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

Abstract

Abstract. We develop a general method for lower bounding the variance of sequences in arithmetic progressions mod q, summed over all q ≤ Q, building on previous work of Liu, Perelli, Hooley, and others. The proofs lower bound the variance by the minor arc contribution in the circle method, which we lower bound by comparing with suitable auxiliary exponential sums that are easier to understand.As an application, we prove a lower bound of (1 − ǫ)QN log(Q 2 /N ) for the variance of the von Mangoldt function (Λ(n))Previously such a result was only available assuming the Riemann Hypothesis. We also prove a lower bound ≫ k,δ QN (log N )

A lower bound for the variance of generalized divisor functions in arithmetic progressions

“…This is essentially due to the fact that the peaks of the exponential sum with coefficients f (n) over small neighbourhoods of such fractions might be amplified by the effect of a higher power inside the integral. The lower bound (1.2) has been proven to hold when f (n) = (n) and f (n) = d k (n), for k ≥ 2 a positive integer and Q in the range N 1/2+δ ≤ Q ≤ N , for any small δ > 0, by Theorems 1 and 2 in [12]. Here (n) indicates as usual the Von Mangoldt function.…”

“…The present paper studies the question of finding a lower bound for the quantity V (Q, f ) when considering suitable generalizations f of the divisor functions introduced above. The main reference on this problem is the Harper and Soundararajan's paper [12], in which the authors set up the bases for the study of lower bounds of variances of complex sequences in arithmetic progressions. More precisely, they showed that for a wide class of functions with a controlled growth we can lower bound the variance (1.1) with the L 2 -norm of the exponential sum with coefficients f (n) over a large portion of the circle, namely, the so-called union of minor arcs.…”

“…The aim of this paper is to show that the new powerful method introduced in [12] can be used to prove the validity of (1.2) for every α-fold divisor function d α (n), defined as the n-th coefficient in the Dirichlet series of ζ(s) α on the half-plane (s) > 1, except for some specific values of α. The result is contained in the following theorem.…”

“…Clearly, this last set occupies almost the totality of the circle, depending on K , Q and Q 0 , and it consists of real numbers well approximated by rational fractions with large denominator. The idea exploited in [12] was to connect the variance in arithmetic progressions with the minor arc contribution of the exponential sum with coefficients f (n). This point of view was already widespread and present in the literature, like for example in the works of Liu [20,21] and Perelli [25].…”

“…This point of view was already widespread and present in the literature, like for example in the works of Liu [20,21] and Perelli [25]. However, as explained in [12], the previous arguments relied on the connection between character sums and exponential sums (similarly as in the usual deductions of the multiplicative large sieve inequality), which can only be made to work for the -function sequence or for other sequences without small prime factors. In contrast, as pointed out in [12], Harper and Soundararajan avoided the use of Dirichlet characters in favour of Hooley's approach, connecting the variance of f (n) in arithmetic progressions with the variance of the exponential sums n≤N f (n)e(na/q).…”