Research proof index

Let $\gamma_3(n)=H_n-\log n-\frac{1}{2n}+\frac{1}{12n^2}-\frac{1}{120n^4},$ where $H_n$ is the $n$th harmonic number. Then $\gamma_3(n)<\gamma_3(n+1)$ for every integer $n\ge1$.

Let $\gamma_3(n)=H_n-\log n-\frac{1}{2n}+\frac{1}{12n^2}-\frac{1}{120n^4}.$ For every integer $n\ge1$, this corrected harmonic approximation is a strict lower bound for the Euler–Mascheroni constant: $\gamma_3(n)<\gamma.$

The discriminant Δ_fun = 1728⁻¹(E₄³ - E₆²) equals the standard discriminant Δ.

Θ₂(It) has positive real part for t > 0. Proof: Each term Θ₂_term n (It) = exp(-π(n+1/2)²t) is a positive real. The sum of positive reals is positive.

Each term Θ₂_term n (I*t) has positive real part equal to exp(-π(n+1/2)²t) for t > 0.

Each term Θ₄_term n (I*t) has zero imaginary part for t > 0.