On the distribution of the divisor function and Hecke eigenvalues

Abstract

Abstract. We investigate the behavior of the divisor function in both short intervals and in arithmetic progressions. The latter problem was recently studied byÉ. Fouvry, S. Ganguly, E. Kowalski, and Ph. Michel. We prove a complementary result to their main theorem. We also show that in short intervals of certain lengths the divisor function has a Gaussian limiting distribution. The analogous problems for Hecke eigenvalues are also considered.

On the variance of sums of divisor functions in short intervals

“…However, this upper bound is most likely not sharp and the true order of magnitude is probably of size (X 1− 1 k /L) · (log L) k 2 −1 . More precise estimates than (1.2) in the case that k = 2 are given by Jutila [8] and Ivić [7] (see also [1] and [9]). In particular, Ivić [7] derives an explicit asymptotic formula for the variance of sums of d 2 (n) = d(n) in short intervals with x < n ≤ x + h and…”

Sums of divisor functions in $$\mathbb {F}_q[t]$$ F q [ t ] and matrix integrals

“…where p 3 is a polynomial of degree 3 with positive leading coefficient. See also the recent papers by Fouvry, Ganguli, Kowalski, Michel [14] and by Lester and Yesha [30] discussing higher moments. For k ≥ 3, Kowalski and Ricotta [27] considered smooth analogues of the divisor sums S d k ;X;Q (A), and among other things computed the variance 1 for Q k− 1 2 +ǫ < X < Q k−ǫ .…”

“…for a certain cubic polynomial F 3 . In that regime, Lester and Yesha [30] showed that ∆ 2 (x, H), normalized to have unit mean-square using (1.6), has a Gaussian value distribution, at least for a narrow range of H below X 1/2 , the conjecture being that this should hold for X ǫ < H < X 1/2−ǫ for any ǫ > 0. For k ≥ 3, Milinovich and Turnage-Butterbaugh [31, p. 182] give an upper bound, assuming RH, of In concurrent work, Lester [29] shows that for k ≥ 3, assuming the Lindelöf Hypothesis, if h(x) = ( x X ) 1− 1 k X δ , (1.8)…”

Lower Bounds for the Variance of Sequences in Arithmetic Progressions: Primes and Divisor Functions

“…For KQ 0 < q ≤ R(≤ √ N ≤ Q/(2K)) and 1 ≤ a ≤ q − 1 with (a, q) = 1, note that the intervals ( a q − 1 2qR , a q + 1 2qR ) are all disjoint, and do not overlap with any major arc. Thus these intervals are all contained in the minor arcs, and therefore (19) Since |a n | ≪ ǫ N ǫ by assumption, this is ≪ ǫ BRN 1 2 +ǫ . Using the triangle inequality, the main term from Lemma 3 contributes to the right side of (19) an amount (20) ≥ N The main term above, when inserted in (20) leads to the main term of our proposition.…”

“…Thus these intervals are all contained in the minor arcs, and therefore (19) Since |a n | ≪ ǫ N ǫ by assumption, this is ≪ ǫ BRN 1 2 +ǫ . Using the triangle inequality, the main term from Lemma 3 contributes to the right side of (19) an amount (20) ≥ N The main term above, when inserted in (20) leads to the main term of our proposition. The remainder term above contributes to (20) an amount…”