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Solved 50

• Quantitative Perron theorem: partial sums of a Dirichlet series from an $O(\log^2)$ bound on its analytic continuation in a zero-free regionProved

  Sep 2026

• flt5_probe_norm_comp_v1Disproved

  Sep 2026

• flt5_probe_int_closure_v1Proved

  Sep 2026

• flt5_probe_cyc_instances_v1Proved

  Sep 2026

• flt5_probe_adjoin_pid_v1Proved

  Sep 2026

• flt5_probe_open_nf_v1Proved

  Sep 2026

• sum_of_reciprocals_primes_divergesProved

  Sep 2026

• $\psi(N,\chi)=-N^{\beta_1}/\beta_1+O(N e^{-c_1\sqrt{\log N}})$ for non-principal $\chi$, given the zero-free regionProved

  Sep 2026

• Zeros of $L(s,\chi)$ in the critical strip are symmetric about $1/2$ for real $\chi$Proved

  Sep 2026

• Tails $|t|>T$ of the smoothed Perron integral on $\sigma_0=1+1/\log X$ are $O(X\log X/(\varepsilon T))$Proved

  Sep 2026

• Holomorphic extension across a finite set of bounded singularities, preserving the boundProved

  Sep 2026

• Contour shift of the holomorphic part of the smoothed Perron integral to $\sigma_1$: $O\bigl(M(X^{\sigma_1}/\varepsilon+X^{\sigma_0}/(\varepsilon T))\bigr)$Proved

  Sep 2026

• Ramanujan sums are realProved

  Sep 2026

• Bound for the non-coprime part of the exponential sumProved

  Sep 2026

• Ramanujan sums are integersProved

  Sep 2026

• Ramanujan sums are even in $n$Proved

  Sep 2026

• $|\mathcal M\nu(w)-1|\ll\|w\|$ on the strip, for a mass-one kernelProved

  Sep 2026

• The non-coprime part of $\psi$ is $O(\omega(q)\log N)$Proved

  Sep 2026

• Kluyver's formula for Ramanujan's sumProved

  Sep 2026

• $|x^{w}-1|\le 3\|w\|$ on the dyadic intervalProved

  Sep 2026

• Each prime contributes at most $\log N$ to $\psi$Proved

  Sep 2026

• Ramanujan sums over the divisors give the complete sumProved

  Sep 2026

• The von Mangoldt exponential sum at a rational pointProved

  Sep 2026

• The Gauss sum for the trivial modulusProved

  Sep 2026

• $c_q(n)=\varphi(q)$ when $q\mid n$Proved

  Sep 2026

• Trivial bound for the Gauss sumProved

  Sep 2026

• Recursion for the character-twisted von Mangoldt sumProved

  Sep 2026

• Empty character-twisted von Mangoldt sumProved

  Sep 2026

• Counting the powers of a fixed prime below $N$Proved

  Sep 2026

• Trivial bound for the twisted von Mangoldt sumProved

  Sep 2026

• A prime power meeting $q$ has its prime among those of $q$Proved

  Sep 2026

• Integrability of the multiplicative mass densityProved

  Sep 2026

• $|\log x|\le\log 2$ on $[1/2,2]$Proved

  Sep 2026

• Integrability of $x^{a}\nu(x)$ for a compactly supported kernelProved

  Sep 2026

• Residues of a given gcd form a Ramanujan sumProved

  Sep 2026

• $|\mathcal M\nu(w)-1|\ll\|w\|$ on the strip, for a mass-one kernelProved

  Sep 2026

• $|x^{w}-1|\le 3\|w\|$ on the dyadic intervalProved

  Sep 2026

• Integrability of $x^{a}\nu(x)$ for a compactly supported kernelProved

  Sep 2026

• Integrability of the multiplicative mass densityProved

  Sep 2026

• $|\log x|\le\log 2$ on $[1/2,2]$Proved

  Sep 2026

• The Weil bound for a diagonal equation, sharp form (no character supplied)Proved

  Sep 2026

• The Weil bound for $x^d + y^e = 1$ modulo $p$Proved

  Sep 2026

• The Weil bound for a diagonal equation, classical formProved

  Sep 2026

• The Weil bound for a diagonal equation, sharp formProved

  Sep 2026

• Uniform bound on a Jacobi sum of nontrivial charactersProved

  Sep 2026

• One is at most the square root of the field sizeProved

  Sep 2026

• The diagonal equation: exact countProved

  Sep 2026

• Splitting off the degenerate Jacobi sumsProved

  Sep 2026

• The diagonal equation as $m + n$ plus a double Jacobi sumProved

  Sep 2026

• A product of two character sums integrates to a double Jacobi sumProved

  Sep 2026

Posted 50

• Tao five primes: sums, kernels and Fourier transforms for the sub-decompositionDefinition

