ACTA ARITHMETICA XCII.2 2 The major arcs approximation of an exponential sum over primes by D. A. Goldston an Jose, CA. Introduction. Let. α = Λnenα, eu = e 2πiu, where Λn is the von Mangoldt function defined by Λn = log p if n = p m, m, p a prime, and Λn = otherwise. Parseval s euation and the prime number theorem provide the L 2 estimate.2 α 2 dα = Λ 2 n N log N as N, and therefore by the Cauchy chwarz ineuality we have, for any ε > and N sufficiently large,.3 α dα + ε N log N. In this paper we obtain the following improvement on this L upper bound. Theorem. For any ε > and N sufficiently large we have 2 N.4 α dα 2 + ε log N..5 Vaughan [3] has shown that α dα N, 2 Mathematics ubject Classification: Primary L2. Research supported in part by NERC Grant A523 and NF grants. [69]
7 D. A. Goldston and has also made the conjecture that there exists a constant c such that, as N,.6 α dα c N log N. This conjecture may be very difficult to prove, but it might be possible to obtain the lower bound.7 α dα N log N. I have proved that.7 follows from a strong form of either the Goldbach or the twin prime conjectures. Further, certain approximations of α also satisfy.7. These results will be presented in a later paper. Notation. We use the following notation. We take n,, r, k, j to always be positive integers, and in general summation signs will start with if a lower limit is not specified. We will use the notation = and =. a a a,= Q Q a 2. The major arc approximation. Hardy and Littlewood [], [2] introduced the method used for the analysis of α. For α near a fraction a/, α = a/ + β is large and can be approximated well by the simple expression where 2. Iβ = µ Iβ, N enβ. n= Here α = a/ + β where β needs to be small so that there are no other fractions with denominators within a distance of β of a/. This reuirement leads to the Farey decomposition of the unit interval. The Farey fractions of order Q are given by F Q = {a/ : Q, a, a, = }. We define the Farey arcs around each of these fractions, except / which we exclude, as follows. Let a / < a/ < a / be consecutive fractions in
Major arcs approximation 7 the Farey decomposition of order Q, and let a + a M Q, a = +, a + ] a + for a, a, 2.2 M Q, = ] ] Q +,,. Q + These intervals are disjoint and their union covers the interval [, ]. We will sometimes shift these intervals to the origin, which we denote by a + a 2.3 θ Q, a = + a, a + a + a ] ] = +, + when, and θ Q, = /Q +, /Q + ]. ince Q < + < 2Q and similarly for +, we see ] 2.4 θ Q, a = Q + µ,, Q + ν for integers < µ, ν < Q which depend on a,, and Q. In particular, 2.5 2Q, θ Q, a 2Q Q,. Q Finally we define the characteristic function of M Q, a by { 2.6 χ Q α;, a = if α MQ, a, otherwise. With this preparation, our approximation to α is given by 2.7 J Q α = µ I α a χ Q α;, a. Q a We will call J Q α the major arcs approximation for α. This approximation has the advantage that each term in the approximation is orthogonal to every other on the unit interval, which greatly simplifies the computation of various means. One expects that J Q α will become a better approximation of α as Q is increased as a function of N. For the L 2 norm this is the case when Q N, but for N < Q N the approximation degrades because the support χ Q α;, a becomes so small that the terms with N/Q no longer make any contribution. Because of this we define { Q if Q N, 2.8 Q = minq, N/Q = N/Q if N Q N, and we interpret J Q α as actually being an L 2 approximation to α of length Q rather than Q. This disadvantage of J Q α may be corrected by simply deleting the factor χ Q α;, a in the definition of J Q α, but the resulting approximation is much harder to analyze when Q > N.
