Sums of singular series and primes in short intervals in algebraic number fields

Abstract

Gross and Smith have put forward generalizations of Hardy–Littlewood twin prime conjectures for algebraic number fields. We estimate the behaviour of sums of a singular series that arises in these conjectures, up to lower-order terms. More exactly, where \(\mathfrak {S}(\eta )\) is the singular series, we find asymptotic formulas for smoothed sums of \(\mathfrak {S}(\eta )-1\). Based upon Gross and Smith’s conjectures, we use our result to suggest that for large enough ‘short intervals’ in an algebraic number field K, the variance of counts of prime elements in a random short interval deviates from a Cramér model prediction by a universal factor, independent of K. The conjecture over number fields generalizes a classical conjecture of Goldston and Montgomery over the integers. Numerical data are provided supporting the conjecture.