Explicit formula: explicit work with general smoothing?

Question

The following is a literature question, in the sense that I already know how to do what I am asking about, and in fact have already done it; now I'd like to write a brief historical overview as an introduction.

What are some (any) examples of explicit work based on an explicit formula for $\zeta$ or $L(s,\chi)$, using a non-polynomial smoothing? For that matter, is there explicit work where a general, unspecified smoothing function is used for as long as possible, and the choice of smoothing (polynomial or not) is made only in the final estimations? Nearly everybody seems to follow Rosser (1941) in using a polynomial smoothing.

(Note to non-analytic-number-theorists: the use of "explicit" twice in the above is not a mistake - it means two different things: an explicit formula means an expression for a sum of an arithmetical function $f(n)$ in terms of the complex zeroes of $\zeta(s)$ or $L(s,\chi)$; explicit in "explicit work" means just what it usually does - namely, work where all bounds have fully worked-out constants, rather than expressions such as $O()$ or $\ll$.)

Answer 1

Some remarks on your question:

Technically, Rosser & Schoenfeld 1976 uses repeated integration which is not exactly the same as a polynomial smoothing.

People studying links between degrees and discriminants have used several smoothings, this encompasses Serre, Odlyzko and Poitou (1976 : Minoration de discriminants). Serre uses exp(-x^2/4b) and Odlyzko a more fancy smoothing. Hugh Montgomery also if my memory serves me well.

Roger Heath-Brown 1992 on Linnik's constant has used very fancy smoothings, borrowed by Habiba Kadiri 2005. Heath-Brown smoothes only the contribution of zeroes at distance O(1/log q), while Kadiri smoothes them all. I would believe Stark used some explicit formulae with the aim of computing explicit estimates.

I (Ramaré) with Yannick Saouter have used a polynomial smoothing for small gaps between primes 2003: but it is because the optimisation leads to this choice. The same applies to Faber & Kadiri 2013, though they explain why.

That's all that comes off the top of my hat :) ! Good hunting! Best, Olivier

Answer 2

One example of a standard non-polynomial smoothing which one could make explicit with relative ease is the approximation of

$\displaystyle\sum_{n\leq x} a(n)$

by

$\displaystyle\sum_{n\geq 1}a(n) e^{-n/x} = \frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\Big(\sum_{n\geq 1}\frac{a(n)}{n^s}\Big)x^s \Gamma(s)ds$.

However, the relation $\Gamma(s) = \Gamma(s+1)/s$ ensures that at best, one can achieve a polynomial rate of convergence in the integral, assuming the Dirichlet series is itself polynomially bounded. For other non-polynomial weights, the situation is largely the same. Thus it is not especially advantageous to smooth polynomially vs. non-polynomially from the perspective of convergence. In light of this, it is often (though not always) better to smooth polynomially because we have a better understanding of how to extract good explicit bounds for the associated Mellin transforms. Molteni and others have used this to get good explicit bounds on the error term in the Chebotarev density theorem under GRH.

Choosing a weight such that either the weight or its Mellin transform is compactly supported can be useful in many circumstances. For instance, this is crucial in bounds for the least prime in an arithmetic progression. But now the choice of kernel depends highly on what your goal with the explicit formula is.

ADDED: I'll add that in my experience, it is perhaps a bit easier to unsmooth using a polynomial smoothing than a non-polynomial smoothing while maintaining good constants. This might also influence your choice.