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Solved 50

• The $p\mid m$ part of $S(p^h,a)$ equals $p^{h-1}$ for $h\le k$Proved

  Sep 2026

• The $p\mid m$ part of $S(p^h,a)$ is $p^{k-1}S(p^{h-k},a)$ for $h\ge k$Proved

  Sep 2026

• Probe: universe-polymorphic binderProved

  Sep 2026

• Probe: universe-free binderProved

  Sep 2026

• Ramanujan sums as a sum over the unit groupProved

  Sep 2026

• Interval integrability of linear phasesProved

  Sep 2026

• Gauss sum expansion of $e(b/q)$ over Dirichlet charactersProved

  Sep 2026

• All local factors are positive at odd $n$Proved

  Sep 2026

• Evaluation of $c_p(n)$ at a prime modulusProved

  Sep 2026

• $|c_q(n)|\le\varphi(q)$Proved

  Sep 2026

• $c_q(0)=\varphi(q)$Proved

  Sep 2026

• The first term of the singular series is $1$Proved

  Sep 2026

• Explicit trivial bound $|S(\alpha,N)|\le N\log N$Proved

  Sep 2026

• Empty von Mangoldt exponential sumProved

  Sep 2026

• Recursion in the length of the von Mangoldt exponential sumProved

  Sep 2026

• The von Mangoldt exponential sum is $1$-periodic in $\alpha$Proved

  Sep 2026

• Nonnegativity of $\psi$Proved

  Sep 2026

• The exponential sum at the origin is $\psi(N)$Proved

  Sep 2026

• Trivial bound $|S(\alpha,N)|\le\psi(N)$Proved

  Sep 2026

• Conjugation symmetry of the von Mangoldt exponential sumProved

  Sep 2026

• Empty prime exponential sumProved

  Sep 2026

• Trivial bound for the prime exponential sumProved

  Sep 2026

• Chebyshev bound $\psi(N)\le N\log N$Proved

  Sep 2026

• Ramanujan sum of modulus zeroProved

  Sep 2026

• $c_1(n)=1$Proved

  Sep 2026

• Counting reduced residues gives Euler's totientProved

  Sep 2026

• Reduced residues modulo a primeProved

  Sep 2026

• $e(a/q)$ depends only on $a$ modulo $q$Proved

  Sep 2026

• Ramanujan sums are periodic in $n$ modulo $q$Proved

  Sep 2026

• Conjugation symmetry of Ramanujan sumsProved

  Sep 2026

• Reduced residues as the unit group of $\mathbb Z/q$Proved

  Sep 2026

• Representative of a residue classProved

  Sep 2026

• Ramanujan sums are invariant under unit twists of the argumentProved

  Sep 2026

• Ramanujan sums are multiplicative in the modulusProved

  Sep 2026

• The local factor at $2$ vanishes for even $n$Proved

  Sep 2026

• The local factor at $2$ equals $2$ for odd $n$Proved

  Sep 2026

• Positivity of the local factors at odd primesProved

  Sep 2026

• The $p$-th term of the singular series is the local densityProved

  Sep 2026

• The terms of the singular series are multiplicativeProved

  Sep 2026

• The singular series term at a squarefree modulusProved

  Sep 2026

• Euler product for the three primes singular seriesProved

  Sep 2026

• The truncated singular series factors over the primesProved

  Sep 2026

• Positivity of the product of local densitiesProved

  Sep 2026

• The three primes singular series vanishes at even $n$Proved

  Sep 2026

• The three primes singular series is nonzero at odd $n$Proved

  Sep 2026

• $e(b/q)$ depends only on $b$ modulo $q$Proved

  Sep 2026

• A character-twisted von Mangoldt sum is supported on reduced residuesProved

  Sep 2026

• Splitting the von Mangoldt exponential sum at a modulusProved

  Sep 2026

• Major-arc expansion of the von Mangoldt exponential sumProved

  Sep 2026

• Continuity of the additive characterProved

  Sep 2026

Posted 50

