18,466 research outputs found
On almost universal mixed sums of squares and triangular numbers
In 1997 K. Ono and K. Soundararajan [Invent. Math. 130(1997)] proved that
under the generalized Riemann hypothesis any positive odd integer greater than
2719 can be represented by the famous Ramanujan form ,
equivalently the form represents all integers greater than
1359, where denotes the triangular number . Given positive
integers we employ modular forms and the theory of quadratic forms to
determine completely when the general form represents
sufficiently large integers and establish similar results for the forms
and . Here are some consequences of our main
theorems: (i) All sufficiently large odd numbers have the form
if and only if all prime divisors of are congruent to 1 modulo 4. (ii) The
form is almost universal (i.e., it represents sufficiently large
integers) if and only if each odd prime divisor of is congruent to 1 or 3
modulo 8. (iii) is almost universal if and only if all odd prime
divisors of are congruent to 1 modulo 4. (iv) When , the form
is almost universal if and only if all odd prime divisors of
are congruent to 1 modulo 4 and , where is the
2-adic order of .Comment: 35 page
Dihedral Gauss hypergeometric functions
Gauss hypergeometric functions with a dihedral monodromy group can be
expressed as elementary functions, since their hypergeometric equations can be
transformed to Fuchsian equations with cyclic monodromy groups by a quadratic
change of the argument variable. The paper presents general elementary
expressions of these dihedral hypergeometric functions, involving finite
bivariate sums expressible as terminating Appell's F2 or F3 series.
Additionally, trigonometric expressions for the dihedral functions are
presented, and degenerate cases (logarithmic, or with the monodromy group Z/2Z)
are considered.Comment: 28 pages; trigonometric expressions added; transformations and
invariants moved to arxiv.org/1101.368
Linear Congruences with Ratios
We use new bounds of double exponential sums with ratios of integers from
prescribed intervals to get an asymptotic formula for the number of solutions
to congruences with
variables from rather general sets
On Congruences with Products of Variables from Short Intervals and Applications
We obtain upper bounds on the number of solutions to congruences of the type
modulo a prime with variables from some short intervals. We give some
applications of our results and in particular improve several recent estimates
of J. Cilleruelo and M. Z. Garaev on exponential congruences and on
cardinalities of products of short intervals, some double character sum
estimates of J. B. Friedlander and H. Iwaniec and some results of M.-C. Chang
and A. A. Karatsuba on character sums twisted with the divisor function
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