ) = and we have seen that the distribution deends only on m h). This is true also for distribution in Z / k Z, and so it is enough to look at Z / m h)+ Z. Set l = m h), and let us look at airs a, a + h) in Z / l+ Z. 5

It is easily verified that 4 of the airs are of the form 0 + 4m), 0 + 4m) + l) and similarly for all 0 i l we have of the airs are of the i+ form i + 4m), i + 4m) + l). The air l, l + l) is not reresentable since l + l = 3 l. The last two airs we must take into consideration are 0, l ) and l, 0). As shown in roosition 5 the density of reresentable lifts of these airs to Z / k Z for k is. We can now comute the density of reresentable airs: l n= n + l+ = l+ 3 l+. For l = 0, the sum does not contribute and so the density is 4 and 8 resectively, giving the desired result. As before the events that a air m, m + h), m n is reresentable in all the rime ower modulo rings are not indeendent, a deendency which gives rise to a term Y h n) which must be taken into consideration. The density of reresentable airs is thus: Y h n) W h) mh)+) +. Following Connors-Keating we extract the asymtotic term deending on n from the above exression using our calculation for the mean density of reresentable integers: 5.) Y h n) W h) ) yn) log n Y h n) log n yn) Y h n) mh)+) + Ln) Mn)/yn) ) W h) mh)+)) ) + ) + ) ) W h) mh)+). Notice that for such that m h) = 0 the roduct is, and since we are interested in n, we can assume n h and so the roduct is over all 3 4) such that h. The conjecture resented by Connors-Keating is thus equivalent to the conjecture that for all h yn) Y h n), as n, 6 )

a conjecture for which we resent some numerical comutations in Section 7. Assuming the validity of this conjecture the density of reresentable airs is given by T h = log n W h) h mh)+). 6. Generalization to Other Binary Quadratic Forms 6.. Preliminaries. Let us look at the following family of binary quadratic forms qd; x, y) = x + dy. Definition 0. We say that d N is a convenient idoneal) number if there is finite set of rimes S, an integer N and congruence classes c,..., c k mod N such that for all rimes S = x + dy c,..., c k mod N. Examle. For d =, S = {}, c = and N = 4 we have Fermat s result for sums of two squares. We focus here on convenient d s such that the form x + dy is of class number which are d =,, 3, 4, 7. In these cases one can fully determine if an integer n is reresentable by the form simly by making sure that the rimes which are not reresentable aear with an even multilicity in the integer s rime factorization. Again we are first interested in the mean density of reresentable integers, and we can calculate the densities in the modulo rings in the exact same way that we did for d = and thus generalize 3.). In [SS] Shanks roduces Landau s constants β, β, β 3, β 4, β 7 for which # { n N nis of the form x + dy } β d N log N and so we can again calculate the ratio between the roduct and the analytic exressions as was done in Section 4 for sums of squares. First let us recall the following classical results: Theorem. An integer n is reresentable by the form x + dy if and only if: If d =, m n) is even for all rimes 3 4). If d =, m n) is even for all rimes 5, 7 8). If d = 3, m n) is even for all rimes 3). If d = 4, m n) is even for all rimes 3 4) and m n). If d = 7, m n) is even for all rimes 3, 5, 6 7) and m n). The conditions for reresentation by these forms bare obvious resemblance. 7

Definition 3. For convenience reasons we divide the rimes into the following sets: Let Q d denote rimes such that if n is of the form x + dy then m n) is even. Notice that by Dirichlet s theorem this consists of aroximately half of the rimes. Let R d denote rimes such that if n is of the form x + dy then m n) has some constraint which is not that m n) is even, or the N for which = x + dy c,..., c k mod N is such that, N). Notice that R d is a finite set. Let P d denote rimes not in any of the above sets, or more directly rimes of the form x + dy such that, N) =. Again this set consists of aroximately half of the rimes. Examle 4. Q = { rime: 3 4)}, P = { rime: 4)}, R = {}. Q 7 = { rime: 3, 5, 6 7)}, P 7 = { rime:,, 4 7), }, R 7 = {, 7}. 6.. Ratio between the roduct density and Landau s density. Following the same methods established in Section 3 for our roduct calculation of Mn) we can now give a roduct exression for M d n), which stands for the roduct formula for the mean density of integers of the form x + dy. Notice that for all the above d s the condition for being reresentable by the form is over half of the rimes lus some local conditions, and so again we have an exression of the form M d n) = + ) R d w d ) Q d with w d ) the mean density of reresentable element in Z / k Z, k, for R d. The rimes P d do not articiate here since similarly to the case of sum of two squares, the mean density of reresentable elements in Z/ k Z, k, is. These roducts can be comuted using Merten s formula for arithmetic rogressions, as was done in the revious section for d = : Q d + ) e γ/ log n Q d ) + ) P d R d Again we are interested in the analogue of.9), that is in the ratio between these roducts and the leading term of the analytic exression given by the generalization of Landau s theorem as shown in [SS]: 6.) L d n) = β u log n, β d = δ d g d 8 ) Ld ) d πϕ d ) ).

