On the Gauss mean-value formula for class number

Abstract

Abstract. In his masterwork Disquisitiones Arithmeticae, Gauss stated an approximate formula for the average of the class number for negative discriminants. In this paper we improve the known estimates for the error term in Gauss approximate formula. Namely, our result can be written as iV" 1 ]ζ n<iV H(-n) = 4πv / ΪV/(21C(3)) -2π 2 + O(ΛT 15 / 44+e ) for every e > 0, where H(-n) is, in modern notation, h (-4n). We also consider the average of h(-n) itself obtaining the same type of result.Proving this formula we transform firstly the problem in a lattice point problem (as probably Gauss did) and we use a functional equation due to Shintani and Dirichlet class number formula to express the error term as a sum of character and exponential sums that can be estimated with techniques introduced in a previous work on the sphere problem. §1. Introduction and statement of the main result Gauss noted in Art. 302 of [Ga] that the average of the class number for negative discriminants increases very regularly, and he gave (without proof) an approximate formula for this average, checking its accuracy up to unbelievably large values. (He, in fact, computed not only the class number but also the subdivision into genera for several thousands of discriminants).Probably Gauss proved that his formula gives the correct asymptotics but the first known proof of this fact is due to Lipschitz in 1865. In this century I. M. Vinogradov studied the error term in Gauss' approximation using the method of trigonometric sums. In a series of papers, along fifty years, he gave several upper bounds that also hold for the "sphere problem" (the three-dimensional analogue of the circle problem).In [Ch-Iw] we improved the results of Chen [Ch] and Vinogradov [Vi] on the sphere problem. The key point in our method is the double interpretation of the sphere problem; the first one leads us to trigonometric sums and the second one to character sums. The purpose of this paper is to apply

The mean value of the product of class numbers of paired quadratic fields. I

“…This conjecture was proved by Lipschutz [19] in the case of imaginary quadratic fields and by Siegel [23] in the case of real quadratic fields, and much work has been done on the error term estimate also (see Shintani [22], pp. 44, 45 and Chamizo-Iwaniec [2], for example). However, each quadratic field has infinitely many orders and so we must filter out this repetition in order to obtain the density of class number times regulator for quadratic fields.…”

On the Density of Unnormalized Tamagawa Numbers of Orthogonal Groups I

“…The real case was proved by Siegel [27]. Mertens [19], Vinogradov [29], Shintani [26] and Chamizo-Iwaniec [2] worked on the error term estimates for these cases. Shintani estimated the error term using the zeta function theory of prehomogeneous vector spaces.…”

“…This Z(s) is not the zeta function of the prehomogeneous vector space (1.1). In Part II, we shall express Z(s) as a sum of two Euler products by a technique used in [11], and prove that Z(s) 2 has the rightmost pole at s = is related to the generalized Riemann hypothesis. So it seems difficult to obtain any error term estimate.…”

“…We now explain the significance of Proposition 6.6 in comparison with Theorem 6.2. Theorem 6.2 contains a global condition on d v , whereas Proposition 6.6 contains only a local condition for S v except for the condition (2)…”

Koecher-Maass series for positive definite Fourier coefficients of real analytic Siegel-Eisenstein series of degree 2

“…161-162], the analysis of a convolution product of Eisenstein series of half-integral weight was suggested. We also refer the reader to [6]. Shintani's functional equation was used as a tool to obtain a good estimate of the mean value of the class numbers of binary quadratic forms for negative discriminants.…”