Representation of numbers by quadratic forms in algebraic number fields

Let K be the totally real field of algebraic numbers of degree n=[k2e] with the discriminant D=D(K); t=t(x₁, ..., x_s) a totally positive quadratic form of the determinant d>0 over the ring of integers from the field K; S4. Let be the number of representations over of the number m by the form a complete singular series. It is proved that for given s and n, there exists a constant c such that for N(d)>0 it is not true that for all m with m totally positive.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 151, pp. 68–77, 1986.     

Representation of integers by positive-definite quadratic forms in totally real algebraic number fields

The results of Gundlach (K. B. Gundlach, Acta Math.,92, 309–345 (1954)) are extended to forms with an odd number of variables. We obtain for alln4 asymptotic formulae for the number of integral representations of numbers by totally positiven-ary forms in totally real algebraic number fields. The remainder term is of the form . The necessary concepts and results about Hilbert modular forms of half integral weight are developed.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 121, pp. 32–46, 1983.     

The polylogarithm in algebraic number fields

Based on Kummer's 2-variable functional equations for the second through fifth orders of the polylogarithm function, certain linear combinations, with rational coefficients, of polylogarithms of powers of an algebraic base were discovered to possess significant mathematical properties. These combinations are designated “ladders,” and it is here proved that the ladder structure is invariant with order when the order is decreased from its permissible maximum value for the corresponding ladder. In view of Wechsung's demonstration that the functions of sixth and higher orders possess no functional equations of Kummer's type, this analytical proof is currently limited to a maximum of the fifth order. The invariance property does not necessarily persist in reverse—increasing the order need not produce a valid ladder with rational coefficients. Nevertheless, quite a number of low-order ladders do lend themselves to such extension, with the needed additional rational coefficients being determined by numerical computation. With sufficient accuracy there is never any doubt as to the rational character of the numbers ensuing from this process. This method of extrapolation to higher orders has led to many quite new results; although at this time completely lacking any analytical proof. Even more astonishing, in view of Wechsung's theorem mentioned above, is the fact that in some cases the ladders can be validly extended beyond the fifth order. This has led to the first-ever results for polylogarithms of order six through nine. A meticulous attention to the finer points in the formulas was necessary to achieve these results; and a number of conjectural rules for extrapolating ladders in this way has emerged from this study. Although it is known that the polylogarithm does not possess any relations of a polynomial character with rational coefficients between the different orders, such relations do exist for some of the ladder structures. A number of examples are given, together with a representative sample of ladders of both the analytical and numerically-verified types. The significance of these new and striking results is not clear, but they strongly suggest that polylogarithmic functional equations, of a more far-reaching character than those currently known, await discovery; probably up to at least the ninth order.     

Character sums in algebraic number fields

The Pólya-Vinogradov inequality is generalized to arbitrary algebraic number fields K of finite degree over the rationals. The proof makes use of Siegel's summation formula and requires results about Hecke's zeta-functions with Grössencharacters. One application is to the problem of estimating a least totally positive primitive root modulo a prime ideal of K, least in the sense that its norm is minimal.     

Permutation polynomials over algebraic number fields

Let K be an algebraic number field. It is known that any polynomial which induces a permutation on infinitely many residue class fields of K is a composition of cyclic and Chebyshev polynomials. This paper deals with the problem of deciding, for a given K, which compositions of cyclic or Chebyshev polynomials have this property. The problem is reduced to the case where K is an Abelian extension of $\text{Q}$. Then the question is settled for polynomials of prime degree, and the Abelian case for composite degree polynomials is considered. Finally, various special cases are dealt with.     

Cubic polynomials over algebraic number fields

We prove that every cubic form in 16 variables over an algebraic number field represents zero, generalizing the corresponding result of Davenport for cubic forms over the rationals. (This has already been proved for cubic forms in 17 or more variables by Ryavec.) We present this result as a special case of a “local-implies-global” theorem for cubic polynomials.     

Normal binomials over algebraic number fields

In this paper, normal and weakly normal binomials over an arbitrary algebraic number field will be characterized. Explicit results on the possible degrees of such binomials are given. Several examples conclude the paper.     

Caliber number of real quadratic fields

We obtain lower bound of caliber number of real quadratic field using splitting primes in K. We find all real quadratic fields of caliber number 1 and find all real quadratic fields of caliber number 2 if d is not 5 modulo 8. In both cases, we don't rely on the assumption on ζK(1/2).     

On sign determinations in real algebraic number fields

Summary Given an ordered fieldF and a finite ordered extensionE. If it is possible to perform constructively the four rational operations inF and to determine the sign for an arbitrary element ofF, a rule is given for the sign determination of an arbitrary element ofE.     

On the large sieve in algebraic number fields

In the present note Bombieri's central theorem concerning the average distribution of the prime numbers in arithmetic progressions is generalized to arbitrary algebraic number fields.Translated from Matematicheskie Zametki, Vol. 2, No. 6, pp. 673–680, December, 1967.Finally, I express my profound gratitude to B. V. Levin for setting the problem and the help he rendered and to A. I. Vinogradov for valuable suggestions.