with ϕ the Euler totient function and g d = ) ) = and d L d s) = δ d = ) d n s = n odd n, d =, 3, d = 3 3 4, d = 4, 7 d ) = s ) Notice that for d =,, 3, 4, 7 ) d = P d ) d = Q d { ) unless d = 3, in which case Q 3 = : d Reformulating the roducts above we have g d = ) = γ d Q d {, d =,, 4, 7 where γ d = 3, and, d = 3 L d ) = ) P d = P d d } = {}. Q d Q d ) λ d, d =,, 4, 7 where λ d = 3., d = 3 The ratio in question is therefore given by lim y M d n) dn) = lim n n L d n) = lim R d w d ) δ d π e γ ϕ d ) d n Q d γ d λ d R d ) ) = + s ) + ) + ) M d n) β d / log n = ). Recall ϕn) n = n ). For d =,, 4, 7 we have d R d and so all the roducts cancel each other. 9 For d = 3 we have d and

R 3, so we are left with the term ) =. since 3 3 = 3 we can write lim y dn) = n π e γ s d w d ) δ R d d {, d =,, 4, 7 where s d =. 3, d = 3 We can now examine this result for different values of d: Theorem 5. For d =,, 3, 4, 7 the ratio between the roduct of densities in the modulo rings and Landau s density of integers reresentable by the forms x + dy converges to π e as n, that is γ lim y M d n) dn) = lim n n L d n) = π e γ = y. Proof. We comute the ratio case by case: d =. We have already seen in the revious section that R = {}, and that w ) =, since the reresentable elements in Z / k Z are of the form j + 4n) and the density of such elements aroaches as k. In addition δ = and s d = and so M n) lim n L n) = π e γ. d =. Again R = {}. Here w ) = because reresentable elements in Z / k Z are of the form j + 8n) or j 3 + 8n) and the density of such elements aroaches as k. Also δ = and s d = and so M n) lim n L n) = π e γ. d = 3. R 3 = {3}, and w 3 3) = since the reresentable elements in Z /3 k Z are of the form 3 j + 3n) and the density of such elements aroaches as k. Here δ 3 = 3 and s d = 3 therefore the ratio is given by M 3 n) lim n L 3 n) = 3 π e γ 3 = π e γ. d = 4. Once again R 4 = {}. This case is similar to that of d = only here the reresentable elements in Z / k Z are of the form j + 4n) with j. The density of such elements is 3 8 as k. So w 4) = 3 8, δ 4 = 3 4 and s 4 = M 4 n) lim n L 4 n) = 3 8 4 π 3 e γ = π e γ. d = 7. Here R 7 = {, 7}. Reresentable elements in Z /7 k Z are of the form 7 j + 7n), 7 j + 7n) or 7 j 4 + 7n) and the density such elements is as k. The reresentable elements in Z / k Z are of the form j + 4n) or 0

j 3 + 4n) with j, and the density of such elements aroaches 3 4 as k. We thus have w 7 ) = 3 4, w 77) =, δ 7 = 3 4 and s = M 7 n) lim n L 7 n) = 3 4 4 π 3 e γ = π e γ. 6.3. Pair correlation conjecture. We can now roose a conjecture for the air correlation function for the forms x + dy with d =,, 3, 4, 7, generalizing.6) and.4). Denote by W d, h) the density of reresentable airs a, a + h) in Z / k Z, k, for R d, and Y d,h n) the deendance term which must be taken into consideration. We extract the asymtotic term deending on n as was done in 5.): Y d,h n) Y d,h n) R d W d, h) R d W d, h) Q d ) y log n d n) W d,h) Y d,h n) w R d d ) Q d mh)+) + mh)+) Q d + Let us first look at the roducts at hand. As before Q d mh)+) + ) g d = Q d h Q d L d n) M d n)/y d n) ) mh)+) + ) δd gd Ld) d πϕ d ) mh)+) + ) mh)+) S d Q d ). {, d =,, 4, 7 where S d = 3. 4, d = 3 Again the conjecture is that for all h and so 6.) T d,h = c d yd n), as n Y d,h n) R d W d, h) Q d h mh)+)