72 D. A. Goldston We comment that the term major arcs has different meanings depending on the problem the circle method is being applied to as well as individual taste. Our function J Q α is not the only possible choice for an approximation. One complication in applying J Q α is that in some situations one needs to take account of the exact positions of the endpoints of the Farey arcs, which introduces Kloosterman sums. This can often be avoided by using arcs that do not depend on a, such as the intervals in 2.5 that envelope θ Q, a or sometimes even intervals that do not depend on. In this paper however there is no problem with using the Farey arcs in our approximation. The idea for the proof of the Theorem is uite simple. We write 2.9 α = J Q α + T Q α, and refer to T Q α defined by 2.9 as the minor arcs part of α. The major arcs approximation J Q α consists of spikes which are amplified when we take higher powers. As we will see in the next section, J Q α makes only a contribution of at most N in the L norm, but on suaring it contributes half of the amount on the right hand side of.2 when Q = N. Further, in ection 4 we show J Q α and T Q α are essentially orthogonal in L 2, so that T Q α contributes the other half of the amount in.2. Thus, in L the size of α is determined by T Q α, which satisfies an L 2 bound one half as large as the bound in.2. 3. Means of J Q α. We prove in this section some results on means of J Q α. Lemma. For Q N and Q = minq, N/Q, we have 3. J Q α dα Q log 2N Q 2. In particular, 3.2 J Q α dα N. Lemma 2. For Q N, Q = minq, N/Q, and k a positive integer, we have 3.3 J Q α 2k dα = µ2 φ 2k ν kn 2 + OQ 2k Q, n kn where 3.4 ν k n = ν k n, N = In particular, for k = we have n,...,n k N n +...+n k =n.
3.5 Major arcs approximation 73 J Q α 2 dα = N log Q + ON. P r o o f of Lemma. By 2.7, J Q α dα = µ 2 3.6 Q ince we have, by 2.5, 3.7 Using 3.8 /Q θ Q,a = Q µ 2 M Q,a θ Q,a α I a dα Iβ dβ. N + β sinπnβ Iβ = e 2 sinπβ, Iβ dβ = sinπn β sinπβ = 2 θ Q,a /Q sinπn β sinπβ dβ sinπn β sinπβ N/Q dβ sinπu u du dβ + O. { N/Q if N/Q, log2n/q if N/Q, together with 3.6 and 3.7, for Q N /2 we have J Q α dα /Q µ 2 sinπn β sinπβ dβ + O Q Q µ 2 log 2N Q Q log 2N Q 2 = Q log 2N Q 2, since Q = Q in this case, while for Q > N /2, J Q α dα µ 2 log 2N Q + µ 2 N Q Q < Q Q log 2N Q Q + N Q log Q Q log 2Q = Q log 2N, Q Q which completes the proof of the lemma. Q 2
74 D. A. Goldston Now 3.9 P r o o f of Lemma 2. We have J Q α 2k dα = µ 2 φ 2k Q M Q,a = α I a 2k dα θ Q,a n kn M Q,a Iβ 2k dβ = Iβ 2k dβ = ν k nenβ 2 dβ + O = n kn ν 2 kn + OQ 2k. α I a 2k dα. /2 /2Q We also have the trivial bound 3. α I a 2k dα N 2k Q M Q,a [,]θ Q,a dβ β2k Iβ 2k dβ which becomes smaller than the error in 3.9 provided N/Q. We conclude J Q α 2k dα = µ2 φ 2k ν kn 2 + OQ 2k a n kn N 2k µ 2 + O Q φ 2k Q < Q = µ2 φ 2k ν 2 kn n kn + OQ 2k Q + O N 2k QQ 2k where the dash indicates this term only occurs if Q > N /2. We thus see that for Q N /2 the error term above is Q 2k Q as stated in Lemma 2, while for Q > N /2 the error is NQ 2k 2 = Q 2k Q and the lemma follows. In the case k = the result follows from the well known formula [4] 3. Q µ 2 = log Q + O.,
