Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.
Updated
Irrationality
L16_totient_kernel_separation/: Every nonzero real absolutely summable kernel of shifted phi(n)/n has positive absolute limsup on each fixed arithmetic progression; a fresh prime gives an explicit pair of separating refinements.
L189_lcm_series_cone_no_product/: For three or more primes, the lattice-point set whose weighted count is the running-lcm reciprocal series is not a product of two lower-dimensional sets after any unimodular change of variables; for two primes it is.
L190_three_prime_no_finite_planar_partition/: The expanded lattice set for three distinct primes is not a finite real linear combination of indicators of polyhedra whose facet coefficients span at most two dimensions over the rationals.
L218_sparse_perturbation_any_value/: Changing Euler's totient only on a set with gaps at most g, by multiples of 2^s and by at most the square root except at the first change, makes sum psi(n)/2^n equal any real number when s+g is at most half of log2 n.
L219_dilation_design_prime_count/: A signed family of prime dilations, with admissible shifts and primes above K, that cancels levels 1 to K for every function uses at least K+1 distinct primes, whose product therefore exceeds K^(K+1).
L220_partial_multiplicativity_interval/: Positive integer sequences arbitrarily close to the totient, agreeing at every prime power and obeying its multiplication laws at any fixed finite set of primes, can have any binary sum in an interval about the totient sum.
L228_sparse_prime_position_interpolation/: An interval of binary-series values is attained by increasing integer sequences agreeing with the nth prime outside one density-zero set, preserving nested congruences of modulus o(log n), and changing halved gaps by o(log n). In particular, such a sequence can have a dyadic rational sum. The altered terms need not be prime.
L229_critical_modulus_interpolation/: Increasing integer sequences that agree with the nth prime at density one and preserve nested congruences of modulus at least c log n have a set of binary-series values of Hausdorff dimension zero. Together with L228, this distinguishes the logarithmic and sublogarithmic regimes for interval interpolation; it excludes no individual value.
L230_sparse_interpolation_support/: The binary-series values of increasing integer sequences that change only o(N/log log N) of the first N prime positions form a set of Hausdorff dimension zero. No congruence condition is required. Thus an interval interpolation cannot have a common exceptional set this sparse, although an individual rational value is not excluded.
These claims concern arithmetic series and tools for studying their values. The separation theorem for normalized totient kernels supports the binary totient research, but does not settle Problem 249.
Two families test which coefficient information can force irrationality. L218 and L220 construct totient comparisons with prescribed binary sums; the latter keeps prime-power values and multiplication laws at any fixed finite set of primes. L228 constructs increasing integer comparisons agreeing with almost all prime positions and preserving sublogarithmic nested congruences. L229 and L230 limit intervals of attainable values when the congruences or support bounds are strengthened. Their preserved classes are different: neither full totient multiplicativity nor primality of every modified position is established. All these claims are author-recorded at tier 0 and leave both catalog irrationality questions open.