Odd moments in the distribution of primes

Abstract

Montgomery and Soundararajan showed that the distribution of ψpx `Hq ψpxq, for 0 ď x ď N , is approximately normal with mean " H and variance " H logpN {Hq, when N δ ď H ď N 1´δ . Their work depends on showing that sums R k phq of k-term singular series are µ k p´h log h `Ahq k{2 `Ok ph k{2´1{p7kq`ε q, where A is a constant and µ k are the Gaussian moment constants. We study lower-order terms in the size of these moments. We conjecture that when k is odd, R k phq -h pk´1q{2 plog hq pk`1q{2 . We prove an upper bound with the correct power of h when k " 3, and prove analogous upper bounds in the function field setting when k " 3 and k " 5. We provide further evidence for this conjecture in the form of numerical computations.

Sums of singular series along arithmetic progressions and with smooth weights

“…In the function field case of [4], the simplified and stronger bounds correspondingly yield a simplified proof of the analog of Theorem 1.3. The smooth weights f i make the exponential sums here cleaner, but because the function field-style bounds are not available for the sums in Lemma 5.1, the simplified proof of the analog of Theorem 1.3 in the function field case also fails to apply.…”

“…Secondly, restricting sums of singular series in this manner may shed light on other questions about sums of singular series. We show in Theorem 1.2 that the asymptotics for (4) are governed by incidences among the c i 's mod r. As discussed in [4], we do not yet know the asymptotic average size of sums of S 0 (H) when |H| is odd. The results of these more refined averages of singular series may clarify where the main term should be coming from for sums of singular series with an odd number of terms.…”

“…In the case of analogous exponential sums in the function field case of polynomials over F q [t], as explored in [4], the sum analogous to µ mod m E µ m + α 2 can be bounded by the analog of mh whenever α is large, i.e. whenever α is a rational function of sufficiently large degree (see Lemma 3.4 of [4]). The equivalent statement here would be that µ mod m E µ m + α 2 is bounded by mh whenever α is far away from an integer.…”