Let [Formula: see text] be the argument of the Riemann zeta-function at the point [Formula: see text]. For [Formula: see text] and [Formula: see text] define its antiderivatives as [Formula: see text] where [Formula: see text] is a specific constant depending on [Formula: see text] and [Formula: see text]. In 1925, Littlewood proved, under the Riemann Hypothesis (RH), that [Formula: see text] for [Formula: see text]. In 1946, Selberg unconditionally established the explicit asymptotic formulas for the second moments of [Formula: see text] and [Formula: see text]. This was extended by Fujii for [Formula: see text], when [Formula: see text]. Assuming the RH, we give the explicit asymptotic formula for the second moment of [Formula: see text] up to the second-order term, for [Formula: see text]. Our result conditionally refines Selberg’s and Fujii’s formulas and extends previous work by Goldston in [Formula: see text], where the case [Formula: see text] was considered.
We prove several results about integers represented by positive definite quadratic forms, using a Fourier analysis approach. In particular, for an integer $\ell\ge 1$, we improve the error term in the partial sums of the number of representations of integers that are a multiple of $\ell$. This allows us to obtain unconditional Brun–Titchmarsh-type results in short intervals and a conditional Cramér-type result on the maximum gap between primes represented by a given positive definite quadratic form.
We prove several results about integers represented by positive definite quadratic forms, using a Fourier analysis approach. In particular, for an integer ℓ ≥ 1, we improve the error term in the partial sums of the number of representations of integers multiple of ℓ. This allows us to obtain unconditional Brun-Titchmarsh-type results in short intervals, and a conditional Cramér-type result on the maximum gap between primes represented by a given positive definite quadratic form.
Let S(t) = 1 π arg ζ 1 2 + it be the argument of the Riemann zeta-function at the point 1 2 + it. For n ≥ 1 and t > 0 define its antiderivatives aswhere δn is a specific constant depending on n and S 0 (t) := S(t). In 1925, J. E. Littlewood proved, under the Riemann Hypothesis, thatfor n ≥ 1. In 1946, Selberg unconditionally established the explicit asymptotic formulas for the second moments of S(t) and S 1 (t). Assuming the Riemann Hypothesis, we give the explicit asymptotic formula for the second moment of Sn(t) up to the second-order term, for n ≥ 1. Our result conditionally refines Selberg's formula for S 1 (t) and makes explicit the estimates given by Littlewood. This extends previous work by Goldston in 1987, where the case n = 0 was considered.
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