• Vaughan Theorem 4.2: $|S(q,a)|\ll_{k,\varepsilon} q^{1-1/k+\varepsilon}$Open

  Sep 2026

• $|S(p^h,a)|\le k^{3}\,(p^h)^{1-1/k}$ at prime powersOpen

  Sep 2026

• $|S(p,a)|\le (k-1)\sqrt{p}$ for prime moduliOpen

  Sep 2026

• Prime-power base case $S(p^h,a)=p^{h-1}$ for $h\le k$Open

  Sep 2026

• Prime-power recursion $S(p^h,a)=p^{k-1}S(p^{h-k},a)$Open

  Sep 2026

• The $p\mid m$ part of $S(p^h,a)$ equals $p^{h-1}$ for $h\le k$Proved

  Sep 2026

• Probe: universe-polymorphic binderProved

  Sep 2026

• Probe: universe-free binderProved

  Sep 2026

• The $p\mid m$ part of $S(p^h,a)$ is $p^{k-1}S(p^{h-k},a)$ for $h\ge k$Proved

  Sep 2026

• The part of $S(p^h,a)$ over residues prime to $p$ vanishesProved

  Sep 2026

• The three primes counting identityProved

  Sep 2026

• Fourier coefficient of the cube of a weighted exponential sumProved

  Sep 2026

• Interval integrability of linear phasesProved

  Sep 2026

• Continuity of the additive characterProved

  Sep 2026

• Major-arc expansion of the von Mangoldt exponential sumProved

  Sep 2026

• Splitting the von Mangoldt exponential sum at a modulusProved

  Sep 2026

• A character-twisted von Mangoldt sum is supported on reduced residuesProved

  Sep 2026

• Gauss sum expansion of $e(b/q)$ over Dirichlet charactersProved

  Sep 2026

• $e(b/q)$ depends only on $b$ modulo $q$Proved

  Sep 2026

• The three primes singular series is nonzero at odd $n$Proved

  Sep 2026

• The three primes singular series vanishes at even $n$Proved

  Sep 2026

• Positivity of the product of local densitiesProved

  Sep 2026

• All local factors are positive at odd $n$Proved

  Sep 2026

• The truncated singular series factors over the primesProved

  Sep 2026

• Euler product for the three primes singular seriesProved

  Sep 2026

• The singular series term at a squarefree modulusProved

  Sep 2026

• The terms of the singular series are multiplicativeProved

  Sep 2026

• The first term of the singular series is $1$Proved

  Sep 2026

• The $p$-th term of the singular series is the local densityProved

  Sep 2026

• Positivity of the local factors at odd primesProved

  Sep 2026

• The local factor at $2$ equals $2$ for odd $n$Proved

  Sep 2026

• The local factor at $2$ vanishes for even $n$Proved

  Sep 2026

• Ramanujan sums are multiplicative in the modulusProved

  Sep 2026

• Ramanujan sums are invariant under unit twists of the argumentProved

  Sep 2026

• The exponent inequality behind the minor-arc boundProved

  Sep 2026

• Interpolating a high moment between a sup bound and a low momentProved

  Sep 2026

• Representative of a residue classProved

  Sep 2026

• Ramanujan sums as a sum over the unit groupProved

  Sep 2026

• Reduced residues as the unit group of $\mathbb Z/q$Proved

  Sep 2026

• Conjugation symmetry of Ramanujan sumsProved

  Sep 2026

• Ramanujan sums are periodic in $n$ modulo $q$Proved

  Sep 2026

• $e(a/q)$ depends only on $a$ modulo $q$Proved

  Sep 2026

• Evaluation of $c_p(n)$ at a prime modulusProved

  Sep 2026

• Reduced residues modulo a primeProved

  Sep 2026

• $|c_q(n)|\le\varphi(q)$Proved

  Sep 2026

• $c_q(0)=\varphi(q)$Proved

  Sep 2026

• Counting reduced residues gives Euler's totientProved

  Sep 2026

• $c_1(n)=1$Proved

  Sep 2026

• Ramanujan sum of modulus zeroProved

  Sep 2026

• Explicit trivial bound $|S(\alpha,N)|\le N\log N$Proved

  Sep 2026