Major arcs approximation 75 4. Inner product of J Q α and α. The pointwise behaviour of α is complicated and the known results are much weaker than what is expected to be true. However when one averages α over the reduced fractions with denominator one easily obtains an asymptotic formula. It is useful to obtain a result of this type due to Vaughan who used it in proving.5. Lemma 3. We have 4. a + β = µβ + O log2n log2. a P r o o f. We have a + β = Λnenβ an e = Λnc nenβ, a a where c n is Ramanujan s sum. ince c n = µ if n, =, and trivially c n, we have Λnc nenβ = µ Λnenβ + O Λn 4.2 n,= = µβ + O n,> n,> Λn = µβ + O log2n log2. We next evaluate the inner product of α and J Q α. Lemma 4. For Q N and Q = minq, N/Q, we have P r o o f. We have 4.3 αj Q α dα = N log Q + ON log2n. αj Q α dα = Q = µ + For the main term we have = µ 4.4 a θ Q,a θ Q,a Q < Q a + β Iβ dβ = + 2. a + β Iβ dβ
76 D. A. Goldston Now = µ a µ = 3 4. a a + β Iβ dβ [,]θ Q,a a + β Iβ dβ a + β Iβ dβ = na Λne enβiβ dβ = na a Λne =, and therefore by Lemma 3, 3 = µ a a = µ2 + O µ 2 log2n log2 = ψn µ2 + OQ log 2 2N, where ψx = n x Λn. By the prime number theorem ψx = x + Oxlog x A for any A >. Hence, by 3., 4.5 3 = N + O N log A N = N log Q + ON. To estimate 4, we have 4.6 where 4 = µ /2Q N = = 5 6, a µ2 + OQ log 2 2N /2Q N a + β Iβ dβ [ ] [ 2Q, Q + ν Q + µ, ] θ Q, a 2Q where ν and µ are the numbers in 2.4 which depend on a,, and Q. By
Major arcs approximation 77 Cauchy chwarz we have 6 µ2 a 2 4.7 + β dβ a N N a 2 + β dβ a N a Q a µ 2 φ 2 M Q,a µ2 N Iβ 2 dβ α 2 dα /Q /2Q /2 α 2 dα β 2 dβ Q Q QN log2n N log2n. For 5 we have, by Lemma 3, 4.8 5 = µ = µ2 /2Q /2Q /2Q /2Q a a + β Iβ dβ βiβ dβ + O log 2 2N /2Q µ 2 Iβ dβ µ2 /2Q /2Q + O log 2 2N µ 2 /2Q β 2 dβ /2 /2Q /2Q dβ β /2Q Iβ 2 dβ Iβ 2 dβ
78 D. A. Goldston µ2 /2 /2 α 2 dα /2Q dβ β 2 + OQ log 3 2N NQQ log2n + ON /2 log 3 2N N log2n. Finally we return to 2. For Q N /2 we have 2 =. Assuming Q > N /2 we have Q = N/Q and 2 = µ a 4.9 + β Iβ dβ Q < Q Q < Q θ Q,a Q a µ 2 a θ Q,a a Iβ 2 dβ M Q,a Q < Q a θ Q,a α 2 dα µ 2 φ 2 /2 α 2 dα Q < a 2 + β dβ /Q /Q µ 2 N 2 Q N 3 log2n = N log2n. QQ Lemma 4 now follows from 4.3 through 4.9. Iβ 2 dβ 5. Proof of the Theorem. As a simple conseuence of Lemma 4 we can determine how closely J Q α approximates α in L 2. Lemma 5. For Q N and Q = minq, N/Q, we have 5. α J Q α 2 dα = N logn/q + ON log2n. P r o o f. On multiplying out we find the left hand side of 5. is α 2 dα 2R αj Q α dα + J Q α 2 dα. By.2, Lemma 2, and Lemma 4 the result follows.
Major arcs approximation 79 Proof of the Theorem. By Cauchy chwarz we have 5.2 α J Q α dα α J Q α 2 dα = N logn/q + ON /2 log /4 2N. By the reverse triangle ineuality and Lemma the left hand side of 5.2 is 5.3 α dα J Q α dα α dα ON /2. The Theorem now follows on taking Q = Q = N /2 in 5.2. References [] G. H. Hardy and J. E. Littlewood, Note on Messrs. hah and Wilson s paper entitled: On an empirical formula connected with Goldbach s theorem, Proc. Cambridge Philos. oc. 9 99, 245 254. [2],, ome problems of Partitio Numerorum : III. On the expression of a number as a sum of primes, Acta Math. 44 923, 7. [3] R. C. Vaughan, The L mean of exponential sums over primes, Bull. London Math. oc. 2 988, 2 23. [4] D. R. Ward, ome series involving Euler s function, J. London Math. oc. 2 927, 2 24. Department of Mathematics and Computer cience an Jose tate University an Jose, CA 9592, U..A. E-mail: goldston@mathcs.sjsu.edu Received on 4..999 354