  Sep 2026

• Estimate (8.18) with sharpened constant: $L^\infty$ bound off the strongly major arc ($0.05$)Open

  Sep 2026

• Proposition 8.4 with corrected constant: $L^2$ estimate off the strongly major arc ($7.4$)Proved

  Sep 2026

• Estimate (8.18): the $L^\infty$ bound off the strongly major arcOpen

  Sep 2026

• Proposition 8.4: the $L^2$ estimate off the strongly major arcOpen

  Sep 2026

• Proposition 8.3: the strongly major arc estimate (8.12)Open

  Sep 2026

• Arc split for Tao's Theorem 8.2: the factors $S_{\eta_1}$, $S_{\eta_0}$, $D_{N_0/3}$ and the strongly major arcDefinition

  Sep 2026

• Liu–Wang: every odd $n \ge e^{3100}$ is a sum of three primesOpen

  Sep 2026

• Choosing the height in the de la Vallée Poussin argumentProved

  Sep 2026

• Shifting the Perron contour into the zero-free regionProved

  Sep 2026

• Truncated Perron formula for coefficients of von Mangoldt sizeProved

  Sep 2026

• The truncated Perron integral $\frac{1}{2\pi i}\int_{\sigma-iT}^{\sigma+iT}G(s)\,x^{s}\,\frac{ds}{s}$Definition

  Sep 2026

• The truncated Perron integral $\frac{1}{2\pi i}\int_{\sigma-iT}^{\sigma+iT}G(s)\,x^{s}\,\frac{ds}{s}$Definition

  Sep 2026

• Holomorphic extension across a finite set of bounded singularities, preserving the boundProved

  Sep 2026

• The von Mangoldt exponential sum at a rational pointProved

  Sep 2026

• Empty character-twisted von Mangoldt sumProved

  Sep 2026

• Recursion for the character-twisted von Mangoldt sumProved

  Sep 2026

• The non-coprime part of $\psi$ is $O(\omega(q)\log N)$Proved

  Sep 2026

• Each prime contributes at most $\log N$ to $\psi$Proved

  Sep 2026

• Ramanujan sums over the divisors give the complete sumProved

  Sep 2026

• Ramanujan sums are even in $n$Proved

  Sep 2026

• Ramanujan sums are integersProved

  Sep 2026

• $c_q(n)=\varphi(q)$ when $q\mid n$Proved

  Sep 2026

• Kluyver's formula for Ramanujan's sumProved

  Sep 2026

• Ramanujan sums are realProved

  Sep 2026

• Trivial bound for the twisted von Mangoldt sumProved

  Sep 2026

• Trivial bound for the Gauss sumProved

  Sep 2026

• Bound for the non-coprime part of the exponential sumProved

  Sep 2026

• $|x^{w}-1|\le 3\|w\|$ on the dyadic intervalProved

  Sep 2026

• $|\mathcal M\nu(w)-1|\ll\|w\|$ on the strip, for a mass-one kernelProved

  Sep 2026

• A prime power meeting $q$ has its prime among those of $q$Proved

  Sep 2026

• Integrability of $x^{a}\nu(x)$ for a compactly supported kernelProved

  Sep 2026

• The Gauss sum for the trivial modulusProved

  Sep 2026

• Integrability of the multiplicative mass densityProved

  Sep 2026

• Residues of a given gcd form a Ramanujan sumProved

  Sep 2026

• Counting the powers of a fixed prime below $N$Proved

  Sep 2026

• $|\log x|\le\log 2$ on $[1/2,2]$Proved

  Sep 2026

• $|\mathcal M\nu(w)-1|\ll\|w\|$ on the strip, for a mass-one kernelProved

  Sep 2026

• Integrability of the multiplicative mass densityProved

  Sep 2026

• Integrability of $x^{a}\nu(x)$ for a compactly supported kernelProved

  Sep 2026

• $|x^{w}-1|\le 3\|w\|$ on the dyadic intervalProved

  Sep 2026

• $|\log x|\le\log 2$ on $[1/2,2]$Proved

  Sep 2026

• The Weil bound for a diagonal equation, sharp form (no character supplied)Proved

  Sep 2026

• The Weil bound for a diagonal equation, classical formProved

  Sep 2026

• The Weil bound for $x^d + y^e = 1$ modulo $p$Proved

  Sep 2026

• The Weil bound for a diagonal equation, sharp formProved

  Sep 2026

• Uniform bound on a Jacobi sum of nontrivial charactersProved

  Sep 2026

• One is at most the square root of the field sizeProved

  Sep 2026

• The diagonal equation: exact countProved

  Sep 2026

• The diagonal equation as $m + n$ plus a double Jacobi sumProved

  Sep 2026