bibkey: berndt2025thetasum authors: Bruce C. Berndt, Raghavendra N. Bhat, Jeffrey L. Meyer, Likun Xie, Alexandru Zaharescu year: 2025 title: “An Arithmetic Sum Associated with the Classical Theta Function” doi: null url: https://arxiv.org/html/2501.03234v3 claim: “S’(h,k) = sum_{j=1}^{k-1} (-1)^(j+1+floor(hj/k)), S(k) = sum_{h=1}^{k-1} S’(h,k). Conjecture 1.1: S(k) > 0 for each odd prime k > 5. Conjecture 3.1: S(k) > 2k for each prime k > 233. Conjecture 3.2: S(k) > 3k for each prime k > 3119. Conjecture 3.3: for arbitrary fixed C1,C2 > 0 and epsilon > 0, every sufficiently large prime k satisfies C1k < S(k) < C2*k^(1+epsilon).” strata_touched: [] license: citation-only triage: anchor
Berndt, Bhat, Meyer, Xie and Zaharescu, arithmetic sum of the theta function
Version 3, revised 2026-01-09, presents the four statements quoted above as conjectures.
Whether later work proves or refutes any of them is not verified here. None is proved in this
repository, and no declaration about S or S' exists here. This note records what was
measured during an earlier attempt on Conjecture 1.1, so that a later attempt does not repeat
eliminated routes; those computations are retained historical evidence and were not rerun for
this source correction.
Structure that is established by computation
Two facts reduce the conjecture. Both were verified across the first 45 odd primes with no violations, and both are bind-only in the sense of the repository’s admission rule — they follow from the pinned library by instantiation and normalisation, so they belong inside a proof and not as declarations.
S'(h,k) = 0 for every odd h, when k is an odd prime. Pair j with k - j: since k is
prime and 1 <= j <= k-1, the quotient h*j/k is never an integer, so
floor(h*(k-j)/k) = h - 1 - floor(h*j/k). The two exponents sum to k + h + 1, odd exactly
when h is odd, so the paired terms cancel; k odd means j is never its own partner.
Primality is needed: S'(3,9) = -2, so the unrestricted odd-composite form is false.
S'(k-1,k) = k-1, since floor((k-1)*j/k) = j - 1 makes every exponent even.
Hence S(k) = (k-1) + rest(k), where rest(k) sums S'(h,k) over the even h in
[2, k-3]. Thus S(k) > 0 is equivalent to rest(k) > -(k-1); the stronger condition
rest(k) >= 0 is sufficient and yields S(k) >= k-1. Measured values of rest for the first
eighteen odd primes: 0, 0, 4, 4, 8, 8, 20, 8, 16, 56, 56, 40, 52, 60, 32, 48, 104, 132 —
always nonnegative, zero only at k = 3 and k = 5, and always divisible by four.
Inversion invariance: for even h whose inverse modulo k is also even,
S'(h,k) = S'(h^{-1} mod k, k). Verified on 2381 such pairs across the 63 odd primes from 7
through 317, and on 2383 pairs across 65 primes when 3 and 5 are included. h = k-1 is a fixed
point, consistent with the value above. This is most likely the standard Dedekind-sum property
s(h,k) = s(h^{-1},k) seen through a Dedekind-type expression for S'.
S'(2,k) = 2 when k = 3 (mod 4) and 0 when k = 1 (mod 4), over the first twelve odd
primes.
Routes eliminated, with the counterexample that kills each
Termwise nonnegativity of the even-h terms: false. S'(6,17) = -4 and S'(4,11) = -2.
Nonnegativity of inversion-orbit sums: false. The first negative orbit is {14, 24} at
k = 67, contributing -4 against rest(67) = 132. Enlarging the orbits by adjoining
x -> 1 - x does not repair it: at k = 23 the orbit {3, 8, 11, 13, 16, 21} contributes
-4 through its even elements.
Endpoint-only compensation after signed inversion cancellation: false. Writing R(k) for the
even h in [2, k-3] with even inverse and Bminus(k) for the sum of the negative parts of
S'(h,k) over R(k), the bound S(k) >= k - 1 + Bminus(k) first becomes nonpositive at
k = 15727, where Bminus = -16056 and the bound gives -330 against the actual
S = 94450. All 1831 odd primes through 15727 were scanned, two independent integer algorithms
agreed on 3372322 values, and direct evaluation of 123653538 original summands at 15727
confirmed the obstruction.
The two-step Euclidean remainder bound from Dedekind reciprocity: insufficient. It first gives
a negative bound for rest at k = 19, namely -4 against the actual 20, and first fails
to establish S > 0 at k = 31, where the bound gives rest >= -32 against the actual 56.
Independent absolute-value estimates for the Dedekind terms are also insufficient. The general
reciprocity route is not eliminated; only these estimates are.
The reformulation that survives
S(k) = (k-1)^2/2 - 4*N(k), where N(k) counts the pairs of even residues whose product
modulo k is even. Conjecture 1.1 is then exactly N(k) < (k-1)^2/8: strictly fewer than half
of the even-even pairs have even product. That inequality is not proved here, and it is the
form in which a further attempt should start.
Verified locator
- Version: v3, revised 2026-01-09
- URL: https://arxiv.org/html/2501.03234v3