Generations of Correlation Averages

Abstract

We give a general link between weighted Selberg integrals of any arithmetic functionfand averages offcorrelations in short intervals, proved by the elementary dispersion method (our version of Linnik’s method). We formulate conjectural bounds for the so-called modified Selberg integral of the divisor functionsdk(n), gauged by the Cesaro weight in the short intervaln∈x-H,x+Hand improved by these some recent results by Ivić. The same link provides, also, an unconditional improvement. Then, some remarkable conditional implications on the 2kth moments of Riemann zeta function on the critical line are derived. We also give general requirements onfthat allow our treatment forfweighted Selberg integrals.

Sieve functions in arithmetic bands

“…Thus, if f = g Q * 1 is gauged by a weight w in the short interval [x − H, x + H] (i.e. H = o(N ), as N → ∞), then it is natural to take the expected mean value of w H (n − x)f (n) for N < x ≤ 2N to be (compare [CopLap14])…”

“…Indeed, a basic tool for the study of the distribution of the sieve function f in short intervals is its weighted Selberg integral Observe that the trivial bound for J w,f (N, H) is N 1+ε H 2 , because f is essentially bounded. In [CopLap14] and [CopLap16] we have investigated and exploited the link between J w,f (N, H) and the correlation…”

“…Moreover, Lemma 3.3 also shows that if f 1 and f 2 are sieve functions, then J w,(f 1 ,f 2 ) (N, H) can be expressed in terms of arithmetic bands of f 1 or f 2 . Such favorable circumstances allow us to apply Theorem 1.1 in order to obtain both a so-called first generation formula (consistently with the terminology of [CopLap14]) for the correlation of the sieve functions f 1 and f 2 (more precisely, here we use (1..1) that is a consequence of Theorem 1.1 as it is showed in Remark 1.2) and an estimate of J w,(f 1 ,f 2 ) (N, H) once f 1 and f 2 are gauged by a good weight w.…”

Finite Ramanujan expansions and shifted convolution sums of arithmetical functions, II

“…H = o(N ), as N → ∞. The short intervals expected mean-value vanishes, here in the f symmetry integral, whatever the f : N → C is (and this by definition).We know (from Lemma 7 of[CL1]) that, since f H is bounded, hereafter H = o(N ),J fH , sgn (N, H) = h W H (h) N <n≤2N f H (n)f H (n − h) + O(H 3 ),where the weight W H (h) is the correlation of sgn function,[CL2], namelyW H (h) def = −H≤h 1 , h 2 ≤H h 2 −h 1 =h sgn(h 1 )sgn(h 2 ),…”

“…We first recall the definition [CL1,CL2,CL3]: a sieve function f : N → C of range Q (that is an unbounded parameter, depending on other variables, see the following) may be written as…”