where c d = δu L u ) u πϕ u ) w R d d )S d It is left to comute c d and W d, h) case by case. Dirichlet s class number formula see [SW]) gives 6.3) L ) = π 4, L ) = π, L 3) = π 3, L 4) = π 4, L 7) = π 7 and so lugging all the different term we have 6.4) c =, c =, c 3 = 3, c 4 =, c 7 = 7 3. In addition, calculations similar to those shown for sums of squares in Section 5 give, m h) = 0 W, h) = 4 mh)+ 3 m, m h) h)+, m h) = 0, W, h) = 4 mh) 3 m, m h) h)+ W 7, h) = W 3,3 h) = 3m 3h)+ 3 m 3h)+ 8, m h) = 0 0, m h) = W 4, h) = 5 6, m h) = 3 m h) 3, m m h)+ h) 3 {, m h) = 0, 3 4, m h), W 7,7 h) = 7m7h)+ 4 7 m. 7h)+ 6.4. Distribution in short intervals - the second moment. We wish to generalize our result from Section concerning the second moments of the distribution of reresentable integers in short intervals. Theorem 6. Let X d n) be the number of integers which can be reresented by the form x + dy, d =,, 3, 4, 7, in the interval n, n + α d ), α d λ β d log N. Denote Pl α d, N) the number of integers n N for which the interval n, n + α d ) contains exactly l such integers. Assuming 6.) the second moment of P l α d, N) is Poisson.

Proof. Following the stes described in Section it is enough to show that T d,h = βdh + oh). h H In order to do that we normalize T d,h by defining a d h) = T d,h T d, which is now multilicative, and show that D d s) = a d h)n s has a simle ole at s = with residue β d T d,, as was done for sums of squares. Following the exact same stes we have where R d s) = R d P d s) = P d Q d s) = Q d n= D d s) = R d s)p d s)q d s) + k= s ) a k ) ) ks s + s+) ) s+). s It can be shown D d s) = A d s)ζs) with A d s) analytic, and so to calculate the residue of Ds) at s = it is left to calculate A d ) which gives Recall D d s) A d ) = lim s ζs) = + a k ) k= k R d ) ). Q d βd = δd L d ) d πϕ d ) Q d ) and so it remains to show that indeed for d =,, 4, 7 R d + k= a k ) k ) = δ d L d ) d πϕ d ) T d, Following the stes detailed in section and lugging in all the relevant constants, all comuted above, we arrive at the desired result. 7. Numerical Comutations The aroach taken in [CK] as well as ours to the air correlation conjecture for integers reresentable as the sum of two squares, stated in.6), is essentially heuristic, and so some numerical comutations are in lace in order to suort our conjecture. The conjecture as stated here is that yn) Y h N), as n 3

which, as shown in 5.), can be calculated by taking the ratio between the numeric density of airs and the conjectured air correlation function: yn) Y h N) = N # {n N n and n + h are reresentable} mh)+) log N W h) h In Figure 7. we resent some calculations of this ratio for various h : Figure 7.. yn) Y h N) for h 5 at N = 06,0 8 Examining different values of h for which the rimes and 3 4) aear with equal multilicity, such as h =, 5, 7, 5 or h = 4, 0, one can see they take very similar values. This was checked for many more values of h which are not shown here and so strengthens our belief that the air correlation deends only on the multilicity of these rimes in h. One can also see that the fluctuations between different values of h diminish for larger N, where the eaks in the above grah are obtained at values of h for which m h) =, or m 3 h) =, since the small rimes are the most dominant in our comutations. We must not be discouraged by the extremely slow decay to, for it is consistent with the large error term which aears in Landau s theorem in.3). In fact the convergence imlied in Landau s theorem, or more recisely β # {n N n is reresentable} N) = β N, N log N 4

shows similar behavior as shown in Figure 7., in which the values for the ratio yn) Y h N) are calculated for h =. The reason we comare the rate of convergence to that of β N) and not to βn) is that we look at airs of reresentable integers. The values for the ratio yn) Y h N) are calculated for h =. Figure 7.. yn) Y N) and Landau s β convergence The generalizations resented in Section 6 for integers of the form x +dy show similar numeric results. Figure 7.3 is the equivalent of Figure 7. for integers reresentable by x + y. 5

