×

Hecke series values of holomorphic cusp forms in the centre of the critical strip. (English) Zbl 0983.11021

Győry, Kálmán (ed.) et al., Number theory in progress. Proceedings of the international conference organized by the Stefan Banach International Mathematical Center in honor of the 60th birthday of Andrzej Schinzel, Zakopane, Poland, June 30-July 9, 1997. Volume 2: Elementary and analytic number theory. Berlin: de Gruyter. 675-689 (1999).
Let \(\chi\) be an odd primitive Dirichlet character to modulus \(N\), and write \(r(m)= \sum_{k|n}\chi(k)\). Then it is easy to show that if \((d,N)=1\) then \[ \sum_{m=0}^d r(m)r(d-m)= A(\chi)\sigma(d)+ \sum_{f\in O_2(N)} c(f)\lambda_f(d) \] for some constant \(A(\chi)\), where \(O_2(N)\) is an orthogonal basis of Hecke eigenfunctions of weight 2 for \(\Gamma_0(N)\), and \(\lambda_f(d)\) are their Fourier coefficients. When \(N=4\) the set \(O_2(4)\) is empty, and the above formula recovers Jacobi’s result on the number of representations of \(N\) as a sum of four squares.
The first result of the paper gives the constants \(A(\chi)\) and \(c(f)\) explicitly, the latter being expressed in terms of the central critical values \({\mathcal L}_f(1,1/2)\) and \({\mathcal L}_f(\chi,1/2)\), where \({\mathcal L}_f(\psi,s)= \sum_{n=1}^\infty \lambda_f(n)\psi(n)n^{-s}\). The second result generalizes the first to forms of higher weight. When \(O_{2k}(N)\) contains a single form \(f\), these results lead to formulae for \[ {\mathcal L}_f(1,1/2) {\mathcal L}_f(\chi,1/2)/ \langle f,f\rangle_{2k}. \] Further results are given involving \({\mathcal L}_f(\chi_1,1/2)\) and \({\mathcal L}_f(\chi_2,1/2)\) for a pair of characters \(\chi_1,\chi_2\).
For the entire collection see [Zbl 0911.00018].

MSC:

11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11F11 Holomorphic modular forms of integral weight
11F25 Hecke-Petersson operators, differential operators (one variable)
11F30 Fourier coefficients of automorphic forms
PDFBibTeX XMLCite