Figure 7.3. y N) Y,h N) for h 5 at N = 06, 0 8 We have obtained results of this tye for the other forms in question where the main difference between the forms is the location of the eaks, which occur at values of h with small m h) for small rimes Q d R d. To conclude we have arrived with numerical results which are consistent with our exectation regarding the deendency on the rime decomosition of h, and regarding the rate of convergence. Note that the numerical data resented here imroves revious comutations by a factor of 0 for h 5 as aears in Figure 7. and by 000 for h = as aears in Figure 7.3. 8. further Directions The work resented here may be exanded by roducing conjectures for k- correlation functions for the set of reresentable airs for k 3, as described in.4). For examle, following the methods resented for the calculation of the mean density and the air correlation one can derive the following result for the density of reresentable trilets of the form n, n +, n + ) for n N, given by 8.) log 3 N 8β n N ) 0.698 log 3 N It is ossible to generalize this result for trilets n, n + h, n + h ) and so on for higher degrees, though it seems difficult to obtain a general k -correlation function this way because of the inductive element of our aroach. Also when comaring the exression for the density of reresentable trilets 8.) to the exression for the density of reresentable airs.6) 6

one can easily notice that the roduct for the latter deends only on rimes dividing h, where in the case of the trilets the roduct is over all rimes 3 4) and so the maniulation of such exressions is bound to be more comlicated. A second direction, assuming a k -correlation function is obtained, is to rove.5), which is a version of Gallagher s Lemma.)for sums of two squares. Gallagher s aroach in [G], and similarly the aroach taken by Ford when roving the Lemma in the case of the rimes, would aarently not do in the case of sums of two squares. Proving this would assert that assuming a k -correlation conjecture the distribution of reresentable integers in short intervals is Poissonian, and here is the lace to state that we believe that this is indeed the case. The main difference between the case of the set of rimes and the case of the set of integers reresentable as a sum of two squares is the k-correlation conjectures. While Hardy and Littlewood s conjecture for rimes.) deends only on the ν d ) which stands for the number of residue classes modulo occuied by d,..., d k, which in the case of k = is equivalent to whether or not divides d d or whether or not if m d d ) is 0, in the case of the sums of squares Connors and Keating s conjecture.6) and the numerical work resented in Section 7) rovide evidence of deendence also on the values of m d d ). Our roof of Gallagher s Lemma for sums of squares and k = uses Dirichlet analysis, though these methods become extremely difficult in higher dimensions. We hoe this work encourages the reader to further exlore the set of integers reresentable by sums of squares an other forms, where there is a lot yet to be done. 7

References [CK] Connors, R. D.; Keating, J. P. Two-oint sectral correlations for the square billiard. J. Phys. A 30 997), no. 6, 87 830. [C] Cox, D. A. Primes of the form x + ny. Fermat, class field theory and comlex multilication. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 989). [F] Ford, K. Simle roof of gallgher s singular series sum estimate, arxiv:08.386v [math.nt] [G] Gallagher, P. X. On the distribution of rimes in short intervals. Mathematika 3 976), no., 4 9. [HL] Hardy, G. H.; Littlewood, J. E. Some roblems of Partitio numerorum ; III: On the exression of a number as a sum of rimes. Acta Math. 44 93), no., 70. [L] Landau, E. Uber die Einteilung der ositiven ganzen Zahlen in vier Klassen nach der Mindeszahl der zu ihrer additiven Zusammensetzung erforderlichen Quadrate, Archiv der Math. und Physik 3), v. 3, 908),. 305-3. [LZ] Languasco A.; Zaccagnini A. A note on Mertens formula for arithmetic rogressions. J. Number Theory, 7:37 46, 007). MR3566. [M] Murty, M. R. Problems in analytic number theory. Graduate Texts in Mathematics, 06. Readings in Mathematics. Sringer-Verlag, New York, 00).. 43-46. [SS] Shanks, D.; Schmid, L. P. Variations on a theorem of Landau. I. Math. Com. 0 966) 55 569. [SW] Shanks, D.; Wrench, J. W. Jr. The calculation of certain Dirichlet series. Math. Com. 7 963) 36 54. Raymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel E-mail address: yotamsky@yahoo.com 8